Dissertation am Fachbereich Mathematik der Freien Universität Berlin An analysis of dimensional theoretical properties of some affine dynamical systems von Dipl.-Math. Jörg Neunhäuserer betreut durch Priv.-Doz. Dr. Jörg Schmeling eingereicht im Juni 1999 Contents 1. Introduction 4 2. Affine dynamical systems 7 2.1. Self-affine repellers 7 2.2. Attractors of piecewise affine maps 12 3. Applying symbolic dynamics 16 3.1. Shift codings 16 3.2. Representation of ergodic measures 19 4. Calculation of box-counting dimension 24 5. Dimension formulas and estimates for ergodic measures 27 5.1. Lyapunov exponents and charts 27 5.2. Exact dimensionality and Ledrappier Young formula 30 5.3. Some consequences 33 6. Overlapping self-similar measures 38 6.1. Main results 38 6.2. Absolute continuity 39 6.3. An upper bound on the dimension 44 7. Generic dimensional theoretical properties of the systems 47 8. Extension of some results to Markov chains 54 9. Erdös measures 60 9.1. Singularity 61 2 9.2. Garsia entropy 63 9.3. An upper bound on the dimension of Erdös measures 68 9.4. Construction of an Erdös measure with full dimension 71 10. Number theoretical peculiarities 74 10.1. Ergodic Measures 74 10.2. Invariant Sets 76 Appendix A: General facts in dimension theory 82 Appendix B: Pisot-Vijayarghavan numbers 85 Notations 86 References 89 Zusammenfassung der Ergebnisse (in deutscher Sprache) 92 Lebenslauf (in deutscher Sprache) 93 3 1. Introduction In this work we study dimensional theoretical properties of some affine dynamical systems. By dimensional theoretical properties we mean Hausdorff dimension and boxcounting dimension of invariant sets and ergodic measures on theses sets. Especially we are interested in two problems. First we ask whether the Hausdorff and boxcounting dimension of invariant sets coincide. Second we ask whether there exists an ergodic measure of full Hausdorff dimension on these invariant sets. If this is not the case we ask the question, whether at least the variational principle for Hausdorff dimension holds, which means that there is a sequence of ergodic measures such that their Hausdorff dimension approximates the Hausdorff dimension of the invariant set. It seems to be well accepted by experts that these questions are of great importance in developing a dimension theory of dynamical systems (see the book of Pesin about dimension theory of dynamical systems [PE2]). Dimensional theoretical properties of conformal dynamical systems are fairly well understood today. For example there are general theorems about conformal repellers and hyperbolic sets for conformal diffeomorphisms (see chapter 7 of [PE2]). On the other hand the existence of two different rates of expansion or contraction forces problems that are not captured by a general theory this days. At this stage of development of the dimension theory of dynamical systems it seems natural to study non conformal examples. This is the first step to understand the mechanisms that determine dimensional theoretical properties of non conformal dynamical systems. Affine dynamical systems represent simple examples of non conformal systems. They are easy to define, but studying their dimensional theoretical properties does nevertheless provide challenging mathematical problems and exemplify interesting phenomena. We consider here a special class of self-affine repellers in dimension two, depending on four parameters (see 2.1.). Furthermore we study a class of attractors of piecewise affine maps in dimension three depending on four parameters as well. The last object of our work are projections of these maps that are known as generalized Baker's transformations (see 2.2.). The contents of our work is the following: In chapter two we give an overview about some main results in the area of dimension theory of affine dynamical systems and define the systems we study in this work. We will explain, what is known about the dimensional theoretical properties of these systems and describe what our new results are. In chapter three we then apply symbolic dynamics to our systems. We will introduce explicit shift codings 4 and find representations of all ergodic measures for our systems using these codings. From chapter four to chapter eight we study dimensional theoretical properties, which our systems generally or generically have. In chapter four we will prove a formula for the box-counting dimension of the repellers and the attractors (see theorem 4.1.). Then in chapter five we apply general dimensional theoretical results for ergodic measures found by Ledrappier and Young [LY] and Barreira, Schmeling and Pesin [BPS] to our systems. These results relate the dimension of ergodic measures to metric entropy and Lyapunov exponents. Using this approach we will be able to reduce questions about the dimension of ergodic measures in our context to questions about certain overlapping and especially overlapping self-similar measures on the line. These overlapping self-similar measures are studied in chapter six. Our main theorem extends a result of Peres and Solomyak [PS2] concerning the absolute continuity resp. singularity of symmetric self-similar measures to asymmetric ones (see theorem 6.1.3.). In chapter seven we bring our results together. We prove that we generically (in the sense of Lebesgue measure on a part of the parameter space) have the identity of box-counting and Hausdorff dimension for the repellers and the attractors. (see theorem 7.1.1. and corollary 7.1.2.). This result suggest that one can expect that the identity of box-counting dimension and Hausdorff dimension holds at least generically in some natural classes of non conformal dynamical systems. Furthermore we will see in chapter seven that there generically exists an ergodic measure of full Hausdorff dimension for the repellers. On the other hand the variational principle for Hausdorff dimension is not generic for the attractors. It holds only if we assume a certain symmetry (see theorem 7.1.1.). For generalized Baker's transformations we will find a part of the parameter space where there generically is an ergodic measure of full dimension and a part where the variational principle for Hausdorff dimension does not hold (see theorem 7.1.3.). Roughly speaking the reason why the variational principle does not hold here is, that if there exists both a stable and an unstable direction one can not generically maximize the dimension in the stable and in the unstable direction at the same time. In an other setting this phenomenon was observed before by Manning and McCluskey [MM]. In chapter eight we extend some results of the last section to invariant sets that correspond to special Markov chains instead of full shifts (see theorem 8.1.1.). In the last two chapters of our work we are interested in number theoretical exceptions to our generic results. The starting point of our considerations in section nine are results of Erdös [ER1] and Alexander and Yorke [AY] that establish singularity and a decrease of dimension for infinite convolved Bernoulli measures under special conditions. Using a generalized notion of the Garsia entropy ([GA1/2]) we are able 5 to understand the consequences of number theoretical peculiarities in broader class of overlapping measures (see theorem 9.1.1.). In chapter ten we then analyze number theoretical peculiarities in the context of our dynamical systems. We restrict our attention to a symmetric situation where we generically have the existence of a Bernoulli measure of full dimension and the identity of Hausdorff and box-counting dimension for all of our systems. In the first section of chapter ten we find parameter values such that the variational principle for Hausdorff dimension does not hold for the attractors and for the Fat Baker's transformations (see theorem 10.1.1.). These are the first known examples of dynamical systems for which the variational principle for Hausdorff dimension does not hold because of number theoretical peculiarities of parameter values. For the repellers we have been able to show that under certain number theoretical conditions there is at least no Bernoulli measure of full Hausdorff dimension; the question if the variational principle for Hausdorff dimension holds remains open in this situation. In the second section of chapter ten we will show that the identity for Hausdorff and box-counting dimension can drops because there are number theoretical peculiarities. In the context of Weierstrass-like functions this phenomenon was observed by Przytycki and Urbanski [PU]. Our theorem extends this result to a larger class of sets, invariant under dynamical systems (see theorem 10.2.1). At the end of this work the reader will find two appendices, a list of notations and the list of references. In appendix A we introduce the notions of dimension we use in this work and collect some general facts in dimension theory. In appendix B we state the facts about Pisot-Vijayarghavan number, we need in our analysis of number theoretical peculiarities. The list of notations contains general notations and a table with a summary of notations we use to describe the dynamical systems that we study. Acknowledgments I wish to thank my supervisor Jörg Schmeling for a lot of valuable discussion and all his help. Also thanks to Luis Barreira for his great hospitality in Lisboa and many interesting comments. This work was done while I was supported by "Promotionstipendium gem. NaFöG der Freien Universität Berlin". 6 2. Affine dynamical systems 2.1. Self-affine repellers Self-affine repellers are simple examples of non conformal dynamical systems. We introduce them now. Let T1, . . . , Tp : D −→ D be affine contractions of a domain D in IRm. Assume that the sets Ti(D) are disjoint . From Hutchinson [HU] we know there is an unique compact self-affine subset Λ of D satisfying: Λ = p ⋃ i=1 Ti(Λ). Define a map T on ⋃p i=1 Ti(D) by T (x) = T−1i (x) if x ∈ Ti(D). Clearly T is a smooth expanding map. Λ is invariant and a repeller for T , which means that there is an open neighborhood V of Λ such that Λ = {x ∈ V |f n(x) ∈ V ∀n ≥ 0} (see chapter 20 of [PE2]). We call Λ a self-affine repeller. There is one generic result about the dimension of large classes of self-affine sets. Theorem 2.1.1. Let L1, . . . , Lp be linear contractions of IR m with ||Li|| < 1/2 and let b1, . . . , bm ∈ IRm. If Λ is the compact self-affine set satisfying Λ = m⋃ i=1 Li(Λ) + bi then the identity dimB Λ = dimH Λ holds for almost all (b1, . . . , bm) ∈ IRmp in the sense Lebesgue measure and the common value is independent of (b1, . . . , bm). Falconer [FA2] proved this theorem in the case ||Li|| < 1/3 and Solomyak [SO2] extended the proof to the case ||Li|| < 1/2. Moreover Solomyak showed that the statement does not longer hold if we replace 1/2 by 1/2 + δ. Of course 2.1.1. leaves many questions open. First of all the question about the existence of an ergodic measure of full Hausdorff dimension remains open. Moreover one would like to have some information about classes of self-affine repellers with larger expansion rates and there may be natural subclasses that fall in the exceptional set of 2.2.1. . 7 Let us discuss a very natural family of self-affine repellers that is completely understood today and proved to fall in the exceptional class of 2.1.1. . Given integers l ≥ m ≥ 2 choose a set A of pairs of integers (i, j) with 0 ≤ i < l and 0 ≤ j < m. Denote the cardinality of A by a. Now let Tk for k = 1 . . . a be affine maps given in the following way: if k enumerates the element (i, j) ∈ A then let Tk([0, 1] 2) = [i/l, (i+ 1)/l]× [j/m, (j + 1)/m]. Figure 1: The images of the affine maps inducing a self-affine carpet with l = 8 m = 4 and A = {(4, 0), (2, 1), (6, 1), (7, 1), (0, 2), (1, 2), (2, 2), (5, 2), (1, 3), (3, 3)} Let ΛA be the self-affine set generated by these affine contractions. A set of this type is known as general Sierpinski carpet. We remark that ΛA viewed as a subset of the Torus is invariant under the toral endomorphism given by: T : (x, y) −→ (lx,my) mod 1. Dimensional theoretical questions are answered by the following theorem of McMullen: 8 Theorem 2.1.2. [MC] Let tj be the number of those i for which (i, j) ∈ A and let r be the number of those j for which there is some i such that (i, j) ∈ A. We have: dimH ΛA = logm( m−1∑ j=0 t logl m j ) and dimB ΛA = logm r + logl(a/r). Moreover there exists a Bernoulli measure of full Hausdorff dimension on ΛA. We remark that it is easy to see that a Bernoulli measure on the carpet is in fact an ergodic measure with respect to the map T on the torus or the expanding map T associated with the affine contractions. Note that the theorem we implies that the Hausdorff and box-counting dimension of a general Sierpinski carpet coincide if and only if the carpet is self-similar (l=m) or the number of rectangles is constant or zero in every raw (tj = 0 or tj = const. for all j). There are some generalizations of 2.1.2. . Kenyon and Peres [KP] extended the result to analogous subsets of higher dimensional cubes, which they called self-affine Sierpinski sponges. Using this result they where also able to show the existence of an ergodic measure of full Hausdorff dimension on all compact invariant sets for endomorphisms of the d-Torus with integer eigenvalues. Gatzouras and Lalley [GL] extended the result on self-affine carpets in another direction. They considered affine contractions which map the unit square to rectangles with height greater than width such that these rectangles are lined up in rows. They calculated the Hausdorff and box-counting dimension of the limit set and found ergodic measures of full dimension. Now we define the class of self-affine repellers we will study in our work. Let P 4all = {(β1, β2, τ1, τ2) ∈ (0, 1)4|β1 + β2 ≥ 1 and τ1 + τ2 < 1} be the set of all parameters we consider. Given θ ∈ P 4all we define two affine contractions T1,θ and T−1,θ of the square [−1, 1]2 by T1,θ(x, z) = (β1x+ (1− β1), τ1z + (1− τ1)) T−1,θ(x, z) = (β2x− (1− β2), τ2z − (1− τ2)). Let Λθ be unique compact self-affine subset of [−1, 1]2 satisfying Λθ = T1,θ(Λθ) ∪ T−1,θ(Λθ) 9 and let Tθ be the smooth expanding transformation on T1,θ([−1, 1]2)∪T2,θ([−1, 1]2) defined by Tθ(x) = (Ti,θ) −1(x) if x ∈ Ti,θ([−1, 1]2) for i = 1, 2. The set Λθ is an invariant repeller for the transformation Tθ. T−1,θ([−1, 1]2) T1,θ([−1, 1]2) 2τ2 2τ1 2β2 2β13⁄4 ? 6 3⁄4 - - ? 6 Figure 2: The transformations T1,θ and T−1,θ on [−1, 1]2 We will now describe what is known about dimensional theoretical properties of the systems (Λθ, Tθ). The symmetric situation θ = (β, β, τ, τ) ∈ P 4all has been studied by Pollicott and Weiss [PW]. We need one definition to state the result. We say that β ∈ (0, 1) is a Garsia-Erdös number if ∃ C > 0 ∀x ∈ IR : card{(s0, . . . , sn−1)| n−1∑ k=0 skβ k ∈ [x, x+ βn)} ≤ C(2β)n ∀n ≥ 0. Examples of Garsia-Erdös numbers are the numbers 1n√2 for n ≥ 0. Furthermore we know from appendix 3 of [PW] that for some ρ almost all β ∈ (1 − ρ, 1) are Garsia-Erdös numbers. 10 Theorem 2.1.3. [PW] If θ = (β, β, τ, τ) ∈ P 4all then we have dimB Λθ = 1 + log(2β) log(1/τ) . If β is in addition a Garsia-Erdös number then we have dimB Λθ = dimH Λθ and the equal-weighted Bernoulli measure on Λθ has full Hausdorff dimension. Now let us say what our new results about the dimensional theoretical properties of the systems (Λθ, Tθ) are and where in our work the corresponding theorems can be found: New results First of all in theorem 4.1.1. we will find a formula for dimB Λθ for all θ ∈ P 4all. In fact the box-counting dimension is given by the unique positive solution of the equation β1τ x−1 1 + β2τ x−1 2 = 1. Furthermore we show or almost all θ ∈ P 4trans := {(β1, β2, τ1, τ2) ∈ P 4all|β2 ≤ β1 ≤ 0.649} in the sense of four dimensional Lebesgue measure the identity dimB Λθ = dimH Λθ and and the existence of an ergodic measure of full Hausdorff dimension for the system (Λθ, Tθ); see corollary 7.2. . The restriction of this generic result depends on the technique we use and is due to a certain transversality condition; see chapter six. In fact our main generic result in theorem 7.1. is little bit stronger than corollary 7.2. and takes special cases into consideration. We will see that the statements in the second part of Pollicott and Weiss theorem holds for almost all β ∈ (0.5, 1) in the sense of one dimensional Lebesgue measure. Our technique is different from the arguments of Pollicott and Weiss and the condition we have for the identity of Hausdorff and box-counting dimension is not the number theoretical Garsia-Erdös condition (see the remarks after 7.2.). Let us now for a moment consider the case θ = (β, β, τ, τ) with τ = 0.5. In this situation the set Λθ coincides (up to a countable number of points) with the graph of Weierstrass-like function studied by Przytycki and Urbanski [PU]. Przytycki and Urbanski were able to show that the Hausdorff dimension of these graphs is less than their box-counting dimension if β is the reciprocal of a Pisot-Vijayarghavan number (short PV number). The reader will find the definition and examples of PV numbers in appendix B. 11 In our work number theoretical peculiarities are also one point of main effort. We will show that if θ = (β, β, τ, τ) ∈ P 4all and β is the reciprocal of a PV number then we have no Bernoulli measure of full Hausdorff dimension for the system (Λθ, Tθ) (see 10.1.1.(3)) and the inequality dimH Λθ < dimB Λθ holds (see 10.2.1.). The arguments we need to get this result in our situation with τ < 0.5 are very different from the arguments of [PU]. 2.2. Attractors of piecewise affine maps Attractors of piecewise affine maps provide simple examples of generalized hyperbolic attractors. Especially the Belykh attractors raised great interest in the literature (see [PE1]). We want to introduce them here. Consider piecewise affine transformations on the square [−1, 1] given by fk,ρ1,ρ2β1,β2 (x, y) = { (β1x+ (1− β1), ρ1y + (1− ρ) if y ≥ kx (β2x− (1− β2), ρ2y − (1− ρ) if y < kx where β1, β2 ∈ (0, 1) k ∈ (−1, 1) and ρ1, ρ2 ∈ (1, 2/(|k|+ 1). It is easy to see that there is a global attractor called Belykh attractor for all of these maps. The definition we used here is due to Pesin [PE1]. Belykh [BE] himself only considered the case β1 = β2 and ρ1 = ρ2. Dimensional theoretical properties of the systems in this special case were studied by Schmeling [SCH]. In our work we are interested in another special case. We set k = 0 and ρ1 = ρ2 = 2 and obtain transformations fβ1,β2(x, y) := f 0,1,1 β1,β2 (x, y) = { (β1x+ (1− β1), 2y − 1) if y ≥ 0 (β2x− (1− β2), 2y + 1) if y < 0 . of the square [−1, 1]2 for all β1, β2 ∈ (0, 1). We call these maps generalized Baker's transformations. If β = β1 = β2 we write fβ instead of fβ,β. Alexander and Yorke [AY] called fβ a Skinny Baker's transformation if β < 0.5 and a Fat Baker's transformation if β > 0.5. f0.5 is known as the Baker's transformation. The attractor for fβ1,β2 is given by Qβ1,β2 := closure( ∞⋂ k=0 fnβ1,β2([−1, 1]2)). In the case β1 + β2 < 1 dimensional theoretical properties of the dynamical system (Qβ1,β2 , fβ1,β2) are well known: 12 Theorem 2.2.1. Let β1 + β2 < 1 and d be the unique positive number satisfying β d 1 + β d 2 = 1 then dimB Qβ1,β2 = dimH Qβ1,β2 = d+1 and there is an ergodic measure of full Hausdorff dimension for the system (Qβ1,β2 , fβ1,β2). This result seems to be folklore in the dimension theory of dynamical systems. In fact the attractor in the non-overlapping situation is a product of a standard Cantor set in the line with the interval [−1, 1]. The ergodic measure of full dimension is a product of a Cantor measure (a Bernoulli measure on the standard Cantor set) with the normalized Lebesgue measure on [−1, 1]. We refer to chapter 23 of [PE2] for theses facts. We consider in this work the overlapping situation, which means (β1, β2) ∈ P 2olapp := {(β1, β2)|β1 + β2 ≥ 1}. fβ1,β2 2β1 2β2 -3⁄4 -3⁄4 Figure 3: The action of fβ1,β2 on the square [−1, 1]2 where β1 + β2 > 1 If (β1, β2) ∈ P 2olapp the attractor of the map fβ1,β2 is obviously the hole square [−1, 1]2 with Hausdorff and box-counting dimension equal to two. The interesting problem is whether there exist an ergodic measure of full Hausdorff dimension resp. whether the variational principle for Hausdorff dimension holds for ([−1, 1]2, fβ1,β2) if β1 + β2 ≥ 1. 13 New results From the work of Alexander and Yorke [AY] and a result of Solomyak (see 6.1.1.) it is easy to deduce that for almost all β ∈ (0.5, 1) there is an ergodic measure of full dimension for the Fat Baker's transformation ([−1, 1]2, fβ). This measure is given by the product of an infinite convolved Bernoulli measure with the normalized Lebesgue measure on [−1, 1] (see 7.3.). Our main result about the Fat Baker's transformation is that the variational principle for Hausdorff dimension does not hold for ([−1, 1]2, fβ) if β is the reciprocal of a PV number (see 10.1.1. (1)). This result is of great interest. Its is the first known example showing that the variational principle for Hausdorff dimension can fail to hold because of number theoretical peculiarities. Beside this we have new results in the asymmetric situation. On the one hand we will show that if β1β2 < 0.25 then the variational principle does not hold for the systems ([−1, 1], fβ1,β2). On the other hand we will see that for almost all (β1, β2) ∈ P 2trans := {(β1, β2) ∈ P 2olapp|β2 ≤ β1 ≤ 0.649} with β1β2 > 0.25 there exists an ergodic measure of full Hausdorff dimension for the system ([−1, 1], fβ1,β2). The last class of dynamical systems we study in this work is given by piecewise affine maps in dimension three: fθ : [−1, 1]3 7−→ [−1, 1]3 fθ(x, y, z) = { (β1x+ (1− β1), 2y − 1, τ1z + (1− τ1)) if y ≥ 0(β2x− (1− β2), 2y + 1, τ2z − (1− τ2)) if y < 0 where θ = (β1, β2, τ1, τ2) ∈ P 4all. fθ fθ(A) fθ(B) A B Figure 4: The action of fθ on [−1, 1]3. 14 We see that the projection of fθ onto the (x, y)-plane is fβ1,β2 for θ = (β1, β2, τ1, τ2) ∈ P 4all. Obviously the maps fβ1,β2 are non invertible if β1 + β2 ≥ 1 but their lifts fθ are invertible. Furthermore we notice that the attractor Λθ of fθ is a product of the self-affine set Λθ in the (x, z)-plane with the interval [−1, 1] on the y-axis: Λθ = closure( ∞⋂ k=0 fnθ ([−1, 1]3)) = {(x, y, z)|(x, z) ∈ Λθ, y ∈ [−1, 1]}. New results By the product structure of the sets Λθ and proposition A5 we have dimH/B Λθ = dimH/B Λθ+1. Thus all our results about Hausdorff dimension and box-counting dimension of the self-affine sets Λθ have an analogon for Λθ (see 4.1.,7.1.,7.2., 10.2.1.). Very interesting is the question whether the variational principle for Hausdorff dimension holds. We will show in 7.1. that in the generic situation (for almost all θ ∈ P 4trans) it can only hold for (Λθ, fθ) if we have logτ1 log(2β1) = logτ2(2β2). Thus the variational principle for Hausdorff dimension is not generic on the parameter set P 4trans if we consider (Λθ, fθ) but it is generic if we consider (Λθ, Tθ). This phenomenon is related to fact that the map fθ is hyperbolic; it has both a stable and an unstable direction (see also the first remark after 7.2.). For the symmetric systems (Λβ,β,τ,τ , fβ,β,τ,τ ) the situation is different. There exists an ergodic measure of full dimension for almost all β ∈ (0.5, 1) (see 7.1.). But again there are number theoretical peculiarities. If β ∈ (0, 5) is the reciprocal of a PV number and τ is small then variational principle does not hold for (Λβ,β,τ,τ , fβ,β,τ,τ ) (see 10.1.1.(2)) 15 3. Applying symbolic dynamics 3.1. Shift Codings Let Σ = {−1, 1}ZZ and Σ+ = {−1, 1}IN0 . With the product metric defined by d(s, t) = ∞∑ k=−∞ resp. 0 |sk − tk|2−|k| for s = (sk) and t = (tk) Σ (resp. Σ+) becomes a perfect, totally disconnected and compact metric space; see proposition 7.6. of [DGS]. A cylinder set in Σ (resp. Σ+) is given by [t0, t1 . . . , tu]v := {(sk)|sv+k = tk for k = 0, . . . , u}. The forward shift map σ on Σ (resp. Σ+) is given by σ((sk)) = (sk+1). The backward shift σ−1 is defined on Σ and given by σ((sk)) = (sk−1). We will use (Σ, σ) resp. (Σ, σ−1) and (Σ+, σ) to describe the dynamics of the systems defined in the previous chapter symbolically by coding the points of the invariant set. We begin with the class of self-affine repellers defined in 2.1. . Given s ∈ Σ+ we denote by ]k(s) the cardinality of {si|si = −1 i = 0 . . . k}. For γ1, γ2 ∈ (0, 1) we define a map π∗γ1,γ2 : Σ+ −→ [ −γ21−γ2 γ1 1−γ1 ] by π∗γ1,γ2(s) = ∞∑ k=0 skγ ]k(s) 2 γ k−]k(s)+1 1 . We scale this map so that it is into [−1, 1]. Let Lγ1,γ2 be the affine transformation on the line that maps −γ2 1−γ2 to −1 and γ1 1−γ1 to 1 and let πγ1,γ2 = Lγ1,γ2 ◦ π ∗ γ1,γ2 . For θ = (β1, β2, τ1, τ2) ∈ P 4all we set πθ = (πβ1,β2 , πτ1,τ2). Proposition 3.1.1. The systems (Σ+, σ) and (Λθ, Tθ) are homeomorph conjugated via πθ. Proof It is obvious that πγ1,γ2 is continuous since d(s, t) ≤ 1 2n ⇒ sk = tk for k = 0, . . . , n⇒ |πγ1,γ2(s)− πγ1,γ2(t)| ≤ γn+11 1− γ1 + γn+12 1− γ2 . hence πθ is continuous. Just looking at the definition of πγ1,γ2 we see that π∗γ1,γ2((sk+1)) = { γ−11 π ∗ γ1,γ2 ((sk))− 1 if s0 = 1 γ−12 π ∗ γ1,γ2 ((sk)) + 1 if s0 = −1 . 16 Hence we have πγ1,γ2(σ(s)) = Lγ1,γ2(π ∗ γ1,γ2 ((sk+1))) = { Lγ1,γ2(γ −1 1 π ∗ γ1,γ2 ((sk))− 1) if s0 = 1 Lγ1,γ2(γ −1 2 π ∗ γ1,γ2 ((sk)) + 1) if s0 = −1 = { γ −1 1 πγ1,γ2(s) + (1− γ−11 ) if s0 = 1 γ−12 πγ1,γ2(s)− (1− γ−12 ) if s0 = −1 (∗). Since πθ(s) ∈ T1,θ([−1, 1]2) if s0 = 1 and πθ(s) ∈ T−1,θ([−1, 1]2) if s0 = −1 this implies πθ(σ(s)) = { T−11,θ(πθ(s)) if s0 = 1 T−1−1,θ(πθ(s)) if s0 = −1 = Tθ(πθ(s)). This means that σ and Tθ are conjugated via πθ. Furthermore we see by induction that πθ(s) = Ts0,θ ◦ . . . ◦ Tsn−1,θ(πθ(σn(s))) ∈ Ts0,θ ◦ . . . ◦ Tsn−1,θ([−1, 1]2) and thus πθ(s) = lim n−→∞Ts0,θ ◦ . . . ◦ Tsn−1,θ([−1, 1] 2). So πθ is onto Λθ and invertible since T1,θ([−1, 1]2) ∩ T−1,θ([−1, 1]2) = ∅. The continuity of the inverse map follows from compactness. 2 Now we examine the systems ([−1, 1]2, fβ1,β2) for (β1, β2) ∈ P 2olapp. Define ς from Σ− := {−1, 1}ZZ− onto [-1,1] by ς(s) = ∞∑ k=1 s−k2 −k where s = (sk)k∈ZZ− ∈ Σ−. This function is well known. It maps the signed dyadic expansion of a point in [−1, 1] to this point. ς is continuous and one to one restricted to (Σ−\{(sk)|∃k0∀k ≤ k0 : sk = 1}) ∪ {(1)}. Let Σ = (Σ\{(sk)|∃k0∀k ≤ k0 : sk = 1}) ∪ {(1)}. For (β1, β2) ∈ P 2olapp we now define πβ1,β2 : Σ 7−→ [−1, 1]2 by πβ1,β2((sk)) = (πβ1,β2((sk)k∈IN0), ς((sk)k∈ZZ−)) Proposition 3.1.2. πβ1,β2 is continuous, surjective and conjugates the backward shift σ −1 and fβ1,β2 on Σ. 17 Proof It is obvious that the map is continuous and surjective since the components are continuous and onto [−1, 1]. Let s = (sk) ∈ Σ. We have (sk+1)k∈Z− 6= (. . . , 1, 1,−1) and hence ς((sk+1)k∈ZZ−) = ∞∑ k=1 s−k+12 −k ≥ 0⇔ s0 = 1. Thus fβ1,β2 ◦ πβ1,β2((sk+1)) = { (β1πβ1,β2((sk+1)k∈IN0) + (1− β1), 2ς((sk+1)k∈ZZ−)− 1) if s0 = +1 (β2πβ1,β2((sk+1)k∈IN0)− (1− β2), 2ς((sk+1)k∈ZZ−) + 1) if s0 = −1 . In view of (*) in the proof of 3.1.1. and the definition of ς we now see that fβ1,β2 ◦ πβ1,β2((sk+1)) = πβ1,β2((sk)). σ as a map of Σ is invertible and we get fβ1,β2 ◦ πβ1,β2(s) = πβ1,β2(σ −1(s)). 2 Now we have a look at the lifts (Λθ, fθ). For θ = (β1, β2, τ1, τ2) ∈ P 4all we define πθ : Σ −→ Λθ by πβ1,β2((sk)) = (πβ1,β2((sk)k∈IN0), ς((sk)k∈ZZ−), πτ1,τ2((sk)k∈IN0))) Proposition 3.1.3. πθ is continuous and surjective. Moreover it is bijective from Σ onto Λθ and conjugates the backward shift map σ−1 and fθ on Σ. Proof It is obvious that πθ is continuous. Treating the third component in the same way as the first we see that πθ conjugates σ −1 and fθ on Σ using the arguments of the proof of 3.1.2. . That the map is onto Λθ and one to one restricted to Σ follows from proposition 3.1.1. and the properties of the map ς. 2 Given a shift coding it is easy deduce interesting properties of a dynamical system. We say that a topological dynamical system (A, T ) has strange dynamics, if it has the following properties: (1) There are periodic orbits of all periods for T in A (2) The set of periodic points of T is dense in A (3) There are orbits of T which are dense in A. Property (3) is known as topological transitivity of the system (A, T ). From the propositions of these section we get the following corollary: 18 Corollary 3.1.4 The dynamical systems ([−1, 1]2, fβ1,β2), (Λθ, fθ) and (Λθ, Tθ) have strange dynamics. Proof It is easy to see that the systems (Σ+, σ), and (Σ, σ−1) have strange dynamics. Since all our coding maps are surjective and continuous it follows that all our systems have properties (2) and (3). Since πθ and the restriction of πθ to Σ are bijective (1) holds for the systems (Λθ, fθ) and (Λθ, Tθ). We observe that the points of a periodic orbit in Σ have different images under πβ1,β2 since the component given by ς is different. Thus (1) holds for ([−1, 1]2, fβ1,β2) as well. 2 3.2. Representation of ergodic measures Given a compact metric space X we denote by M(X) the set of all Borel probability measures on X. With the weak∗ topology M(X) becomes a compact, convex and metricable space. If T is a Borel measurable transformation on X we call a measure μ T -invariant if μ ◦ T−1 = μ. The set of all invariant measures forms a compact, convex and nonempty subset of M(X). A invariant measure μ is called ergodic if T−1B = B ⇒ μ(B) ∈ {0, 1} holds for all Borel sets B in X. M(X,T ) := {μ ∈ M(X)|μ T -ergodic} is compact, convex and nonempty. It consists of the extreme points of the set of invariant measures. By bp for p ∈ (0, 1) we denote the Bernoulli measure on Σ resp. Σ+, which is the product of the discrete measure giving 1 the probability p and −1 the probability (1 − p). We write b for the equal-weighted Bernoulli measure b0.5. The Bernoulli measures are ergodic with respect to forward and backward shifts. For these basic facts in ergodic theory we refer to the book of Denker, Grillenberger and Sigmund [DGS]. We will need one more definition. Given bp on {−1, 1}ZZ− we define the corresponding Bernoulli measure `p on [−1, 1] by `p = bp ◦ ς−1. ` := `0.5 is the normalized Lebesgue measure on [-1,1]. We will now introduce the measures we study in the context of our dynamical systems. Let μ ∈M(Σ+, σ) and γ1, γ2 ∈ (0, 1). We define two Borel probability measures on the real line by μ∗γ1,γ2 = μ ◦ (π∗γ1,γ2)−1 and μγ1,γ2 = μ ◦ (πγ1,γ2)−1. 19 The measure μγ1,γ2 is just μ ∗ γ1,γ2 scaled on the interval [−1, 1] by the transformation Lγ1,γ2 : μγ1,γ2 = μ ∗ γ1,γ2 ◦ L−1γ1,γ2 . If γ1 + γ2 ≥ 1 we say that μγ1,γ2 is overlapping, if not, we say that this measure is non-overlapping. For γ ∈ (0, 1) we write μγ instead of μγ,γ and call this measure symmetric. bpγ1,γ2 for a Bernoulli measure b p is a self-similar measure in the sense of the following proposition: Proposition 3.2.1. For all p ∈ (0, 1) and all γ1, γ2 ∈ (0, 1). the relation bpγ1,γ2 = pbpγ1,γ2 ◦ S1 + (1 − p)bpγ1,γ2 ◦ S2 holds with S1(x) = γ−11 x+ (1− γ−11 ) and S2(x) = γ−12 x− (1− γ−12 ). Proof bpγ1,γ2(B) = b p(π−1γ1,γ2(B)) = bp({s|s1 = 1 ∧ πγ1,γ2(s) ∈ B}) + bp({s|s1 = −1 ∧ πγ1,γ2(s) ∈ B}) = bp({s|s1 = 1∧S−11 ◦ πγ1,γ2 ◦ σ(s) ∈ B})+ bp({s|s1 = −1∧S−12 ◦ πγ1,γ2 ◦ σ(s) ∈ B}) = bp({s|s1 = 1 ∧ σ(s) ∈ π−1γ1,γ2(S1(B))}) + bp({s|s1 = −1 ∧ σ(s) ∈ π−1γ1,γ2(S2(B))}) = p bpγ1,γ2 ◦ S1(B) + (1 − p) bpγ1,γ2 ◦ S2(B) holds for all Borel subsets B of the real line. 2 The symmetric self-similar measures bpγ are usually called infinite convolved Bernoulli measures because of the following fact: Proposition 3.2.2. The measures bpγ are given by the infinite convolution of the discrete measures b p,n γ , which give (1− γ)γn the probability p and −(1− γ)γn the probability (1− p). Proof bpγ is obviously the distribution of the random variable Y p γ = ∑∞ n=0X p,n(1 − γ)γn where Xp,n are independent random variables taking the values 1 and −1 with probability p resp. 1−p. It is well known that the distribution of the sum of independent random variables is the convolution of the distributions of these random variables. But the distribution of Xp,n(1− γ)γn is given by the measure bp,nγ . 2 20 We remark that we do not have a convolution structure for the asymmetric measures bpβ1,β2 . We can not write the measure as a distribution of the sum of independent random variables in this case, because the term that is added randomly at the n'th step depends on the terms that were added before. In chapter six and nine we will continue with the discussion of the measures defined here. Now we go to characterize the ergodic measures for the dynamical system (Λθ, Tθ). Proposition 3.2.3. The map μ −→ μθ := μ ◦ π−1θ is a affine homeomorphism from M(Σ+, σ) onto M(Λθ, Tθ). If θ = (β1, β2, τ1, τ2) then the projection of μθ onto the x-axis is μβ1,β2 and the projection onto the z-axis is μτ1,τ2 . Proof The first statement follows from proposition 3.1.1 using proposition 3.11. of [DGS] and the remark on page 24 of [DGS]. The second statement is a direct consequence of the product structure of the map πθ and the definition of the involved measures. 2 We now describe all ergodic measures for the dynamical system ([−1, 1]2, fβ1,β2). We will need pr+, the projection from Σ onto Σ +. Proposition 3.2.4. μ 7−→ μβ1,β2 := μ ◦ π−1β1,β2 is a continuous affine map from M(Σ, σ) onto M([−1, 1]2, fβ1,β2). The projection of μβ1,β2 onto the x-axis is the measure (pr+μ)β1,β2 and bpβ1,β2 is the product of b p β1,β2 with `p. Proof Since πβ1,β2 is surjective and continuous we get from proposition 3.1. of [DGS] that μ 7−→ μβ1,β2 := μ ◦ π−1β1,β2 is a continuous affine map from M(Σ) onto M([−1, 1]2). If μ is shift invariant we obviously have μ(Σ) = 1. Because we know from proposition 3.1.2. that πβ1,β2 conjugates the backward shift and fβ1,β2 on Σ we get that μβ1,β2 is fβ1,β2 ergodic if μ is shift ergodic. It remains to show that the map is onto M([−1, 1]2, fβ1,β2) restricted to M(Σ, σ). So let us choose an arbitrary measure ξ in M([−1, 1]2, fβ1,β2). 21 We first want to show that ξ(πβ1,β2(Σ\Σ)) = 0. Let D be set of all numbers of the form k/2n with n ∈ IN and |k| ≤ n− 1. A direct calculation shows that: πβ1,β2(Σ\Σ) = (D×[−1, 1])∪({1}×[−1, 1)) = ( ∞⋃ k=0 f−kβ1,β2({0}×[1,−1]))∪({1}×[−1, 1)). Recall that the measure ξ is in particular shift invariant. Hence the measure of the first set in union is zero because it is given by a disjunct infinite union of sets with the same measure. The measure of the second set is zero since {1} × [−1, 1) ⊆ f−kβ1,β2({1} × [1− 2βk1 , 1)) ∀ k ≥ 0. Now take μpre ∈M(Σ) such that μpre◦π−1β1,β2 = ξ. μpre is not necessary shift invariant so we define a measure μ as a weak∗ accumulation point of the sequence μn := 1 n+ 1 n∑ i=0 μpre ◦ σ−n. From the considerations above we have μpre(Σ) = 1 and hence: μn ◦ π−1β1,β2 = 1 n+ 1 n∑ i=0 μpre ◦ σ−n ◦ π−1β1,β2 = 1 n+ 1 n∑ i=0 μpre ◦ π−1β1,β2 ◦ f−iβ1,β2 = 1 n+ 1 n∑ i=0 ξ ◦ f−iβ1,β2 = ξ. Thus μβ1,β2 is just the measure ξ and μ is shift invariant by definition. We have thus shown that the set M(ξ) := {μ|μ σ-invariant and μβ1,β2 = ξ} of Borel measures on Σ is not empty. Since the map μ 7−→ μβ1,β2 is continuous and affine on the set of σ-invariant measures we know thatM(ξ) is compact and convex. It is a consequence of Krein-Milman theorem that there exists an extremal point μ of M(ξ). We claim that μ is an extremal point of the set of all σ-invariant Borel measures on Σ and hence ergodic. If this is not the case then we have μ = tμ1 + (1 − t)μ2 where t ∈ (0, 1) and μ1, μ2 are two distinct σ-invariant measures. This implies ξ = t(μ1)β1,β2 + (1− t)(μ2)β1,β2 . Since ξ is ergodic we have (μ1)β1,β2 = (μ2)β1,β2 = ξ and hence μ1, μ2 ∈ M(ξ). This is a contradiction to μ being extremal in M(ξ). Now we calculate the projection: prX μβ1,β2(B) = μβ1,β2(B× [−1, 1]) = μ(π−1β1,β2(B× I)) = μ({−1, 1}ZZ−0 × π−1β1,β2(B)) = μ(pr−1+ (π−1β1,β2(B))) = pr+μ(π−1β1,β2(B)) = (pr+μ)β1,β2(B) Since the measure bp on Σ is the product of bp on {−1, 1}ZZ−0 and bp on Σ+ and ς maps bp on {−1, 1}ZZ−0 to `p we see that bβ1,β2 = bβ1,β2 × `p. 2 Let us now give an analysis of ergodic measures for the lifts (Λθ, fθ). 22 Proposition 3.2.5. The map μ 7−→ μθ := μ ◦ π−1θ is a affine homeomorphism from M(Σ, σ) onto M(Λθ, fθ). The projection of μθ on the (x, z)-plane is (pr+μ)θ. Moreover b p θ is a product of bpθ with ` p. If θ = (β1, β2, τ1, τ2) then the projection of μθ on the (x, y)-plane is μβ1,β2 . Proof By the same reasoning we used in the proof of 3.2.3. we can show that the map is continuous, affine and surjective. We will use the fact that πθ restricted to Σ is a bijection to show that the map is invertible. Given a measure ξ ∈ M(Λθ, fθ) we define a measure Fξ ∈ M(Σ) by Fξ(B) = ξ(πθ(B)) for all Borel sets in Σ. Let μ be in M(Σ, σ) and B be a Borel set in Σ. We have Fμθ(B) = μ(π −1 θ (πθ(B))) = μ(Σ ∩ π−1θ (πθ(B))) = μ(Σ ∩B) = μ(B). Hence F is the inverse map to μ −→ μθ. The continuity of F follows by the compactness of M(Σ, σ). The first statement about the projections follows in the same way as our projection result in 3.2.4. . The second statement is is obvious since prXY πθ = πβ1,β2 . 2 From the proposition of this sections and the propositions of the last section we get a corollary about the metric entropy of the measure theoretical dynamical systems we study. For the definition of the metric entropy hμ(T ) of a dynamical system (X,T ) with an invariant measure μ and a treatment of the properties of this quantity we recommend [WA], [DGS] or [KH]. To get the corollary we cab use for instance proposition 11.14. of [WA]. Corollary 3.2.6. hμθ(Tθ) = hμ(σ) holds for all μ ∈ M(Σ+, σ) and hμθ(fθ) = hμ(σ) holds for all μ ∈M(Σ, σ). We also get the inequality hμβ1,β2 (fβ1,β2) ≤ hμ(σ) as a corollary of 3.1.2. and 3.2.3. . In fact even the identity hμβ1,β2 (fβ1,β2) = hμ(σ) holds. This is easy to see for Bernoulli measures by projecting the systems onto the y-axis using the product structure of bpβ1,β2 but more difficult if we consider other measures. We will sketch a proof of this identity using conditional measures and dimensions in 5.3.6. . 23 4. Calculation of box-counting dimension Before we discuss the dimension of measures and the Hausdorff dimension of sets in the context of our dynamical systems we calculate here the box-counting dimension of the repellers Λθ and the attractors Λθ defined in chapter two. We refer to appendix A for the definition of the various kinds of dimension and basic facts in dimension theory. Theorem 4.1. If θ = (β1, β2, τ1, τ2) ∈ P 4all and d is the unique positive number satisfying β1τ d 1 + β2τ d 2 = 1 then dimB Λθ = d+ 1 and dimB Λθ = d+ 2. Let us make a few remarks on this theorem: Remarks (1) A simple calculation shows that our theorem is consistent with the result of Pollicott and Weiss [PW] in the special case β1 = β2 =: β and τ1 = τ2 =: τ (see theorem 2.1.3.). (2) Recall from the classical work of Moran [MO] that the Hausdorff and boxcounting dimension of a self-similar Cantor set induced by by transformations with contraction rates τ1 and τ2 is given by the solution of τ d 1 +τ d 2 = 1. There is an analogy to our formula. In our setting of an self-affine set with overlaps in the projections the contraction rates in the second direction induces weights in the dimension formula. (3) The overlapping condition β1 + β2 ≥ 1 is necessary for our formula to hold. In the case β1 + β2 < 1 the Hausdorff and box-Counting dimension of the self-affine set is given by the bigger solution of the equations βx1 + β x 2 = 1 and τ x 1 + τ x 2 = 1. This can be shown by transferring the arguments of Pollicott and Weiss [PW] in the non overlapping symmetric to the non overlapping asymmetric situation . (4) It may be interesting to notice that it follows from our result and the implicit function theorem that the function θ 7−→ dimB Λθ is C∞ on the interior of P 4all. 24 Proof of 4.1. Let f(t) = β1τ t 1 + β2τ t 2. Since f(0) = β1 + β2 ≥ 1 and f is strictly monotonous decreasing with limt−→∞ f(t) = 0 there is an unique positive number d with β1τ d 1 + β2τ d 2 = 1. Fix d. Given a real number r > 0 we define a set of finite sequences by Xr := {(s1, . . . , sk)|min{τ1, τ2}r ≤ τs1τs2 . . . τsk < r where sj ∈ {1, 2} ∀j = 1 . . . k}. Notice that the sequences in Xr have not the same length. Let k(r) be the maximal length of a sequence in Xr. We observe that for every sequence (sj) ∈ {1, 2}k(r) there is an unique k such that (s1, . . . , sk) ∈ Xr. Thus we get ∑ (s1,...,sk)∈Xr βs1βs2 . . . βsk(τs1τs2 . . . τsk) d = ∑ (s1,...,sk)∈Xr βs1βs2 . . . βsk(τs1τs2 . . . τsk) d(β1τ d 1 + β2τ d 2 ) k(r)−k = ∑ (s1,...,sk(r))∈{1,2}k(r) βs1βs2 . . . βsk(r)(τs1τs2 . . . τsk(r)) d = (β1τ d 1 + β2τ d 2 ) k(r) = 1 (1). Beside equitation (1) we need one more fact. Let v be the unique positive number satisfying τ v1 + τ v 2 = 1. Since τ1 + τ2 < 1 we have v ≤ 1 ≤ d+ 1. Consequently ∑ (s1,...,sk)∈Xr (τs1τs2 . . . τsk) d+1 ≤ ∑ (s1,...,sk)∈Xr (τs1τs2 . . . τsk) v = 1 (2). Now we are prepared to begin with the main proof. We define a cover of Λθ by Cr = {πθ([κ(s1), . . . , κ(sk)]0)|(s1, . . . , sk) ∈ Xr} where κ(1) = 1 and κ(2) = −1. Since {[κ(s1), . . . , κ(sk)]0|(s1, . . . , sk) ∈ Xr} is a cover of Σ+ we get from 3.1.1. that Cr is in fact a cover of Λθ. An element of Cr is a rectangle parallel to the axis with x-length 2βs1βs2 . . . βsk and y-length 2τs1τs2 . . . τsk . We cover each of this rectangles by squares parallel to the axis of side length 2τs1τs2 . . . τsk . We choose the squares in a row such that they only intersect in their boundary. So we get for each rectangle a covering by dβs1βs2 ...βsk τs1τs2 ...τsk e squares (here dxe denotes the smallest integer bigger than x). In this way we obtain a new cover Ĉr of Λθ, which consists of squares with side length in (2min{τ1, τ2}r, 2r]. Furthermore the number N(r) of elements in Ĉr is given by N(r) = ∑ (s1,...,sk)∈Xr dβs1βs2 . . . βsk τs1τs2 . . . τsk e. 25 Now we have the following upper estimate, N(r)rd+1 ≤ min{τ1, τ2}−(d+1) ∑ (s1,...,sk)∈Xr dβs1βs2 . . . βsk τs1τs2 . . . τsk e(τs1τs2 . . . τsk)d+1 ≤ min{τ1, τ2}−(d+1)( ∑ (s1,...,sk)∈Xr βs1βs2 . . . βsk(τs1τs2 . . . τsk) d+ ∑ (s1,...,sk)∈Xr (τs1τs2 . . . τsk) d+1) ≤(1)/(2) 2min{τ1, τ2}−(d+1) and the following lower estimate: N(r)rd+1 ≥ ∑ (s1,...,sk)∈Xr dβs1βs2 . . . βsk τs1τs2 . . . τsk e(τs1τs2 . . . τsk)d+1 ≥ ∑ (s1,...,sk)∈Xr βs1βs2 . . . βsk(τs1τs2 . . . τsk) d =(1) 1. Now let N(r) be the minimal cardinality of an arbitrary cover of Λθ with squares parallel to the axis of side length 2r. Obviously we have N(r) ≤ N(r) but we need another argument for an opposite estimate. Let R be a rectangle in the cover Cr. We see that the projection of Λθ ∩ Cr on the x-axis has the full x-length of the rectangle since we assumed β1 + β2 ≥ 1. This implies that the intersection of each square in Ĉr with Λθ is not empty. Thus if we have a cover of Λθ each element of Ĉr has to be intersected by at least one element of the cover. But one square with side length 2r can not intersect more than 9min{τ1, τ2}−2 squares in Ĉr because the squares in Ĉr have side length bigger than 2min{τ1, τ2}r and intersect, if at all, only in the boundary. It follows that N(r) ≥ 1/9min{τ1, τ2}2N(r). Putting our estimates together we obtain 1 9 min{τ1, τ2}2 ≤ N(r)rd+1 ≤ 2min{τ1, τ2}−(d+1) and hence dimB Λθ = lim r−→∞ logN(r) log(2r)−1 = lim r−→∞ logN(r) log r−1 = d+ 1. The formula dimB Λθ = d + 2 follows from the product structure of Λθ and proposition A5 of appendix A. So our proof is complete. 2 An analysis of the Hausdorff dimension of the sets Λθ and Λθ is very difficult. We we will present our results in chapter seven and ten. 26 5. Dimension formulas and estimates for ergodic measures 5.1. Lyapunov exponents and charts In this chapter we want to apply the general dimension theory of ergodic measures that was developed in the last twenty years (see [YO], [LY], [BPS] and references there in) to the systems we study. Our aim is to find formulas and upper bounds for the dimension of ergodic measures for (Λθ, Tθ), (Λθ, fθ) and ([−1, 1]2, fβ1,β2) in terms of Lyapunov exponents, metric entropy and the dimension of the measures μβ1,β2 . In this section we do some preparations, namely we show the existence of Lyapunov exponents and charts related to a measure μθ on Λθ and calculate the exponents. Lemma 5.1.1. There is a subset Ωθ ⊆ Λθ which has full measure for a all μθ ∈ M(Λθ, fθ) such that fθ is a bijection on Ωθ and fθ is differentiable for all x = (x, y, z) ∈ Ωθ with Dxfθ =    β1 0 0 0 2 0 0 0 ττ1    if y > 0 Dxfθ =    β2 0 0 0 2 0 0 0 τ2    if y < 0. Proof Denote by S the singularity [−1, 1]× {0} × [−1, 1] of the system and define the set Ωθ by Ωθ = ∞⋂ n=−∞ fnθ ([−1, 1]3\S). By definition we have fθ(Ωθ) = Ωθ and since fθ is injective it is in fact a bijection on Ωθ. Moreover if x ∈ Ωθ then x 6∈ S and hence fθ is differentiable and has obviously the derivative that we stated in the lemma. We only have to show now that μθ(Ωθ) = 1. By elemental calculations we see that Ωθ = ({(x, y, z) ∈ Λθ|y 6= 1, y 6= −1} ∪ {(1, 1, 1), (−1,−1,−1)})\ ∞⋃ n=0 f−n(S). Since μθ is invariant and the union in the expression above is disjoint it has zero measure. It remains to show that μθ([−1, 1] × {1} × [−1, 1]) = μθ({(1, 1, 1)} and μθ([−1, 1] × {−1} × [−1, 1]) = μθ({(−1,−1,−1)}. But this is obvious since fθ is just a contraction with fixed point (1, 1, 1) resp. (−1,−1,−1) on the sets [−1, 1]× {1} × [−1, 1] resp. [−1, 1]× {1} × [−1, 1] . 27 Now define linear subspaces of IR3 by Eu =<    0 1 0    > Es =<    0 0 1    ,    1 0 0    > Etau =<    0 0 1    > Ebeta =<    1 0 0    > . Given a Borel measure μ on Σ and γ1, γ2 ∈ (0, 1) we set Ξμγ1,γ2 = μ([1]0) log γ1 + μ([−1]0) log γ2. Proposition 5.1.2. Given μ ∈M(Σ, σ) and θ = (β1, β2, τ1, τ2) ∈ P 4all we have for μθ-almost all x ∈ Λθ. lim n−→∞ 1 n log ||Dxfnθ v|| = log 2 ∀v ∈ Eu If Ξμβ1,β2 ≥ Ξμτ1,τ2 : limn−→∞ 1 n log ||Dxfnθ v|| = { Ξμβ1,β2 if v ∈ Es\Etau Ξμτ1,τ2 if v ∈ Etau If Ξμβ1,β2 ≤ Ξμτ1,τ2 : limn−→∞ 1 n log ||Dxfnθ v|| = { Ξμτ1,τ2 if v ∈ Es\Ebeta Ξμβ1,β2 if v ∈ Ebeta Proof By lemma 5.1.1. we have for μθ-almost all x ∈ Λθ log ||Dxfnθ (    0 y 0   )|| = n log 2 + log y ∀n ≥ 0. This implies our claim about Eu. Now we look at Es. By lemma 5.1.1. and proposition 3.1.3. and 3.2.5. we have for μθ-almost all x ∈ Λθ log ||Dxfnθ (    x 0 z   )|| = log √ (xβ n−]n(s)+1 1 β ]n(s) 2 ) 2 + (zτ n−]n(s)+1 1 τ ]n(s) 2 ) 2 ∀n ≥ 0 where s = (sk) = π −1 θ (x) and ]n(s) counts the number of entrys in the set {s0, s−1, . . . , s−n} that are −1. 28 We now have to determine the limit of this expression for μalmost all s ∈ Σ. By Birkhoffs ergodic theorem (see 4.1.2. of [KH]) we have: lim n−→∞ 1 n+ 1 n∑ k=0 f(σ−k(s)) = ∫ fdμ μ− a.e. for all L1 functions f on Σ with respect to μ. Applying this to the functions fbeta(s) = log β1 if s0 = 1 log β2 if s0 = −1 ftau(s) = log τ1 if s0 = 1 log τ2 if s0 = −1 we obtain lim n−→∞ 1 n log β n−]n(s)+1 1 β ]n(s) 2 = Ξ μ β1,β2 and lim n−→∞ 1 n log τ n−]n(s)+1 1 τ ]n(s) 2 = Ξ μ τ1,τ2 μ− a.e. and from this by elemental calculus lim n−→∞ 1 n log √ (xβ n−]n(s)+1 1 β ]n(s) 2 ) 2 + (zτ n−]n(s)+1 1 τ ]n(s) 2 ) 2 = max{Ξμβ1,β2 ,Ξμτ1,τ2} μ− a.e. if x 6= 0 and y 6= 0. This implies our claims about the stable directions. 2 This proposition means that Lyapunov exponents exists almost everywhere for the systems (Λθ, fθ, μθ) if μ is ergodic. E u is the unstable direction with Lyapunov exponent log 2 and Es is the stable direction with exponent Ξμβ1,β2 or Ξ μ τ1,τ2 depending on which quantity is bigger. Accordingly E tau or Ebeta is the strong stable direction with Lyapunov exponent Ξμτ1,τ2 resp. Ξ μ β1,β2 . In order to guarantee the existence of Lyapunov charts associated with the Lyapunov exponents we have to show that the set of points that does not approach the singularity S := [−1, 1] × {0} × [−1, 1] with exponential rate has full measure. Precisely we have: Lemma 5.1.3. Given μ ∈M(Σ, σ) and θ = (β1, β2, τ1, τ2) ∈ P 4all we have for all ε > 0 μθ({x ∈ Λθ|∃l > 0 ∀n > 0 d(fn(x), S) > (1/l)e−εn}) = 1, 29 Proof Fix ε > 0. First note that it is sufficient if we show μθ({x ∈ Λθ|∃(nk)k∈IN −→∞ ∀k > 0 d(fnk(x), S) ≤ e−εnk}) = 0 because if we have for a point x that ∃n0∀n > n0 d(fn(x), S) > e−εn then there exists l > 0 such that d(fn(x), S) > (1/l)e−εn ∀n > 0. By 3.1.3. and the definition of the measure μθ this assertion is equivalent to the following statement about the symbolic system (Σ, σ−1, μ): μ(N) = 0 where N := {s ∈ Σ|∃(nk)k∈IN −→∞ ∀k > 0 d(σ−nk(s), S) ≤ e−εnk} and S = {s ∈ Σ|s−1 = 1 and sk = −1 ∀k < −1}. We will now prove this. If s ∈ N we have d(σ−nk(s), S) ≤ e−εnk∀k > 0 By the definition of the metric d this implies σ−nk(s) ∈ [−1,−1, . . . ,−1, 1 } {{ } dcεnke ]−dcεnke−1 ∀k > 0 where the constant c is independent of ε, nk and s. This gives us: σi(s) 6∈ [1]−2 i = nk, . . . , nk + dcεnke − 1 ∀k > 0. Thus we have: N ⊆ {s|∃(nk)k∈IN −→∞ ∀k > 0 : σi(s) 6∈ [1]−2 i = nk, . . . , nk + dcεnke − 1}. Applying lemma 7.1. of [ST2] for the ergodic system (Σ, σ, μ) (with Y = [1]−2) we obtain μ(N) = 0. 2 By this lemma the systems (Λθ, fθ, μθ) fall into the class of generalized hyperbolic attractors in the sense of Schmeling and Troubetzkoy [ST1,2.1.]. From [ST1,3] it follows that our systems have appropriate Lyapunov charts almost everywhere with respect to the exponents given in 5.1.2. . For the definition and the constructed of these Lyapunov charts we reefer to [KS]. 5.2. Exact dimensionality and Ledrappier Young formula Usually the general theory for the dimension of ergodic measures is stated in the context of C2-diffeomorphisms in order to guarantee the existence of Lyapunov exponents and charts. But invertibility and the existence of Lyapunov exponents and charts almost everywhere is enough to apply this theory. We refer to section 4 of [ST1] for this fact. This is of great importance for us. For the systems (Λθ, fθ, μθ) 30 we have shown invertibility and the existence of Lyapunov exponents and charts almost everywhere in last section. We are thus allowed to apply the general results found in [BPS], [LY] and [YO] in our context. To this end first define partitions W s and W u of [−1, 1]3 in the stable and in the unstable directions of fθ by the partition elements W s(x) = [−1, 1]× {y} × [−1, 1] W u(x) = {x} × [−1, 1]× {z} where x = (x, y, z) ∈ Λθ. Given μθ ∈ M(Λθ, fθ) we have conditional measures μsθ(x) on W s and μuθ(x) on W u. These measures are unique μθ-almost everywhere fulfilling the relations: μθ(B) = ∫ μsθ(x)(B ∩W s(x))dμθ(x) resp. μθ(B) = ∫ μuθ(x)(B ∩W u(x))dμθ(x) for all Borel sets B in [−1, 1]3. We refer to [LY] and [RO] for informations about conditional measures on measurable partitions. Let us define balls in the elements of the partitions by Bsr((x, y, z)) = {(x, ȳ, z)|ȳ = y and (x, z) ∈ Br(x, z)}, Bur ((x, y, z)) = {(x, ȳ, z)|x = x z = z and ȳ ∈ Br(y)}. Now applying the results of Barreira, Schmeling and Pesin [BPS] to the system (Λθ, fθ, μθ) we obtain: Proposition 5.2.1. Let μ ∈ M(Σ, σ), θ = (β1, β2, τ1, τ2) ∈ P 4all and let μsθ(x) be conditional measures on W s and μuθ(x) conditional measures on W u with respect to μθ. We have: ds(x, μsθ(x)) := limr−→∞ log μsθ(x)(B s r(x)) log r = const. =: dim μsθ μθ − a.e. du(x, μuθ(x)) := limr−→∞ log μuθ(x)(B u r (x)) log r = const. =: dim μuθ μθ − a.e. d(x, μθ) := limr−→∞ log μθ(x)(Br(x)) log r = dim μuθ + dim μ s θ =: dim μθ μθ − a.e. An introduction to the local dimension, which is used here, can be found in appendix A. The proposition means that the measure μθ is exact dimensional and that the dimension is given by the sum of the unstable and stable dimension resp. the local dimension of conditional measures on partitions in stable and unstable directions, which is almost everywhere constant. 31 Now we want to have some information about the quantities dim μuθ and dim μ s θ. This is easy for the unstable dimension, because this direction is one dimensional. The next proposition follows from Ledrappier and Young [LY] or from the work of Young [YO]: Proposition 5.2.2. Under the assumptions of 5.2.1. we have dim μuθ = hμθ(fθ)/ log 2. An analysis of dim μsθ is more difficult because we have two unstable directions with different expansion rates. If Ξμβ1,β2 ≥ Ξμτ1,τ2 we have a partition W ss in the strong stable direction given by the partition elements W ss(x) = {x} × {y} × [−1, 1] where x = (x, y, z) ∈ Λθ. Given μθ ∈M(Λθ, fθ) we have conditional measures μssθ (x) onW ss. These measures are unique μθ-almost everywhere fulfilling the relation: μθ(B) = ∫ μssθ (x)(B ∩W ss(x))dμθ(x) for all Borel sets B in [−1, 1]3. From the uniqueness of the conditional measures we have for μθ-almost all x = (x, y, z) μs(x)(B) = ∫ μssθ (x, y, z)(B ∩W ss(x, y, z))dprX μsθ(x)(x) for all Borel sets B in W s(x). This statement means that the transversal measures in the sense of [LY] of the nested partitions W s and W ss are in our context given by prX μ s θ(x). Now let: Bssr ((x, y, z)) = {(x, ȳ, z)|ȳ = y , x = x and z ∈ Br(z)} and Btrans((x, y, z)) = {(x, ȳ, 0)| x = x and ȳ ∈ Br(y)}. Applying the results of [LY] about the local dimensions of conditional measures in the context of dynamical systems we obtain: Proposition 5.2.3. Let μ ∈ M(Σ, σ) and θ = (β1, β2, τ1, τ2) ∈ P 4all with Ξμβ1,β2 ≥ Ξμτ1,τ2 . Let μsθ(x) be conditional measures on W s and μssθ (x) conditional measures on W ss with respect to μθ. We have dss(x, μssθ (x)) := limr−→∞ log μssθ (x)(B ss r (x)) log r = const. =: dim μssθ μθ − a.e. 32 dtrans(x, prX μ s θ(x)) := limr−→∞ log prX μ s θ(x)(B trans r (x)) log r = dim μsθ − dimμssθ =: dim μtransθ μθ − a.e. dim μsθ = hμθ(fθ) −Ξμτ1,τ2 + (1− Ξ μ β1,β2 Ξμτ1,τ2 ) dim μtransθ The last equitation is known in dimension theory of dynamical systems as Ledrappier-Young formula. 5.3. Some consequences We will find here some interesting consequences of the general results of the last section. First we have an upper bound on the dimension of the measures μθ. Proposition 5.3.1. Let μ ∈ M(Σ, σ), θ = (β1, β2, τ1, τ2) ∈ P 4all and let d the unique positive number satisfying β1τ d 1 + β2τ d 2 = 1. We have: dim μθ ≤ hμ(σ) log 2 + d+ 1. Proof Combining 5.2.1. and 5.2.2. with 3.2.6. we have: dim μθ = hμ(σ) log 2 + dim μsθ. Since μθ is a measure on Λθ the measures μ s θ(x, y, z) are by definition concentrated on the set {(x, ȳ, z)|y = ȳ (x, z) ∈ Λθ}. Hence we have dimH μ s θ(x, y, z) ≤ dimH Λθ ≤ dimB Λθ ∀(x, y, z) ∈ Λθ. Using theorem A2 we now get dim μsθ ≤ dimB Λθ . But from 4.1. we know dimB Λθ = d+ 1, which competes the proof. 2 It is well known in the theory of dynamical systems that the equal weighted Bernoulli measure is the unique ergodic Borel measure of maximal entropy log 2 for the system (Σ, σ); see 8.9. of [WA]. Thus the last proposition shows that the only ergodic measures for the attractor (Λθ, fθ) that can have full box-counting dimension is the equal weighted Bernoulli measure bθ. We now present an other upper bound on the dimension of μθ, which is in terms of the dimension of the measures (pr+μ)β1,β2 where pr + as usual denotes the projection from Σ onto Σ+. 33 Proposition 5.3.2. Under the assumption of proposition 5.2.3. we have: dim μθ ≤ hμ(σ) log 2 + hμ(σ) −Ξμτ1,τ2 + (1− Ξ μ β1,β2 Ξμτ1,τ2 ) dimH(pr +μ)β1,β2 . Proof The result follows immediately combining 5.2.1. with 5.2.2. and 5.2.3. if we show the inequality dim μtransθ ≤ dimH(pr+μ)β1,β2 . To see this choose a Borel set B in the line with (pr+μ)β1,β2(B) = 1. Because we know from 3.2.3. that prXνθ = νβ1,β2 ∀ν ∈M(Σ+, σ) we have (pr+μ)θ(B×[−1, 1]) = 1. From 3.2.5. we know prXZμθ = (pr+μ)θ. Hence we get μθ(B× [−1, 1]× [−1, 1]) = 1. By the definition of the conditional measures μsθ(x) we get μ s θ(x, y, z)(B × {y} × [−1, 1]) = 1 μθ-almost everywhere. This implies prX μsθ(x, y, z)(B × {y}) = 1 and hence dimH prX μ s θ(x, y, z) ≤ dimH B μθ-almost everywhere. With 5.2.3. and A2 we now get dim μtransθ ≤ dimH B. This implies the desired inequality since B was an arbitrary Borel set with (pr+μ)β1,β2(B) = 1. 2 For the Bernoulli measures bpθ and b p θ we get explicit dimension formulas in terms of the dimension of the measures self-similar measures bpβ1,β2 Proposition 5.3.3. For all θ = (β1, β2, τ1, τ2) ∈ P 4all and p ∈ (0, 1) with p log β1 + (1 − p) log β2 ≥ p log τ1 + (1− p) log τ2 we have: dim bpθ = −p log p− (1− p) log(1− p) log 2 + p log p+ (1− p) log(1− p) p log τ1 + (1− p) log τ2 +(1− p log β1 + (1− p) log β2 p log τ1 + (1− p) log τ2 ) dimH b p β1,β2 and dim bpθ = p log p+ (1− p) log(1− p) p log τ1 + (1− p) log τ2 + (1− p log β1 + (1− p) log β2 p log τ1 + (1− p) log τ2 ) dimH b p β1,β2 . 34 Proof We know from 3.2.5. that the measure bpθ is the product of the measure b p θ in the (x, z)-plane with the measure `p on the y-axis. From this follows, that the conditional measures (bpθ) s(x) are given by the measure bpθ for b p-almost all x. Furthermore the transversal measures prX(b p θ) s(x) are given by bpβ1,β2 because we have from 3.2.3. prX b p θ = bβ1,β2 . The dimension formulas are now just a consequence of the propositions of section 5.2. and the following explicit formulas: hbp(σ) = −p log p− (1− p) log(1− p) and Ξγ1,γ2bp = p log γ1 + (1− p) log γ1. For the first formula see for instance 12.4. of [DGS]. The second one is obvious. 2 We have the following upper bound on the Hausdorff dimension of the ergodic measure for the projected system ([−1, 1]2, fβ1,β2): Proposition 5.3.4. Let μ ∈M(Σ, σ) and (β1, β2) ∈ P 2olapp. We have: dimH μβ1,β2 ≤ hμ(σ) log 2 + 1. Proof Fix (β1, β2) ∈ P 2olapp and choose τ ∈ (0, 0.5). Let θ = (β1, β2, τ, τ). Applying 5.3.2. we have: dim μθ ≤ hμ(σ) log 2 + log(β1 + β2) log τ−1 + 1. From proposition 3.2.5. we know prXY μθ = μβ1,β2 , which obviously implies dimH μβ1,β2 ≥ dimH μθ. Hence dimH μβ1,β2 ≤ hμ(σ) log 2 + log(β1 + β2) log τ−1 + 1 ∀τ ∈ (0, 0.5). Letting τ −→ 0 we get our result. 2 From this proposition and 3.2.4. it follows, that the only ergodic measures for the attractor ([−1, 1]2, fβ1,β2) that can have full Hausdorff dimension is the equal weighted Bernoulli measure bβ1,β2 . We like to include here an upper bound on the Hausdorff dimension of the measures μβ1,β2 , which can be proved elementary without the results of 5.2. . 35 Proposition 5.3.5. Let μ ∈M(Σ, σ) and (β1, β2) ∈ P 2olapp. We have dimH μβ1,β2 ≤ dimH(pr+μ)β1,β2 + 1 Proof Let B be a Borel set with (pr+μ)β1,β2(B) = 1. By 3.2.4. we have prX μβ1,β2 = (pr+μ)β1,β2 . Hence μβ1,β2(B × [−1, 1]) = 1. Thus dimH μβ1,β2 ≤ dimH(B × [−1, 1]) and by proposition A5 dimH μβ1,β2 ≤ dimH(B) + 1. Since B was an arbitrary Borel set with full (pr+μ)β1,β2 measure we get our result. 2 Our results here show that the study of the measures νβ1,β2 and especially the selfsimilar measures bpβ1,β2 is essential for us. This discussion occupies the next chapter. But before we state a result about the entropy of the measures μβ1,β2 . Proposition 5.3.6. If μ ∈M(Σ, σ) and (β1, β2) ∈ P 2olapp we have hμβ1,β2 (fβ1,β2) = hμ(σ). This fact is not trivial because the system ([−1, 1]2, fβ1,β2 , μβ1,β2) is only a measure theoretical factor of (Σ, σ, μ). We know only a long and quite complicated proof of this proposition using conditional measures and dimensions. Because we do not need this proposition in the main line of our argumentation we think it is enough if we give a sketch of our proof; the details can be found in [NE]. Sketch of proof We first define a partition W u of [−1, 1]2 and a partition W su of [−1, 1]3 by W u(x, y) = {x} × [−1, 1] W su(x, y, z) = {x} × [−1, 1]2. Given μβ1,β2 we have conditional measures μ u β1,β2 (x, y) on the elements of W u and given μθ we have conditional measures μ su θ (x, y, z) on the elements of W su(x, z, z). Using properties and uniqueness of conditional measures it is possible to show that the following relations hold for μθ-almost all (x, y, z) ∈ Λθ: (1) prXY μ su θ (x, y, z) = μ u β1,β2 (x, y) (2) μsuθ (x, y, z)(B∩W su(x, y, z)) = ∫ μuθ(x, ȳ, z)(B∩W u(x, ȳ, z))dμsuθ (x, y, z)(ȳ, z). From 5.2.1. and A2 it follows that dimH μ u θ(x, y, z) = dim μ u θ holds μθ-almost everywhere. But this implies for μθ-almost all (x, y, z) ∈ Λθ (3) dimH μ u θ(x, ȳ, z) = dim μ u θ for μ su θ (x, y, z)-almost all (x, ȳ, z) ∈W su(x, y, z). 36 Let G be the set of all (x, y, z) such that (1),(2) and (3) hold. Fix (x, y, z) ∈ G. Let B be an arbitrary Borel set such that μuβ1,β2(x, y)({x} × B) = 1. With (1) it follows that μsuθ (x, y, z)({x} × B × [−1, 1]) = 1 and with (2) we get from this μuθ(x, ȳ, z)({x} × B × {z}) = 1 for μsuθ (x, y, z) -almost all (x, ȳ, z) ∈ W su(x, y, z). Hence dimH μ u θ(x, ȳ, z) ≤ dimH B for μsuθ (x, y, z) -almost all (x, ȳ, z) ∈ W su(x, y, z) and with (3) dim μuθ ≤ dimH B. Since B was arbitrary, G has full μθ measure and μθ projects to μβ1,β2 this shows: dim μuθ ≤ dimH μuβ1,β2(x, y) μβ1,β2-a.e. Now let us look at the entropy. On the one hand we know dim μuθ = hμ(σ)/ log 2 from 5.2.2. . On the other hand it is by means of [MN] not difficult to see in rather direct way that dimH μ u β1,β2 (x, y) ≤ hμβ1,β2 (fβ1,β2)/ log 2 holds μβ1,β2-a.e. (see [NE]). Hence hμ(σ) ≤ hμβ1,β2 (fβ1,β2). For the opposite inequality see the remark after 3.2.6 . 2 It seems to be plausible that a more direct proof of the last proposition should be possible only working with the entropy of conditional measures and without using dimensions at all. But we have not elaborated this. 37 6. Overlapping self-similar measures 6.1. Main results In this section we begin to study the self-similar measure bpγ1,γ2 defined in 3.2. for γ1, γ2 ∈ (0, 1) and p ∈ (0, 1). If γ1+ γ2 < 1 the measure is concentrated on a Cantor set and hence singular. We will discuss here the overlapping case and thus assume γ1 + γ2 ≥ 1. The overlapping symmetric self-similar measures bpγ are usually called infinitely convolved Bernoulli measures. They raised great interest in the literature. Using Fourier transformation techniques Winter [WI] showed in 1935 that bγ is absolutely continuous if γ = 1n√2 with n ≥ 0 and Erdös [ER2] showed in 1940 that the measure is absolute continuous for almost all γ in a small neighborhood of one. Recently one mayor progress was achieved by Solomyak: Theorem 6.1.1. [SO1] The measure bγ is absolutely continuous with square integrable density for almost all γ ∈ (0.5, 1). We like to inform the reader here that there are parameter values γ with special number theoretical properties such that bγ is singular. We will discuss this issue in detail in chapter nine. Peres and Solomyak [PS1] found a considerably simplified proof of theorem 6.1.1. . Moreover they extended the technique used in this proof to the measures bpβ, which have different weights. They proved: Theorem 6.1.2. [PS2] Let p ∈ (0, 1). The measures bpγ are absolutely continuous for almost all γ ∈ (pp(1 − p)1−p, 0.649) and singular if γ < pp(1 − p)1−p. If p ∈ [1/3, 2/3] then the bound 0.649 in this statement can be replaced by 1. As far as we know the overlapping asymmetric self-similar measures bpγ1,γ2 have not been studied jet. This will be our task here. We will prove an analogon of 6.1.2. in the asymmetric situation. Let us first define a subset of the parameters set P 2olapp := {(γ1, γ2) ∈ (0, 1)2|γ1 + γ2 ≥ 1} by P 2trans := {(γ1, γ2) ∈ P 2olapp|γ2 ≤ γ1 ≤ 0.649}. Now we formulate our result: 38 Theorem 6.1.3. Let p ∈ (0, 1) and P 2abs := {(γ1, γ2) ∈ P 2trans|(γ2p)p(γ1(1 − p))1−p ≤ γ1γ2}. The measures bpγ1,γ2 are absolutely continuous for almost all (γ1, γ2) ∈ P 2abs in the sense of two dimensional Lebesgue measure and singular if (γ2p) p(γ1(1− p))1−p > γ1γ2. The first part of this theorem follows from corollary 6.2.2. of the next section using the theorem of Fubini. The singularity assertion is stated in corollary 6.3.2. and follows from a more general upper bound on the box-counting dimension of the measures μγ1,γ2 we will prove in 6.3.1. . We think that it is necessary to make a few remarks on our main result: Remarks (1) Fist note that by the symmetry of the measures in question the assumption of γ2 ≤ γ1 in the definition of P 2trans means no loss of generality. (2) We have to say a few word about the bound 0.649 that appears in 6.1.3. (and also in 6.1.2.). On the first sight this bound seems to be somewhat crude. In the proof we will see that it is due to a certain transversality condition that we need. In fact the bound is given by the infimum of all double zeros of power series with absolute value of the coefficients less equal to one and first coefficient equal to one. 0.649 is an approximation of this quantity. We refer to step 4 of the proof of proposition 6.2.1. for this issue. (3) Peres and Solomyak [PS2] used some additional arguments concerning Fourier transformations to improve the bound to 1 in the symmetric situation if p ∈ [1/3, 2/3]. These arguments do not work if p < 1/3. We have not been able to improve the bound in the asymmetric situation but we do not believe that this bound is really essential. 6.2. Absolute continuity Let us first recall some definitions from chapter three. The measures bp∗γ1,γ2 are given by (π∗γ1,γ2) −1 ◦ bp. bp is the Bernoulli measure on Σ+ = {−1, 1}IN0 with probability distribution (p, 1− p) on {1,−1} and the map π∗γ1,γ2 is given by π∗γ1,γ2(s) = ∞∑ k=0 skγ ]k(s) 2 γ k−]k(s)+1 1 . The quantity ]k(s) counts how often −1 appears in {s0, . . . , sk}. The measures bpγ1,γ2 are just the measures bp∗γ1,γ2 scaled by the affine transformation that maps −γ2 1−γ2 to 39 −1 and γ1 1−γ1 to 1. Now we state our result on absolute continuity and density of the measures at hand. Proposition 6.2.1. Let p ∈ (0, 1), q ∈ (1, 2] and c ∈ (0, 1]. The density of the measures bpγ,cγ is in Lq for almost all γ ∈ [γ0(c, q, p), 0.649] where γ0(c, q, p) = (pq + c1−q(1− p)q) 1 q−1 . The technique we will use in the following proof is similar to argumentations that have been developed in [PS 1/2]. Proof Obviously it is enough if we show that the proposition holds for the unscaled measures bp∗γ,cγ . Fix p, q and c during the proof. 1. Step: An integral condition for the measures to have density in Lq We define the (lower) local density of a measure μ on the real line by D(μ, x) = limr−→0 μ(Br(x)) 2r . If we have ∫ (D(μ, x))q−1dμ(x) <∞ then μ is absolute continuous and has density in Lq. This follows from Mattila [MA,2.12]. Thus it is sufficient for us to show that =(γ0) := ∫ 0.649 γ0 ∫ (D(bp∗γ,cγ , x)) q−1dbp∗γ,cγ(x) dγ <∞ holds for all γ0 > γ0(c, q, p). 2. Step: Some estimates on the integral By applying Fatou's lemma then changing variables using the definition of the measures bp∗γ,cγ and reversing the order of integration we obtain: =(γ0) ≤ limr−→0 1 (2r)q−1 ∫ 0.649 γ0 ∫ (b∗pγ,cγ(Br(x))) q−1 db∗pγ,cγ(x)dγ = limr−→0 1 (2r)q−1 ∫ 0.649 γ0 ∫ Σ+ (b∗pγ,cγ(Br(π ∗ γ,cγ(s)))) q−1 dbp(s)dγ 40 = limr−→0 1 (2r)q−1 ∫ Σ+ ∫ 0.649 γ0 (b∗pγ,cγ(Br(π ∗ γ,cγ(s)))) q−1 dγdbp(s). Applying Hölder's inequality, ∫ fα ≤ C1( ∫ f)α where α ∈ (0, 1] and f ≥ 0, we get =(γ0) ≤ C1limr−→0 1 (2r)q−1 ∫ Σ+ ( ∫ 0.649 γ0 b∗pγ,cγ(Br(π ∗ γ,cγ(s)) dγ) q−1dbp(s). Now note that ∫ 0.649 γ0 b∗pγ,cγ(Br(π ∗ γ,cγ(s))) dγ = ∫ 0.649 γ0 ∫ 1Br(π∗γ,cγ(s))(x) db ∗p γ,cγ(x)dγ = ∫ 0.649 γ0 ∫ Σ+ 1{t| |π∗γ,cγ(s)−π∗γ,cγ(t)|≤r}db p(t)dγ = ∫ Σ+ `({γ ∈ [γ0, 0.649]| |π∗γ,cγ(s)− π∗γ,cγ(t)| ≤ r}) dbp(t). Thus =(γ0) is bounded from above by C1 limr−→0 1 (2r)q−1 ∫ Σ+ ( ∫ Σ+ `({γ ∈ [γ0, 0.649]| |π∗γ,cγ(s)−π∗γ,cγ(t)| ≤ r}) dbp(t))q−1dbp(s). 3. Step: Using the structure of the map π For s = (sk) and t = (tk) in Σ + let |s ∧ t| = min{k|sk 6= tk}. We have: φs,t(γ) := πγ,cγ(s)− πγ,cγ(t) = ∞∑ k=0 (skc ]k(s) − tkc]k(t))γk+1 = γ|s∧t|+1 ∞∑ k=0 (sk+|s∧t|c ]k+|s∧t|(s) − tk+|s∧t|c]k+|s∧t|(t))γk = γ|s∧t|+1(s|s∧t|c ]|s∧t|(s) − t|s∧t|c]|s∧t|(t))(1 + ∞∑ k=1 sk+|s∧t|c ]k+|s∧t|(s) − tk+|s∧t|c]k+|s∧t|(t) s|s∧t|c ]|s∧t|(s) − t|s∧t|c]|s∧t|(t) } {{ } :=ak(s,t) γk) = 1/2(s|s∧t| − t|s∧t|)(1 + c)γ |s∧t|+1c]|s∧t|−1(s)(1 + ∞∑ k=1 ak(s, t)γ k). For the last equitation we used the fact that ]|s∧t|−1(s) = ]|s∧t|−1(t) 1. Now setting gs,t(γ) = 1 + ∑∞ k=1 ak(s, t)γ k and C2 = 1/2(s|s∧t| − t|s∧t|)(1 + c) we have the formula φs,t(γ) = C2γ |s∧t|+1c]|s∧t|−1(s)gs,t(γ). 1We use the convention that ]n(s) = 0 if n < 0 41 Here the absolute value of C2 does not depend on s and t. We now claim that the absolute value of the coefficients of the polynomials gs,t is less or equal to one: |ak(s, t)| ≤ 1 ∀ k > 0 and s, t ∈ Σ+. Since ]|s∧t|−1(s) = ]|s∧t|−1(t) we can write: |ak(s, t)| = (σ|s∧t|(s))kc σ|s∧t|(s) − (σ|s∧t|(t))kcσ|s∧t|(t) 1 + c . But we have |(σ|s∧t|(s))kcσ |s∧t|(s) − (σ|s∧t|(t))kcσ |s∧t|(t)| ≤ |cσ|s∧t|(s)|+ |cσ|s∧t|(t)| ≤ 1 + c by the definition of |s ∧ t|, which proves our claim. 4. Step: The transversality condition We say that the ρ-transversality condition holds for a C1 function g on a closed interval I if g(x) < ρ⇒ g′(x) > ρ ∀x ∈ I. This means that the graph of the function g crosses all horizontal lines that it meets below height λ transversally with slope at most −ρ. Obviously the transversality condition holds for some ρ on an interval I if and only if g has no double zero on the interval I. If we have the ρ-transversality condition for g on I then `{x ∈ I||g(x)| ≤ r} ≤ 2rρ−1 ∀r > 0. This is easy to see. If r ≥ ρ then the claim is obvious. If r < ρ then g is monotonous decreasing with g ′ < −ρ on the set {x ∈ I||g(x)| ≤ r} by ρ-transversality. But this immediately yields the assertion. From lemma 2 of [PS2] we know that: O := inf{x|x is a double zero of a power series f = 1 + ∞∑ k=1 akx k with |ak| ≤ 1} ≈ 0.649138. It follows that there is a ρ such that the ρ-transversality condition holds for all polynomials f = 1 + ∑∞ k=1 akx k with |ak| ≤ 1 on the Interval [0, O]. Especially ρ transversality holds for all polynomials gs,t defined in the third step of our proof on [0, 0.649]. Thus we get: 42 `{γ ∈ [γ0, 0.649]| |φs,t(γ)| ≤ r} ≤ `{γ ∈ [γ0, 0.649]| |gs,t(γ)| ≤ r|C2|−1γ−|s∧t|−1c−]|s∧t|−1(s)} ≤ 2ρ−1r|C2|−1γ−|s∧t|−10 c−]|s∧t|−1(s) = C3rγ−|s∧t|−10 c−]|s∧t|−1(s) with C3 = 2ρ−1|C2|−1. 5. Step: Integrating We put our estimates of step two and four together and obtain =(γ0) ≤ C4 ∫ Σ+ ( ∫ Σ+ γ −|s∧t|−1 0 c −]|s∧t|−1(s) dbp(t))q−1dbp(s) where C4 = C1C q−1 3 2 1−q. Now we integrate: ∫ Σ+ γ −|s∧t|−1 0 c −]|s∧t|−1(s) dbp(t) = ∞∑ n=0 γ−n−10 c ]n−1(s)bp({t ∈ Σ+| |s ∧ t| = n} = ∞∑ n=0 γ−n−10 c −]n−1(s)pn−]n−1(s)(1− p)]n−1(s)(sn(1/2− p) + 1/2) Using the inequality ( ∑ xi) α ≤ ∑ xαi for α = q − 1 ≤ 1 we continue with: =(γ0) ≤ C4 ∞∑ n=0 ∫ Σ+ (γ−n−10 c −]n−1(s)pn−]n−1(s)(1−p)]n−1(s)(sn(1/2−p)+1/2))q−1dbp(s) = C4 ∞∑ n=0 γ (−n−1)(q−1) 0 ((1−p)q−1p+pq−1(1−p)) n∑ k=0 (c−k(1−p)kpn−k)q−1bp{s ∈ Σ+|]n−1(s) = k} = C4((1− p)q−1p+ pq−1(1− p)) ∞∑ n=0 γ (−n−1)(q−1) 0 n∑ k=0 ( n k ) ((1− p)qc1−q)kpq(n−k) = C4((1− p)q−1p+ pq−1(1− p))γ1−q0 ∞∑ n=0 (γ −(q−1) 0 ((1− p)qc1−q + pq)n. The sum in the last expression converges exactly if γ0 > γ0(c, q, p) = (p q + c1−q(1− p)q) 1 q−1 . So =(γ0) <∞ holds for all γ0 > γ0(c, q, p) and our proof is complete. 2 Proposition 6.2.1. has the following corollary: Corollary 6.2.2. Let c ∈ (0, 1] and p ∈ (0, 1). The measures bpγ,cγ are absolutely continuous for almost all γ ∈ [γ0(c, p), 0.649] where γ0(c, p) = pp((1− p)/c)1−p. 43 Proof One only has to show that limq−→1 γ0(c, q, p) = γ0(c, p). But this is easy to see by taking logarithm and using the rule of l'Hospital. 2 6.3. An upper bound the on dimension We will prove here an general upper bound on the box-counting dimension of the measures μγ1,γ2 defined in 3.2. where γ1, γ2 ∈ (0, 1) are arbitrary and μ is a shift ergodic measures. Applying this general upper bound to the overlapping self-similar measures bpγ1,γ2 implies the singularity assertion of our main theorem 6.1.3. . Recall that Ξμγ1,γ2 = μ([1]0) log γ1 + μ([−1]0) log γ2. Proposition 6.3.1. If μ ∈M(Σ+, σ) and γ1, γ2 ∈ (0, 1) we have: dimBμγ1,γ2 ≤ min{1, hμ(σ) −Ξμγ1,γ2 }. Proof Fix γ1, γ2 and p. First note that it is trivial that the box-counting dimension of the measure in question is less or equal to one since it is defined on the real line. We now define a metric δγ1,γ2 on Σ+ by δγ1,γ2(s, t) = γ |s∧t|−]|s∧t|−1(s) 1 γ ]|s∧t|−1(s) 2 . We first claim that dγ1,γ2(s, bp) := lim ε−→0 logBγ1,γ2ε (s) log ε = hμ(σ) −Ξμγ1,γ2 μ-almost everywhere. Here dγ1,γ2 is the local dimension of the measure bp with respect to metric δγ1,γ2 and accordingly Bγ1,γ2ε is a ball of radius ε with respect to this metric. Applying Birkhoffs ergodic theorem (see 4.1.2. of [KH]) to (Σ+, σ, μ) with the function h(s) = { log γ1 if s0 = 1 log γ2 if s0 = −1 we see that: lim n−→∞ 1 n+ 1 log diamγ1,γ2([s0, . . . , sn]0) = limn−→∞ 1 n+ 1 n+1∑ k=0 h(σk(s)) = ∫ h dμ(s) = Ξμγ1,γ2 44 μ-almost everywhere. By Shannon-McMillan-Breiman theorem (see [DGS] 13.4.) we have: lim n−→∞ − 1 n+ 1 log μ([s0, . . . , sn]0) = hμ(σ) μ-almost everywhere. Thus we see: lim ε−→0 logBγ1,γ2ε (s) log ε = lim n−→∞ log μ([s0, . . . , sn]0) diamγ1,γ2([s0, . . . , sn]0) = hμ(σ) −Ξμγ1,γ2 Of course we can define the box-counting dimension of the measure μ with respect to the metric δγ1,γ2 on Σ in exactly the same way as we define the box-counting dimension of a measure on IRq in appendix A. Furthermore it is not difficult to see that an analogon of A2 holds for Borel probability measures on the metric space (Σ+, δγ1,γ2). Thus we have dimγ1,γ2B μ = hμ(σ) −Ξμγ1,γ2 where the box-counting dimension dimγ1,γ2B has to be calculated using δ γ1,γ2 . Now we claim that the map π∗γ1,γ2 is Lipschitz with respect to the metric δ γ1,γ2 : |π∗γ1,γ2(s)− π∗γ1,γ2(t)| ≤ ∞∑ k=|s∧t| |skγk−]k(s)+11 γ]k(s)2 − tkγk−]k(t)+11 γ]k(t)2 | = γ |s∧t|−]|s∧t|−1(s) 1 γ ]|s∧t|−1(s) 2 ∞∑ k=0 |sk+|s∧t|γk−]k(σ |s∧−t|(s))+1 1 γ ]k(σ |s∧t|(s)) 2 − tk+|s∧t|γk−]k(σ |s∧t|(t))+1 1 γ ]k(σ |s∧t|(t)) 2 | ≤ δγ1,γ2(s, t) 2 1−max{γ1, γ2} . But the map πγ1,γ2 is just π ∗ γ1,γ2 scaled on [−1, 1] and hence Lipschitz with respect δγ1,γ2 as well. Since applying a Lipschitz map to the measures μ does obvious not increase its box-counting dimension, the proof is complete. 2 Let us remark that it is well known that the Hausdorff and box-counting dimension of of μγ1,γ2 equals −hμ(σ)/Ξμγ1,γ2 in the case that γ1 + γ2 < 1; see for instance 13.1. of [PE2]. In our work we are more interested in the overlapping case γ1 + γ2 ≥ 1 . From 6.3.1. we get the following corollary about the self-similar measures bpγ1,γ2 : 45 Corollary 6.3.2. Let γ1, γ2, p ∈ (0, 1). We have dimBb p γ1,γ1 ≤ min{1, p log p+ (1− p) log(1− p) p log γ1 + (1− p) log γ2 }. Moreover bpγ1,γ2 is singular if (γ2p) p(γ1(1− p))1−p > γ1γ2. Proof To see the upper bound just recall that hbp(σ) = −p log p − (1 − p) log(1 − p) and Ξγ1,γ2(b p) = p log γ1 + (1− p) log γ2. From the upper bound we have dimBbpγ1,γ2 < 1 if (γ2p) p(γ1(1− p))1−p > γ1γ2, which clearly implies our singularity assertion. 2 46 7. Generic dimensional theoretical properties of the systems We will now formulate our main generic results about the dimensional theoretical properties of the class of repellers (Λθ, Tθ) and the class of attractors (Λθ, fθ). The term "generic" has to be understood in a special sense referring to the Lebesgue measure on certain subspaces of the parameter space P 4all. The restrictions of our generic results depend on the transversality condition; see chapter six. Recall from this chapter that P 2trans = {(β1, β2)|β1 + β2 ≥ 1 and 0 < β2 ≤ β1 ≤ 0.649}. Theorem 7.1. General case For all p ∈ (0, 1) and almost all (β1, β2) ∈ P 2trans and all τ1, τ2 > 0 with τ1 + τ2 < 1 and log τ2 log p/β1 = log τ1 log(1− p)/β2 we have: dimH b p θ = dimH Λθ = dimB Λθ = log p/β1 log τ1 + 1 and dimH Λθ = dimB Λθ = log p/β1 log τ1 + 2 where θ = (β1, β2, τ1, τ2). Moreover if p = 0.5 then b p θ has full dimension on Λθ and if p 6= 0.5 then the variational principle for Hausdorff dimension does not hold for (Λθ, fθ). Special case β1 = β2 = β For p ∈ (0, 1) set I = (0.5, 1) if p ∈ (1/3, 2/3) and I = (0.5, 0.649) if not. We have for almost all β ∈ I and all τ1, τ2 > 0 with τ1 + τ2 < 1 and log τ2 log p/β = log τ1 log(1− p)/β: dimH b p θ = dimH Λθ = dimB Λθ = log p/β log τ1 +1 and dimH Λθ = dimB Λθ = log p/β log τ1 +2 where θ = (β, β, τ1, τ2). Moreover if p 6= 0.5 (which means τ1 6= τ2) then the variational principle for Hausdorff dimension does not hold for (Λθ, fθ). Special case τ1 = τ2 = τ For almost all (β1, β2) ∈ P 2trans and all τ ∈ (0, 0.5) we have: dimH b p θ = dimH Λθ = dimB Λθ = log(β1 + β2) log τ−1 + 1 47 and dimH Λθ = dimB Λθ = log(β1 + β2) log τ−1 + 2 where θ = (β1, β2, τ, τ) and p = β1/(β1 + β2). Moreover if β1 6= β2 then the variational principle for Hausdorff dimension does not hold for (Λθ, fθ). Special case τ1 = τ2 = τ and β1 = β2 = β For almost all β ∈ [0, 5, 1] and all τ ∈ (0, 0.5) we have dimH b 0.5 θ = dimH Λθ = dimB Λθ = log 2β log τ−1 + 1 and dimH b 0.5 θ = dimH Λθ = dimB Λθ = log 2β log τ−1 + 2 where θ = (β, β, τ, τ). We include a corollary, which states the main general result of the last theorem in a weaker but more straightforward way. Recall that: P 4trans = {(β1, β2, τ1, τ2) ∈ P 4all|(β1, β2) ∈ P 2trans}. Corollary 7.2. For almost all θ ∈ P 4trans we have: dimH b p θ = dimH Λθ = dimB Λθ = d+ 1 and dimH Λθ = dimB Λθ = d+ 2 where θ = (β1, β2, τ1, τ2) and d is the solution of β1τ x 1 + β2τ x 2 = 1 and p = β1τ d 1 . Let us discuss our results: Remarks (1) Corollary 7.2. shows that on the set of parameters P 4trans we have generically the identity of Hausdorff and box-counting dimension for the repellers (Λθ, Tθ) and the attractors (Λθ, fθ). (2) The existence of a measure of full dimension is only a generic property of the repellers. Not even the variational principle for Hausdorff dimension holds generically with respect to the Lebesgue measure on P 4trans for the attractors. It holds for (Λθ, fθ) only if we have logτ1 log(2β1) = logτ2(2β2). In the proof we will see that this phenomenon is due to the fact that one can not maximize the stable and the unstable dimension (resp. the dimension of the corresponding conditional measures) at the same time. In the context of Axiom A diffeomorphisms exactly this 48 was observed by Manning and McCluskey [MM]. It allowed them to show that the variational principle is not generic for Axiom A systems in the topological sense; it only holds on a nowhere dense set of Axiom A systems. (3) Now we comment on the special case β1 = β2 and τ1 = τ2. The condition on β we need to get the identity for Hausdorff and box-counting dimension in this case is dimH bβ = 1. By Solomyak's theorem (6.1.1.) this condition holds almost everywhere in (0, 5, 1). Pollicott and Weiss used the number theoretical condition that β is a Garsia-Erdös number to get this identity (see 2.1.3.). The property to be a Garsia-Erdös number can been shown to be equivalent to the absolute continuity of bβ with uniformly bounded density (see 2.1.3. and 5. of [PW]). This condition seems to be stronger than dim bβ = 1. But in fact we do not know if there are numbers such that dim bβ = 1 and β is not Garsia-Erdös. We will now formulate our result about the generic dimensional theoretical properties of the projected systems ([−1, 1]2, fβ1,β2) including the Fat Baker's transformations fβ. Theorem 7.3. General case For almost all (β1, β2) ∈ P 2trans with β1β2 ≥ 0.25 the measure bβ1,β2 = bβ1,β2 × ` is a measure full dimension for ([−1, 1]2, fβ1,β2). But if β1β2 < 0.25 then the variational principle for Hausdorff dimension does not hold for the system ([−1, 1]2, fβ1,β2). Special case β1 = β2 For almost all β ∈ (0.5, 1) bβ = bβ × ` is a measure full dimension for ([−1, 1]2, fβ) The claim about the Fat Baker's transformation fβ in 7.3. is in fact just a simple consequence of Solomyak's theorem [SO1] and the work of Alexander and Yorke [AY]. Before we begin with the proofs we remark that we have number theoretical exceptions to our generic results in the symmetric situation β1 = β2. These results are formulated in chapter ten. Let us now go into the proofs. Proof of 7.1. General case: Fix p ∈ (0.5, 1). We first claim that for almost all (β1, β2) ∈ P 2trans and τ1, τ2 ∈ (0, 1) with τ1 + τ2 < 1 and log τ2 log p/β1 = log τ1 log(1− p)/β2 the identity dim bpβ1,β2 = 1 49 holds. If we are given (β1, β2) ∈ P 2trans and there exists τ1 + τ2 < 1 with log p/β1log τ1 = log(1−p)/β2 log τ2 =: d we have: (pβ1) p((1− p)β2)1−p = (β1β2τ d1 )p(β1β2τ d2 )1−p = β1β2τ dp1 τ d(1−p)2 < β1β2. Now we see that our claim follows from theorem 6.1.3. with the help of A3. Fix θ = (β1, β2, τ1, τ2) with the properties of our claim and let d be defined as above. We have: β1τ d 1 + β2τ d 2 = β1τ log p/β1 τ1 1 β2τ (1−p)β2 τ2 2 = p+ (1− p) = 1. From 4.1. we thus get: dimB Λθ = d+ 1 and dimB Λθ = d+ 2 Moreover from 5.3.3. we get: dimH b p θ = p log p+ (1− p) log(1− p) p log τ1 + (1− p) log τ2 + (1− p log β1 + (1− p) log β2 p log τ1 + (1− p) log τ2 ) = 1 + β1τ d 1 log β1τ d 1 + β2τ d 2 log β2τ d 2 − (β1τ d1 log β1 + β2τ d2 log β2) β1τ d1 log τ1 + β2τ d 2 log τ2 = 1 + β1τ d 1 log τ d 1 + β2τ d 2 log τ d 2 β1τ d1 log τ1 + β2τ d 2 log τ2 = d+ 1. Just by definition we have dimH b p θ ≤ dimH Λθ ≤ dimB Λθ. Thus we get dimH b p θ = dimH Λθ = dimB Λθ = d+ 1 and with the help of A5 dimH Λθ = dimB Λθ = d+ 2. Our first statement in the general situation is proved. Consider the special case p = 0.5. We get from 5.3.3. dim bθ = hb(σ) log 2 + d+ 1 = d+ 2. This means that bθ is a measure of full dimension. Now consider the opposite case p 6= 0.5. Assume that the variational principle for Hausdorff dimension holds for (Λθ, fθ). Then by 3.2.5. there is a sequence of measures μn ∈M(Σ, σ), such that dimH(μn)θ −→ d+ 2 50 Recall that the equal-weighted Bernoulli measure b is the unique measure inM(Σ, σ) which maximizes the metric entropy with hb(σ) = log 2 and that the metric entropy is upper-semi-continuous on M(Σ, σ). By this facts and 5.3.1. we necessarily have μn −→ b. From 5.3.2. we have the inequality dimH μθ ≤ 1 + hμ(σ) log 2 − hμ(σ) + Ξ μ β1,β2 Ξμτ1,τ2 for all μ ∈M(Σ, σ). With the help of upper semi-continuity of hμ(σ) we thus get: limn−→∞ dimH(μn)θ ≤ 2− log 2 + 0.5 log β1 + 0.5 log β2 0.5 log τ1 + 0.5 log τ2 . We have: − log 2 + 0.5 log β1 + 0.5 log β2 0.5 log τ1 + 0.5 log τ2 = −2 log 2 + log p− d log τ1 + log(1− p)− d log τ2 log τ1 + log τ2 = d− 2 log 2 + log p+ log(1− p) log τ1 + log τ2 < d, which implies limn−→∞ dimH(μn)θ < d + 2. This is a contradiction and the variational principle for Hausdorff dimension does not hold for (Λθ, fθ). Special case β1 = β2 = β: One proves the result by exactly the same arguments that we used in the general situation. The only difference is that one uses the theorem of Peres and Solomyak (6.1.2) for the symmetric self-similar measures instead of theorem 6.1.3. for the asymmetric ones. Special case τ1 = τ2 = τ : Setting p = 1 1+c in 6.2.2. we have for all c ∈ (0, 1] dimH b1/(1+c)β,cβ = 1 for almost all β ∈ [ 1 1+c , 0.649]. Using the theorem of Fubini we get from this dimH b p β1,β2 = 1 with p = β1 β1+β2 for almost all (β1, β2) ∈ P 2trans. Now from 4.1. and 5.3.3. the dimension formula for the (Λθ, Tθ) and with help A5 the dimension formula for Λθ follows. If β1 6= β2 our result about the variational principle can be proved by the same arguments that we used in the general situation if p 6= 0.5. 51 Special case β1 = β2 = β and τ1 = τ2 = τ : The statement is just an obvious consequence of 4.1., Solomyak's theorem 6.1.1. and the dimension formula in 5.3.3. . 2 Proof of 7.2. It follows directly from the general case in 7.1. that for all p ∈ (0, 1) there exists a set A(p) ⊆ P 2trans with `2(A(p)) = `2(P 2trans) such that for all (β1, β2) ∈ A(p) and all τ1, τ2 > 0 with τ1 + τ2 < 1 and log τ2 log p/β1 = log τ1 log(1 − p)/β2 our statement about the dimensions holds. Let G(τ1) be given by the following union: ⋃ p∈(0,1) {(β1, β2, τ2)|(β1, β2) ∈ A(p) , τ1+τ2 < 1 , log τ2 log p/β1 = log τ1 log(1−p)/β2}. It is easy to see that the union ⋃ p∈(0,1) {(β1, β2, τ2)|(β1, β2) ∈ P 2trans , τ1+τ2 < 1 , log τ2 log p/β1 = log τ1 log(1−p)/β2} equals the set {(β1, β2, τ2)|(β1, β2) ∈ P 2trans , τ1 + τ2 < 1}. By the theorem of Fubini we thus have `3(G(τ1)) = ` 3{(β1, β2, τ2)|(β1, β2) ∈ P 2trans , τ1 + τ2 < 1}. Now let G = ⋃ τ1∈(0,1) {(β1, β2, τ1, τ2)|(β1, β2, τ2) ∈ G(τ1)}. Note that we have G ⊆ P 4trans and `4(G) = `4(P 4trans). But by definition our dimension formulas hold for all θ ∈ G. This competes the proof. 2 Proof of 7.3. General case: First recall from 3.2.4. that bβ1,β2 = bβ1,β2 × ` is an ergodic measure for the system ([−1, 1]2, fβ1,β2). It follows directly from 6.1.3. that the measure bβ1,β2 is absolutely continuous for almost all (β1, β2) ∈ P 2trans with β1β2 ≥ 0.25. For these (β1, β2) the measure bβ1,β2 × ` is absolutely continuous as well and thus has dimension two (see A3). This proves our first statement. If β1β2 < 0.25 then we have hb(σ) < −Ξbβ1,β2 . By upper semi continuity of the metric entropy there is a weak∗ neighborhood U of b in M(Σ+, σ) such that 52 −hμ(σ)/Ξμβ1,β2 ≤ c1 < 1 holds for all μ ∈ U . Now note that we have from 5.3.5. and 6.3.1. dimH μβ1,β2 ≤ 1 + dimH(pr+μ)β1,β2 ≤ 1 + −hpr+μ(σ) Ξpr +μ β1,β2 . From these facts we get dimH μβ1,β2 ≤ c + 1 < 2 for all μ ∈ Ũ = (pr+)−1(U). Obviously Ũ is a neighborhood of b in M(Σ, σ). Furthermore we have by 5.3.4. dimH μβ1,β2 ≤ hμ(σ) log 2 + 1. Again by upper semi continuity of metric entropy it follows that dimH μβ1,β2 ≤ c2 + 1 < 2 for all μ ∈M(Σ, σ)\Ū . Putting these facts together we get: dimH μβ1,β2 ≤ max{c1, c2}+ 1 < 2 = dim[−1, 1]2 ∀μ ∈M(Σ, σ). This proves our second statement. Special case β1 = β2: Recall from 3.2.4. that bβ=bβ×` is an ergodic measure for the system ([−1, 1]2, fβ). From 6.1.1. we know that the measure bβ is absolutely continuous for almost all β ∈ (0.5, 1). For these β we know that bβ × ` is absolutely continuous and thus has dimension two (see A3). 2 53 8. Extension of some results to Markov chains This chapter forms a kind of supplement to last four chapters. We will extend some of our main general and generic results to invariant sets for the maps Tθ and fθ that correspond to special Markov chains. Let A = ( a1,1 a−1,1 a1,−1 a−1,−1 ) be a (2, 2)-matrix with entrys aij in {0, 1}. By ΣA (resp. Σ+A) we denote the subset of Σ (resp. Σ +) given by {(sk)|asksk+1 = 1}. These sets are obviously invariant under the shift map σ (resp. σ−1). The systems (Σ+A, σ) and (ΣA, σ−1) are called (1-step) Markov chains (see [KH]). Now we define subsets ΛθA and Λ θ A of [−1, 1]2 by ΛθA = πθ(Σ + A) and Λ θ A = πθ(ΣA) for θ ∈ P 4all. By 3.1.1. the set ΛθA is invariant under the map Tθ and by 3.1.3. the set ΛθA is invariant under the map fθ. If the matrix A is not in { ( 1 1 1 0 ) , ( 0 1 1 1 ) } then the sets ΣA (resp. Σ+A) and consequently the sets ΛAθ (resp. Λ A θ ) are at most countable. Dimensional theoretical properties are trivial in this case. By symmetry we may restrict our attention to the case A = ( 1 1 1 0 ) . Fix this matrix for the rest of this chapter. We remark that the dynamical system (Σ+A, σ) is known as goldenshift (see [SV]). For p ∈ (0, 1) define Markov measures on ΣA (resp. Σ+A) in the following way: Consider the stochastic matrix P = ( p1,1 p−1,1 p1,−1 p−1,−1 ) := ( p 1 1− p 0 ) and the stochastic vector (p1, p−1) = (1/(2 − p), (1 − p)/(2 − p)). Define a measure on the cylinder sets in ΣA (resp. Σ + A) by mp([t0, t1 . . . , tu]v) = pt0 u−1∏ i=0 ptiti+1 . Now extend this measure to a Borel probability measure mp on ΣA (resp. Σ + A). It is well known that mp is ergodic with respect to the shift map (see [DGS]). Define measures mpθ and m p θ by mpθ = m p ◦ π−1θ and mpθ = mp ◦ π−1θ . By 3.2.3. mpθ is an ergodic measure for the system (Λ A θ , Tθ) and by 3.2.5. m p is an ergodic measure for the systems (ΛAθ , fθ). 54 Our main result in this chapter is nothing but an extension of 4.1. and 7.1.2. to the invariant sets ΛAθ and Λθ. Let P 4A = {(β1, β2, τ1, τ2)|β1 + β1β2 ≥ 1, τ1 + τ2 < 1} and P 4A−trans = {(β1, β2, τ1, τ2) ∈ P 4A|β1 ≤ β2 ≤ 0.649}. Theorem 8.1.1. (1) For all θ = (β1, β2, τ1, τ2) ∈ P 4A we have dimB Λ A θ = d+ 1 and dimB Λ A θ = d+ 1 + log(( √ 5 + 1)/2) log 2 where d is unique positive number satisfying β1τ d 1 + β1β2(τ1τ2) d = 1. (2) For almost all θ = (β1, β2, τ1, τ2) ∈ P 4A−trans (in the sense of four dimensional Lebesgue measure) we have dimH m p θ = dimH Λ A θ = dimB Λ A θ and dimH Λ A θ = dimB Λ A θ where p = β1τ d 1 and d is as in (1). Remarks (1) The condition β1 + β1β2 ≥ 1 is necessary. It means that the projection of ΛθA onto the first component has positive length. This fact is essential for our proof (see also remark (3) in chapter four). (2) Note that we can write our dimension formula in the symmetric situation θ = (β, β, τ, τ) ∈ P 4A using the topological entropy htop(σ|ΣA): dimB Λ A θ = 1 + log β + htop(σ|ΣA) log τ and dimB Λ A θ = 1 + log β + htop(σ|ΣA) log τ + log β + htop(σ|ΣA) log 2 (3) Of course the reader will ask the question if there are generalizations of 8.1.1. to n-step Markov chains (see [KH] for definition). Let us first discuss the box-counting dimension. We were not able to prove an analogon of 8.1.1.(1) for all n-step Markov chains. But under certain assumption such a generalization is in fact possible using our methods. Let us discuss this in detail. We say that a Markov chain ΣMarkov 55 (resp. Σ+Markov) has block form if there is a set of finite sequences B={b1, . . . , bk}, where bi have entrys in {1,−1}, such that each element of ΣMarkov (resp. Σ+Markov) can be written as a sequence of elements in B. If ΣMarkov has block form and in addition prXπθ(ΣMarkov) has positive length we get dimB πθ(Σ + Markov) = d+ 1 and dimB πθ(ΣMarkov) = d+ 1 + htop(σ|ΣMarkov) log 2 . Here d is the solution of k∑ i=0 τ ]1(bi) 1 τ ]−1(bi) 2 (β ]1(bi) 1 β ]−1(bi) 2 ) x = 1 where ]1 counts the number of entrys that are 1 and ]−1 counts the number of entrys that are −1 in an element b of B and htop(σ|ΣMarkov) denotes the topological entropy of the Markov chain. The proof of this statement differs from the proof of 8.1.1.(1) only in technical respects; no new idea is needed. We have thus decided not to write down the proof of this assertion. We remark that some but not all Markov chains have block form. For instance the blocks (1) and (−1, 1, 1) induces a 2-step Markov chain. But the 2-step Markov chain which is is given by excluding only the block (2, 1, 2) does not have block form. (4) A generalization of 8.1.1.(2) fails because we need the transversality condition to treat the Hausdorff dimension (see chapter six). To see this again consider the Markov chain induced by the blocks (1) and (−1, 1, 1). The condition for overlapping projections is β1 + β 2 1β2 ≥ 1, which implies β1 ≥ 0.65 or β2 ≥ 0.65. This contradicts the transversality condition β1 ≤ β2 ≤ 0.649. Now we want to give a comprehensive proof of 8.1.1., only elaborating the details that are different from what was done in the last chapters. Proof of 8.1.1. Fix θ = (β1, β2, τ1, τ2) ∈ P 4A and the number d. 1. Step: Calculation of box-counting dimension Let τ3 = τ1τ2 and β3 = β1β2. Given r > 0 we define a set of finite sequences by Xr := {(s1, . . . , sk)|min{τ1, τ3}r ≤ τs1τs2 . . . τsk < r where sj ∈ {1, 3} ∀j = 1 . . . k}. 56 Let Cr = {πθ([κ(s1), . . . , κ(sk)]0)|(s1, . . . , sk) ∈ Xr} where κ(1) = 1 and κ(3) = (−1, 1). Since {[κ(s1), . . . , κ(sk)]0)|(s1, . . . , sk) ∈ Xr} is a cover of Σ+A we have that Cr is a cover of Λ A θ . An element of Cr is a rectangle parallel to the axis with x-length 2βs1βs2 . . . βsk and y-length 2τs1τs2 . . . τsk . We cover each of this rectangles by squares parallel to the axis of side length 2τs1τs2 . . . τsk . We choose the squares in a row such that they only intersect in their boundary. In this way we obtain a new cover Ĉr of Λ A θ , which consists of squares with length in (2min{τ1, τ3}r, 2r]. By exactly the same arguments we used in the proof of 4.1. we see that we have the following estimates for the number of elements Nr in the cover Ĉr: 1 ≤ rd+1N(r) ≤ 2min{τ1, τ3}−(d+1). Now we want to analyze the sets ΛAθ ∩ R where R is a rectangle in Cr. First note that the projection of ΛAθ onto the x-axis is given by the set I fulfilling the relation I = L1(I) ∪ ÃL1 ◦ L2(I) where L1(x) = β1x + (1 − β1) and L2(x) = β2x + (1 − β2). Using the fact that β1 + β1β2 ≥ 1 a direct calculation shows that I is the interval [β1β2−2β1+1 1−β1β2 , 1]. Let l be the length of this interval. We now see that `(prXΛ A θ ) = l. But this implies `(prX(R∩ΛAθ )) = l`(prX(R)). Thus the number of those squares in Ĉr that have nonempty intersection with Λ A θ is bigger than lN(r). One square with side length 2r parallel to the axis can not intersect more than 9min{τ1, τ3}−2 squares in Ĉr because the squares in Ĉr have side length bigger than 2min{τ1, τ3}r and intersect, if at all, only in the boundary. Thus if we have a cover of ΛAθ with square of side length 2r parallel to the axis, this cover has at least 1/9min{τ1, τ3}2lN(r) elements. Let N(r) be the minimal cardinality of an cover of ΛAθ with square of side length 2r parallel to the axis. Putting our estimates together we obtain 1/9min{τ1, τ3}2l ≤ 1/9min{τ1, τ3}2lN(r)rd+1 ≤ N(r)rd+1l ≤ min{τ1, τ3}−(d+1). This shows dimB Λ A θ = d + 1. It remains to deduce the dimension formula for Λ A θ . By the product structure of the map πθ we get: ΛAθ = {(x, y, z)|(x, z) ∈ ΛAθ and y ∈ F} where F = ι(pr−(ΣA)). Here ι is defined in 3.1. and pr − is the projection from Σ onto Σ−. Define ῑ by ῑ((sk)k∈IN0) = ∞∑ k=0 sk2 −k−1. It is easy to see that F = ῑ(Σ+A). But this set is well known in dimension theory and we get from [FU] dimH F = dimB F = htop(σ|Σ+A ) log 2 = log(( √ 5 + 1)/2) log 2 . 57 Using the definition of the box-counting dimension with δ-mesh cubes (see 3.1. of [FA]) we see dimB Λ A θ = dimB Λ A θ + dimB F . This gives us the dimension formula for ΛAθ . Now the first part of theorem 8.1.1. is proved. 2. Step: The dimension of the Markov measures mpθ If we assume Ξmpτ1,τ2 ≤ Ξ mp β1,β2 we get from the Ledrappier-Young formula (see 5.2.3. and 5.3.3.) the following formula for the Markov measures mpθ: dimmpθ = hmp(σ) −Ξmpτ1,τ2 + (1− Ξ mp β1,β2 Ξ mp τ1,τ2 ) dimmpβ1,β1 . Here the measure mpβ1,β2 is given by m p β1,β2 = mp ◦ π−1β1,β2 . Just by definition we have Ξ mp β1,β2 = 1 2− p log β1 + 1− p 2− p log β2 and Ξ mp τ1,τ2 = 1 2− p log τ1 + 1− p 2− p log τ2. Furthermore we know from 4.4. of [KH], hmp(σ) = −( p 2− p log p+ 1− p 2− p log(1− p)). This gives us the formula: dimmpθ = p log p+ (1− p) log(1− p) log τ1 + (1− p) log τ2 + (1− log β1 + (1− p) log β2 log τ1 + (1− p) log τ2 ) dimmpβ1,β1 . 3. Step: Absolute continuity of the measures mpβ1,β2 We claim that an analogon of proposition 6.2.1. holds for the Markov measures mpβ1,β2 on the real line: Claim: Let p ∈ (0, 1) and c ∈ (0, 1]. The measures mpβ,cβ are absolutely continuous with density in L2 for almost all β ∈ [0, 0.649] with p2/β + (1− p)2/(cβ2) ≤ 1. Using the arguments of the first four steps in the proof of 6.2.1. we see that it is enough if we show =(β0) = ∫ Σ+ ∫ Σ+ β −|s∧t|−1 0 c −]|s∧t|−1(s)dmp(t)dmp(s) <∞ for all β0 with p 2/β0+(1− p)2/(cβ20) ≤ 1 . Here all notations are the same as in the proof of 6.2.1. . We integrate: =(β0) = ∞∑ n=0 ∫ Σ+ ∞∑ n=0 γ−n−10 c −]n−1(s)mp({t ∈ Σ+| |s ∧ t| = n)}dmp(s) 58 ≤ max{p, 1− p} ∞∑ n=1 ∫ Σ+ ∞∑ n=0 γ−n−10 c −]n−1(s)mp([s0, . . . sn−1]0)dm p(s) = max{p, 1− p}β−10 ∞∑ n=1 ∑ s∈{−1,1}n β−n0 c −]n−1(s)mp([s0, . . . sn−1]0) 2. For t ∈ {1, (−1, 1)}n we denote by ]1(t) the number of entrys in t that are 1 and by ]−1,1(t) the number of entrys that are (−1, 1). With this notations we have =(β0) ≤ max{p, 1− p} 2− p β −1 0 ∞∑ n=1 ∑ t∈{1,(−1,1)}n γ −]1(t)−2]−1,1(t) 0 c −]−1,1(t)p2]1(t)(1− p)2]−1,1(t) = max{p, 1− p} 2− p β −1 0 ∞∑ n=1 ( p2 β0 )]−1(t)( (1− p)2 cβ20 )]−1,1(t) = max{p, 1− p} 2− p β −1 0 ∞∑ n=1 ( p2 β0 + (1− p)2 β20c )n. Now we see that our claim holds. 4. Step: Conclusion of the proof We will prove the following statement: Claim: For all p ∈ (0, 1) and almost all (β1, β2) ∈ {(β1, β2) ∈ P 2trans|β1 + β1β2 ≤ 1} and all τ1, τ2 with τ1 + τ2 < 1 and log(p/β1) log τ1 = log((1−p)/(β1β2)) log(τ1τ2) we have dimH m p θ = dimH Λ A θ = dimB Λ A θ and dimH Λ A θ = dimB Λ A θ where θ = (β1, β2, τ1, τ2). Using the argumentations in the proof of 7.1.2. this claim implies the second part of our theorem. Therefore we now gone prove this claim: With the help of Fubini's theorem we get from our claim in the third step of this proof: For all p ∈ (0, 1) and almost all (β1, β2) ∈ P 2trans with p2/β1 + (1 − p)2/(β1β2) ≤ 1 the identity dimmpβ1,β2 = 1 holds . If we are given p ∈ (0, 1) and (β1, β2) ∈ {(β1, β2) ∈ P 2trans|β1 + β1β2 ≤ 1} and τ1, τ2 with τ1 + τ2 < 1 and log p/β1 log τ1 = log(1−p)/(β1β2) log(τ1τ2) =: d then we have p = β1τ d 1 , (1− p) = β1β2(τ1τ2)d and p2β1 + (1− p)2/(β1β2) = β1τ 2d1 + β1β2(τ1τ2)2d < 1. 59 Hence for all p ∈ (0, 1) and almost all (β1, β2) ∈ {(β1, β2) ∈ P 2trans|β1 + β1β2 ≤ 1} and all τ1, τ2 with τ1 + τ2 < 1 and log p/β1 log τ1 = log(1−p)/(β1β2) log(τ1τ2) we have dimmpβ1,β2 = 1 and with the help of the dimension formula from the second step of our proof: dimmpθ = p log p+ (1− p) log(1− p) log τ1 + (1− p) log τ2 + (1− log β1 + (1− p) log β2 log τ1 + (1− p) log τ2 ) = 1+ β1τ d 1 log β1β2(τ1τ2) d + β1β2(τ1τ2) d log(β1β2(τ1τ2) d − log β1 − β1β2(τ1τ2)d log β2 log τ1 + β1β2(τ1τ2)d log τ2 = 1 + log τ d1 + β1β2(τ1τ2) d log τ d2 log τ1 + β1β2(τ1τ2)d log τ2 = d+ 1. We know from the first step of the proof that dimB Λ A θ = d+ 1. Hence we get: dimmpθ = dimH Λ A θ = dimB Λ A θ . Moreover we know from the first step of the proof that dimB Λ A θ = dimB Λ A θ+dimB F . On the other hand by the product formula of Falconer [FA1, 7.4.] we have dimH Λ A θ = dimH Λ A θ + dimH F . Thus dimH Λ A θ = dimB Λ A θ holds. This completes the proof of our claim. 2 60 9. Erdös measures 9.1. Singularity In this chapter we usually assume that β ∈ (0, 5, 1) is the reciprocal PisotVijayarghavan number (short: PV number). We refer to appendix B for definition, examples and properties of these algebraic integers. Erdös [ER1] showed in 1939 that the equal-weighted infinitely convolved Bernoulli measure bβ is singular if β ∈ (0, 5, 1) is the reciprocal PV number. Furthermore it follows from a work of Alexander and Yorke [AY] that dimH bβ < 1 in case that β is the reciprocal PV number. This was observed by Przytycki and Urbanski [PU] who also gave their own proof of this fact. It is not known (and perhaps a difficult problem) whether there are other parameters β than reciprocals of PV numbers such that infinitely convolved measures Bernoulli bβ measure is singular. Some information how big the set of exceptions can be maximal follows from a very resent result of Peres and Schlag [PSch]. They have shown the relation dimH{β ∈ (0.5, 1)| dimH bβ < 1} < 1 for infinitely convolved measures Bernoulli bβ. Our objects here are all symmetric overlapping measures μβ where β ∈ (0, 5, 1) is the reciprocal PV number and μ ∈ M(Σ+, σ). We call such a measure μβ an Erdös measures. Our main theorem extends the results mentioned above: Theorem 9.1.1. Let β ∈ (0, 5, 1) be the reciprocal PV number and μ ∈ M(Σ+, σ). dimH μβ < 1 holds if and only if μβ is singular. Moreover the set {μ ∈ M(Σ+, σ)|μβ is singular} is open in the weak∗ topology on M(Σ+, σ) and contains all Bernoulli measures. This theorem will follow from several different propositions we prove in this chapter (see the end of 9.3.). In view of 7.1. it is natural to ask whether there are measures μ ∈ M(Σ+, σ) with dimH μβ = 1 at all. We will answer this question in 9.4. . We remark that the technique, we will develop here, can not be extended to the asymmetric measures μβ1,β2 . We will see where the problems come from. We first state the generalization of Erdös result to all infinitely convolved Bernoulli measures. The proof we give is nothing but an obvious extension of Erdös original argument. 61 Proposition 9.1.2. If β ∈ (0, 5, 1) is the reciprocal PV number then the measures bpβ are singular for all p ∈ (0, 1). Proof By [JW] we know that the Fourier transformation of a convolution is the product of the Fourier transformation of the convolved measures. Consequently by 3.2.1. the Fourier transformation φ of bpγ is given by: φ(bpβ, ω) = ∞∏ n=0 (cos((1− β)βnω) + (2p− 1) sin((1− β)βnω)). We see that: |φ(bpβ, ω)| = ∞∏ n=0 |(cos((1−β)βnω)+(2p−1) sin((1−β)βnω))| ≥ ∞∏ n=0 | cos((1−β)βnω)|. Now let ωk = 2πβ −k/(1− β). We have: |φ(bpβ, ωk)| ≥ ∞∏ n=0 | cos(2πβn−k)| = k∏ n=0 | cos(2πβn−k)| ∞∏ n=k+1 | cos(2πβn−k)| = C k∏ n=0 | cos(2πβ−n)| where C is a constant independent of k and not zero. Now let β be the reciprocal of a PV number. From proposition B1 of appendix B we know that there is a constant 0 < θ < 1 such that: ||β−n||ZZ ≤ θn ∀n ≥ 0 where ||.||ZZ denotes the distance to the nearest integer. This implies |φ(bpβ, ωk)| ≥ Ĉ > 0 for all k > 0. Thus we have that |φ(bpβ, ω)| does not tend to zero with ω −→∞. Hence by Riemann-Lebesgue lemma bpβ can not be absolutely continuous if β is the reciprocal of a PV number. But it follows from the theory of infinity convolutions developed by Jessen and Winter [JW] that bpβ is either absolutely continuous or singular. This completes the proof. 2 We remark that we have no nice product formula for the Fourier transformation of the bp∗β1,β2 in the asymmetric situation. In fact we have: φ(bp∗β1,β2 , ω) = limv−→∞ ∑ s∈{−1,1}v pv−]v(s)(1− p)]v(s) v∏ n=1 esnωβ n−]n(s) 1 β ]n(s) 2 . and the Fourier transformation of bpβ1,β2 is just this function scaled on [−1, 1]. We have not been able to apply the idea used in the proof of 9.1.2. to this Fourier transformation and have thus not been able to find β1 6= β2 with β1 + β2 ≥ 1 such that bpβ1,β2 is singular. 62 9.2. Garcia entropy Garcia [G1/G2] introduced a kind of entropy related to the equal weighted infinitely convolved Bernoulli measures. We will generalize his account here. In contrast to Garsia we will work on the hole shift space Σ+ and consider all measures μβ for μ ∈M(Σ+, σ). Let ∼n,β be the equivalence relation on Σ+ given by i ∼n,β j ⇔ n−1∑ k=0 ikβ k = n−1∑ k=0 jkβ k and define a partition Πn,β of Σ + by Πn,β = Σ +/ ∼n,β. Recall that entropy of a partition Π with respect to a Borel probability measure μ on Σ+ is Hμ(Π) = − ∑ P∈Π μ(P ) log μ(P ). We denote the join of two partitions Π1 and Π2 by Π1 ∨ Π2. This is the partition consisting of all sections A ∩B for A ∈ Π1 and B ∈ Π2. The following lemma is easy to proof but essential for us. Lemma 9.2.1. The partition Πn,β ∨ σ−n(Πm,β) is finer than the partition Πn+m,β and the sequence Hμ(Πn,β) is sub-additive for a shift invariant measure μ on Σ +. Proof We have that σ−n(Πm,β) = Σ +/ 3n,m where 3n,m is given by i 3n,m j ⇔ n+m−1∑ k=n ikβ k = n+m−1∑ k=n jkβ k. and hence Πn,β ∨ σ−n(Πm,β) = Σ+/ ≈n,m where i ≈n,m j ⇔ n−1∑ k=0 ikβ k = n−1∑ k=0 jkβ k and n+m−1∑ k=n ikβ k = n+m−1∑ k=n jkβ k. Now obviously i ≈n,m j ⇒ i ∼n+m,β j. Thus Πn,β ∨ σ−n(Πm,β) is finer than the partition Πn+m,β. Now let μ ∈ M(Σ+, σ). By well known properties of Hμ (see 10.13. of [DGS]) we have: Hμ(Πn+m,β) ≤ Hμ(Πn,β ∨ σ−n(Πm,β)) ≤ Hμ(Πn,β) +Hμ(σ−n(Πm,β)) = Hμ(Πn,β) +Hμ(Πm,β) . 63 We can now define theGarsia entropy Gβ(μ) for a shift invariant Borel probability measure μ on Σ+: Gβ(μ) := lim n−→∞ Hμ(Πn,β) n = inf n Hμ(Πn,β) n . The limit in the definition exists and equals the infimum because the sequence Hμ(Πn,β) is sub-additive. If we have a σ invariant measure μ on the full shift space Σ = {−1, 1}ZZ , we define the Garsia entropy of μ as Gβ(μ) := Gβ(pr+μ), where pr+ is the projection of Σ onto Σ+. Gβ(pr+μ) is well defined because pr+μ is σ invariant if μ is σ invariant. Let us think a moment about the asymmetric case. We could define partitions of Σ+ in the same way as in the symmetric case. But an analogon of Lemma 9.2.1. does not hold and the limit in the definition of the Garsia entropy does not have to exist. The Garsia entropy Gβ(μ) is less equal to the usual metric entropy hμ(σ) for a σ invariant measure μ, since the partition of Σ+ into cylinder sets of length n is finer than Πn,β. If β is not the solution of an algebraic equation with coefficients in {−1, 0, 1}, these partitions are identical and Gβ(μ) = hμ(σ) holds. Also the partitions Πn,β are not in general generated by a transformation we can show that the Garsia entropy G, interpreted as a function on the space of σ invariant Borel probability measures on Σ+, has typical properties of an entropy. Proposition 9.2.2. The function μ 7−→ Gβ(μ) is upper-semi-continuous and affine on the space of σ invariant Borel probability measures on Σ+. Proof We first prove that the function is upper-semi-continuous. Let μ, μn be σ invariant Borel probability measures with μn → μ. Fix ε > 0. From the definition of the Garsia entropy we know there exists an k such that, Gβ(μ) ≥ Hμ(Πk,β) k − ε 2 . The elements of the partition Πk,β are finite unions of cylinder sets in Σ + and hence open and closed. Thus we know that, limn−→∞ μn(P ) = μ(P ) ∀P ∈ Πk,β. Hence there exists an n0 such that for all n ≥ n0 1 k |Hμ(Πk,β)−Hμn(Πk,β)| ≤ ε 2 . 64 Using both inequalitys and 9.2.1. we have: Gβ(μ) ≥ 1 k Hμn(Πk,β)− ε ≥ Gβ(μn)− ε. This proves upper-semi-continuity. Now let μ1, μ2 be σ invariant and μ = pμ1 + (1 − p)μ2 with p ∈ (0, 1). For all partitions Π the inequality 0 ≤ −pHμ1(Π)− (1− p)Hμ2(Π) +Hμ(Π) ≤ log 2 holds (see proposition 10.13. of [DGS]). Thus by the definition of the Garsia entropy we have Gβ(μ) = pGβ(μ1) + (1− p)Gβ(μ2). But this means that G is affine. 2 The next proposition shows the significants of the Garsia entropy in our discussion. Proposition 9.2.3. Let β ∈ (0.5, 1) be the reciprocal of a PV number and μ be a shift invariant Borel probability measure on Σ+. We have Gβ(μ) ≤ log β−1. Moreover if μβ is singular then Gβ(μ) < log β −1 holds. Garsia sketched in [GA1] a proof of the inequality Gβ(b) < log β −1 using a slightly different notion of Garsia entropy. We will adopt some of his ideas in our proof. Proof Fix β. Define πn from Σ + to [−1, 1] by πn((sk)) = ∑n−1 k=0 sk(1 − β)βk and let μn = μ◦π−1n . Let ](n) be the number of distinct points of the form ∑n−1 k=0 ±(1−β)βk and ω(n) be the minimal distance between two of those points. Furthermore denote the points by xni i = 1 . . . ](n) and let m n i be the μ measure of the corresponding elements in Πn,β, which means m n i = μn(x n i ). We first state a property of PV numbers we will have to use here, see proposition B2 of appendix B: β−1 is PV number ⇒ ∃ c : ω(n) ≥ cβn. Since (](n)− 1)ω(n) ≤ 2 we get ](n) ≤ 4ω(n)−1 ≤ cβ−n with c := 4c−1. From this we have that Hμ(Πn,β) ≤ log ](n) ≤ log c + n log β−1 and hence Gβ(μ) ≤ log β−1. 65 Now we assume that μβ is singular. It follows that there exists a constant C such that: ∀ε > 0 ∃ disjoint intervals (a1, b1), . . . , (au, bu) with u∑ l=1 (bl − al) < ε and μβ(O) > C where O := u⋃ l=0 (al, bl). With out loss of generality we may assume μβ(al) = μβ(bl) = 0 for l = 1 . . . u. It is obvious that the discreet distribution μn converges weakly to μβ. Thus we have: ∃n1(ε) ∀n > n1(ε) : μn(O) > C. We now expand the intervals a little bit, so that their length is a multiple of ω(n). kl,n := max{k | kω(n) ≤ al} al,n := kl,nω(n) kl,n := min{k | bl ≤ kω(n)} bl,n := kl,nω(n) Since ω(n) −→ 0 we have: ∃n2(ε) > n1(ε) ∀n > n2(ε) : (al,n, bl,n) disjunct for l = 1 . . . u and u∑ l=1 (bl,n − al,n) < ε and μn(Ō) > C where Ō = u⋃ l=0 (al,n, bl,n). Let ](n) be the number of distinct points xni in Ō. Since in one interval (al,n, bl,n) there are at most kl,n−kl,n points xni we have ω(n)](n) ≤ ε and hence ](n) ≤ εcβ−n. For all n > n2(ε) we can now estimate: Hμ(Πn,β) = − ](n) ∑ i=1 mni logm n i = − ∑ xni ∈Ō mni logm n i − ∑ xni 6∈Ō mni logm n i ≤ μn(Ō) log ](n) μn(Ō) + (1− μn(Ō)) log ](n)− ](n) 1− μn(Ō) ≤ μn(Ō) log ](n) + (1− μn(Ō)) log ](n) + log 2 ≤ μn(Ō) log εcβ−n + (1− μn(Ō)) log cβ−n + log 2 ≤ n log β−1 + C log ε+ log c+ log 2. If ε is small enough we have Hμ(Πn,β)/n < log β −1 for all n ≥ n2(ε). With the sub-additivity of Hμ(Πn,β) the desired result follows. 2 In the special case that β is the golden ratio there exists an explicit formula for the Garsia entropy of the equal-weighted Bernoulli measure found by Alexander and Zagier: 66 Theorem 9.2.4. [AZ] G√5−1 2 (b) = log 2− 1 18 ∞∑ n=1 kn 4n ≈ (log 2√ 5− 1) * 0.995714 . . . where kn = ∑ a1,...,at∈IN a1+...+at=n p/q=〈a1,...,at〉 (p+ q) log(p+ q) and 〈a1, . . . , at〉 denotes the continuous fraction. This theorem was also proved by Sidorov and Vershik [SV]. One needs quit delicate combinatorial considerations to find this formula and it seems to be very hard to prove a formula for other reciprocals of PV numbers. We do not want to include the prove of 9.2.4. here but we like to present a very nice interpretation of the Garsia entropy of Bernoulli measures in the light of the articles [AZ] and [SV]. If β is the reciprocal of a PV number we define an infinite binary graph associated with β. We label the edges of the graph with −1 each left and +1 each right. The vertices at the n'th level of the graph are supposed to correspond to the points x of the form x = ∑n−1 k=0 skβ k with sk ∈ {−1, 1} and paths are the sequences (s0, . . . sn−1) treated as the representations of these points. If β = √ 5−1 2 this graph is called the Fibonacci graph. Now we may think of a random walk on such a graph where we go left with probability (1−p) and right with the probability p. The probability to reach a vertex x at the n-level of the graph is in our notation just bp(P ) where P is the element of the partition Πn,β corresponding to x. The entropy of the random walk, we described, is the Garsia entropy Gβ(b p). What has been done in [AZ] and [SV] is (in some sense) to count the number of paths in the Fibonacci graph that reach a vertex x at the n'th level of the graph. This allows to calculate b(P ) for P ∈ Π n, √ 5−1 2 and hence the Garsia entropy G√5−1 2 (b) resp. the entropy of the random walk with transition probabilities (1/2, 1/2) on the Fibonacci graph. We remark that this approach is not strong enough to calculate G√5−1 2 (bp) if p 6= 0.5. One would have to know not only how many paths reach a vertex, but also how many of them have a given number of steps, say, to the right. It seems awkward to count these quantities. 67 Figure 5: The Fibonacci graph We end these section with a conjecture about Gβ(b p) as a function in p. Conjecture 9.2.5. Gβ(b p) is a continuous unimodal function in p with maximum at p = 0.5. In the case that β = √ 5−1 2 we have some (vague) numerical evidence for this conjecture. Moreover we think that it gets intuitive plausible, if we look at symmetries of the Fibonacci graph. 9.3. An upper bound on the dimension of Erdös measures We will here prove an upper bound on Hausdorff dimension of all measures μβ in terms of the Garsia entropy. In view of our result about the Garsia entropy in 9.2.3., if β is the reciprocal of a PV number, this bound is of special interest if we consider Erdös measures. 68 Because we will operate with Rényi dimension dimR (see appendix A) we are interested in an upper bound on the quantity hμ(ε) = inf{Hμ(Π)|Π a partition with diamΠ ≤ ε} by the entropy of the partitions Πn,β of Σ +. The following lemma plays the crucial role in our argumentation. Lemma 9.3.1. hμβ(2β n) ≤ Hμ(Πn,β) Proof Fix β ∈ (0.5, 1), τ ∈ (0, 0.5), a measure μ on Σ+ and n ∈ IN . We define a partition of Λβ,τ by ℘n = πβ,τ (Πn,β). By definition we have Hμ(Πn,β) = Hμβ,τ (℘n). We should say something about the structure of ℘n. The image of a cylinder set [i0, . . . , in−1]0 in Σ + under πβ,τ is the part of Λβ,τ lying in the rectangle Tin−1 ◦ . . . ◦ Ti0(Q) of x-length 2β n. It is not difficult to check that two cylinder sets lie in the same element of Πn,β if and only if the corresponding rectangles lie above each other. So the projection of an element in ℘n onto the x-axis has length 2β n. The projection onto the x-axis of two elements in ℘n may overlap. Starting with ℘n, we want to construct inductively a partition ℘n of Λβ,τ with non-overlapping projections, in a way that does neither increase length of the projections nor entropy. LetN(℘) be the number of pairs of elements in a partition ℘ that do have overlapping projections onto the x-axis. We now construct a finite sequence ℘kn of partitions. First let ℘0n = ℘n. Now let ℘ k n be constructed and N(℘ k n) > 0. Let P1 and P2 be two elements of ℘ k n with overlapping projections. Without loss of generality we may assume μβ,τ (P1) ≥ μβ,τ (P2) and define: P1 = P1 ∪ (P2 ∩ (prXP1 × [−1, 1])) P2 = P2\(prXP1 × [−1, 1]). We have P1∪P2 = P1∪P2, P1 ⊆ P1 and P2 ⊆ P2. Thus we know: μβ,τ (P1) + μβ,τ (P2) = μβ,τ (P1) + μβ,τ (P2) and μβ,τ (P1) ≥ μβ,τ (P1) ≥ μβ,τ (P2) ≥ μβ,τ (P2). Since the function −x log x is concave, this implies: −(μβ,τ (P1) log μβ,τ (P1) + μβ,τ (P2) log μβ,τ (P2)) ≤ −(μβ,τ (P1) log μβ,τ (P1) + μβ,τ (P2) log μβ,τ (P2)). 69 Hence if we substitute P1, P2 for P1, P2, we get a partition ℘ k+1 n of Λβ,τ with nonincreased entropy. From the definition of P1 and P2 we see that prXP1 = prXP1, prXP2 ⊆ prXP2 and that the projections of P1 and P2 onto the x-axis do not overlap. So the length of the projections are obviously not increased. Furthermore we observe that there cannot be any new overlaps of the projections of P1 or P2 with the projections of other elements in ℘kn, that do not appear, when we consider P1 or P2. Hence N(℘k+1n ) < N(℘ k n). So after a finite number of steps we get a partition ℘n with Hμβ,τ (℘n) ≥ Hμβ,τ (℘n), non-overlapping projections onto the x-axis and diam prX℘n ≤ 2βn. prX℘n is a partition of the interval [−1, 1] and we have Hμβ(prX ℘n) = Hμβ,τ (℘n), since the measure μβ is the projection of μβ,τ onto the x-axis. Now the proof is complete: hμβ(2β n) ≤ Hμβ(prX℘n) = Hμβ,τ (℘n) ≤ Hμβ,τ (℘n) = Hμ(Πn,β). 2 The idea of cutting up overlaps we used here appeared in an other form in the work of Alexander and Yorke [AY]. From our lemma it is easy for us to deduce the following proposition: Proposition 9.3.2. If μ is a shift ergodic Borel probability measure on Σ+ we have: dimH μβ ≤ Gβ(μ)/ log β−1. Proof First we estimate the Rényi dimension: dimRμβ = limε−→∞ hμβ(ε) log ε−1 = limn−→∞ hμβ(2β n) log 0.5β−n = limn−→∞ hμβ(2β n) n log β−1 ≤ lim n−→∞ Hμ(Πn,β) n log β−1 = Gβ(μ) log β−1 . Using part (2) of theorem A2 from appendix A we get: 70 ∀δ > 0 ∃X with μβ(X) > 0 and d(x, μβ) ≤ Gβ(μ)/ log β−1 + δ ∀x ∈ X. But the measure μβ is exact dimensional, because it is the transversal measure in the context of the ergodic dynamical system (Λβ,β,τ,τ , Tβ,β,τ,τ , μβ,β,τ,τ ). This fact was observed by Ledrappier and Porzio, see [LP]. So our estimate must hold μβ-almost everywhere and by part (3) of theorem A2 we get dimH μβ ≤ Gβ(μ)/ log β−1 + δ for all δ > 0. This proves the proposition. 2 Let us here again remark that Alexander and Yorke [AY] have proved the identity dimR bβ = Gβ(b)/ log β −1 for the equal-weighted infinitely convolved Bernoulli measure bβ. In their proof they used the self-similarity of this measure. Our proof of 9.3.2. shows that appealing to self-similarity is not necessary for the upper bound. At the beginning of this section we have formulated our main result in theorem 9.1.1. . We are now able to give the proof of this result. Proof of 9.1.1. Let β ∈ (0, 5, 1) be the reciprocal of a PV number. We have for all μ ∈M(Σ+, σ): μβ is singular⇒9.2.3. Gβ(μ) < log β−1 ⇒9.3.2. dimH μβ < 1⇒ μβ is singular. These implications prove the first statement of our theorem. Now choose an Erdös measure ξβ with dim ξβ < 1. We have Gβ(ξ) < log β −1. By upper-semi-continuity of G (9.2.2.) Gβ(μ) < log β −1 and hence dimμβ < 1 holds for all μ in a hole weak∗ neighborhood of ξ in M(Σ+, σ). Thus the set {μ ∈ M(Σ, σ)|μβ is singular} is open in the weak∗ topology on M(Σ, σ). The set contains the Bernoulli measure by proposition 9.1.2. . 2 9.4. Construction of an Erdös measure with full dimension Let β−1 be a PV number as usual. We will construct here a measure m ∈M(Σ+, σ) (depending on β) such that the Erdös measure mβ has Hausdorff dimension one. From the proof of our main theorem 9.1.1. we know that it is sufficient to find a m ∈ M(Σ+, σ) of full Garsia entropy, which means Gβ(m) = log β−1. In ergodic theory there is a quite natural construction of an invariant measure with maximal metric entropy, see 4.5. of [KH] or 18 of [DGS]. We will use a similar construction for the Garsia entropy. 71 Proposition 9.4.1. Let β ∈ (0.5, 1) be the reciprocal of PV number. There exists a measure m ∈ M(Σ+, σ), such that Gβ(m) = log β −1 and hence dimH mβ = 1. Proof In this prove we will omit the subscript β. We first construct a shift invariant measure m with Gβ(m) = log β −1 and afterwards prove the existence of an ergodic one. Recall that ]β(n) denotes the number of elements of the partition Πβ,n. Now choose measures mn ∈M(Σ+) such that mn(P ) = 1/]β(n) ∀P ∈ Πβ,n and let m be a weak∗ accumulation point of the sequence mn = 1 n n−1∑ i=0 mn ◦ σ−i. By this construction we immediately have that m is invariant under σ. Given two partitions ℘1 and ℘2 on Σ + we write ℘1 1 ℘2 if ℘2 is finer than ℘1. Note that ℘1 1 ℘2 ⇒ σ−k(℘1) 1 σ−k(℘2) and σ−k(℘1 ∨ ℘2) 1 σ−k(℘1) ∨ σ−k(℘2) where ∨ denotes the join as usual. Recall that we know from lemma 9.2.1. Πn+m 1 Πn ∨ σ−n(Πm). From these facts we get by induction Πaq 1 ∨a−1 i=0 σ −iq(Pq). Let bxc be the integer part of x. Given n and q and k with 0 < q < n and 0 ≤ k < q we set a(k) = b(n− k)/qc and write n− k in the form a(k)q+ r with 0 ≤ r < q. We get Σn 1 σ−k(Σn−k) ∨ Σk 1 σ−k(Σa(k)q) ∨ σ−(a(k)q+k)(Σr) ∨ Σk 1 a(k)−1 ∨ i=0 σ−iq+r(Σq) ∨ σ−(a(k)q+k)(Σr) ∨ Σk and hence Hmn(Σn) ≤ a(k)−1 ∑ i=0 Hmn(σ −iq+k(Σq)) +Hmn(σ −(a(k)q+k)(Σr)) +Hmn(Σk) ≤ a(k)−1 ∑ i=0 Hmn(σ −iq+k(Σq)) + 2q log 2. The last inequality follows from the fact, that the partitions Σq and Σr have less than 2q elements. Now summing over k gives qHmn(Σn) ≤ q−1 ∑ k=0 a(k)−1 ∑ i=0 Hmn(σ −iq+k(Σq)) + 2q 2 log 2 ≤ nHmn(Σq) + 2q2 log 2. 72 This implies Hmn(Σn) n ≤ Hmn(Σq) q + 2 log 2 q n . By the definition of the measures mn we have Hmn(Σn) = log ](n) and from proposition B2 of appendix B we know ](n) ≥ Cβ−n. This gives us: log β−1 + log C n ≤ Hmn(Σq) q + 2 log 2 q n . By the definition of m we thus have log β−1 ≤ Hm(Σq)/q, which implies log β−1 ≤ G(m). The opposite inequality has been proved in 9.2.3. . We have shown up to this point that the set M := {μ|μ σ-invariant and G(μ) = log β−1} of Borel measures on Σ+ is not empty. We know from 9.2.2. that G is upper semi-continuous and affine, which implies that M is compact and convex with respect to the weak∗ topology. By Krein-Milman theorem there exist an extremal point m of M . We show that m is extremal in the space of σ invariant Borel probability measure and hence σ ergodic. If this is not the case we have m = pμ1 + (1 − p)μ2 for two distinct σ invariant measures μ1 and μ2 and p ∈ (0, 1). Since μ is extremal in M we have that μ1 or μ2 is not in M . From 9.2.3. it follows that G(μ1) < log β −1 or G(μ2) < log β −1. This implies G(m) < log β−1 because G is affine. This is a contradiction to m ∈M . 2 We remark here that we do not know if the measure of full dimension in the last proposition is unique. The construction describe is not unique. Obviously one can choose the measures mn in different ways. But it is not clear, if this induces different Erdös measures of full dimension. 73 10. Number theoretical peculiarities 10.1. Ergodic Measures Now we study the ergodic measures for the systems (Λθ, Tθ), (Λθ, fθ) and ([−1, 1]2, fβ) in the case that θ = (β, β, τ, τ) ∈ P 4all and β a reciprocal of a PV number. We concentrate on the variational principle for Hausdorff dimension. Theorem 10.1.1. If β is the reciprocal for a PV number we have: (1) The variational principle for Hausdorff dimension does not hold for the Fat Baker's transformation ([−1, 1]2, fβ). (2) The variational principle for Hausdorff dimension does not hold for the attractors (Λθ, fθ) where θ = (β, β, τ, τ) and τ is sufficient small. (3) For the repellers (Λθ, Tθ) with θ = (β, β, τ, τ) and τ sufficient small Bernoulli measures do not have full Hausdorff dimension. Remark This theorem compared with our results in chapter seven shows that the dimensional theoretical properties of a dynamical systems can considerably change because of number theoretical peculiarities. Particular looking at the attractors (Λθ, fθ) for τ small and on the systems ([−1, 1]2, fβ) we see that in situations, where the variational principle for Hausdorff dimension generically holds, it does not have to hold generally because of such peculiarities. Looking at the repellers (Λθ, Tθ) for τ small, we see that generically a Bernoulli measure of full dimension is available but if the parameters have special number theoretical properties then such a measure does not exist. This provides substantial difficulties. We can not decide with our technique whether there exists a measure full dimension for (Λθ, Tθ) if β is the reciprocal of a PV number . Now we want to proof theorem 10.1.1. Proof of 10.1.1. (1) Let μ ∈M(Σ, σ). By 5.3.5. and 9.3.2. we have: dimH μβ ≤ 1 + dimH(pr+μ)β ≤ 1 +Gβ(pr+μ)/ log β−1. By 9.2.2. and 9.2.3. Gβ(pr +μ)/ log β−1 ≤ c1 < 1 holds for all μ in hole weak∗ neighborhood U of b in M(Σ, σ). Hence dimH μβ ≤ c1 + 1 < 2 holds for all μ in U . 74 On the other hand we have by the properties of the metric entropy hμ(σ)/ log 2 ≤ c2 < 1 on the complement of U . With 5.3.4. it follows that dimH μβ ≤ c2 + 1 < 2 holds for all μ ∈M(Σ+, σ)\U . Putting these facts together we obtain: dimH μβ ≤ max{c1, c2}+ 1 < 2 = dim[−1, 1]2 ∀μ ∈M(Σ, σ). But by 3.2.4. all ergodic measures for the system ([−1, 1]2, fβ) are of the form μβ for some μ ∈M(Σ, σ). So the proof is complete. (2) Let μ ∈M(Σ, σ). By 5.3.2. we have: dim μθ ≤ hμ(σ) log 2 + hμ(σ) log τ−1 + (1− log β log τ ) dimH(pr +μ)β. This implies dim μθ ≤ 1 + hμ(σ) log 2 + log 2β log τ−1 and combined with 9.3.2. dim μθ ≤ 1 + Gβ(pr +μ) log 2 + log 2β log τ−1 . By the same arguments we used in (1) we now see dimH μβ ≤ max{c1, c2}+ 1 + log 2β log τ−1 ∀μ ∈M(Σ, σ) where the constants c1, c2 are the same as in (1). If τ is sufficient small we get dimH μβ ≤ c < 2 ≤ dimH Λθ for all μ ∈ M(Σ, σ). But by 3.2.5. all ergodic measures for the system (Λθ, fθ) are of the form μβ for some μ ∈M(Σ, σ). This completes the proof. (3) From 5.3.3. we know dimH b p θ = −p log p− (1− p) log(1− p) log τ−1 + (1− log β log τ ) dimH b p β ∀p ∈ (0, 1). From 9.1.1. we have dimH b p β ≤ c < 1 for all p ∈ (0, 1). This implies dimH b p θ ≤ c+ log 2 + c log β log τ−1 ∀p ∈ (0, 1). If τ is sufficient small, we get our result: dimH b p θ ≤ c < 1 ≤ dimH Λθ ∀p ∈ (0, 1). 2 75 10.2. Invariant Sets We prove here upper bounds on the Hausdorff dimension of the repellers Λθ and attractors Λθ in the symmetric situation θ = (β, β, τ, τ) ∈ P 4all under the assumption, that β−1 is a PV number. An important consequence of this upper bound is the following theorem: Theorem 10.2.1. If β is the reciprocal for a PV number, τ ∈ (0, 0.5) and θ = (β, β, τ, τ) we have: dimH Λθ < dimB Λθ and dimH Λθ < dimB Λθ. Remark If we compare this result with 7.1. we learn that dimensional theoretical properties of invariant sets of a dynamical system can considerably change because of number theoretical peculiarities of parameter values. For our classes of attractors and repellers we generically have the identity for Hausdorff dimension and box-counting dimension, but for parameter values with special number theoretical properties this identity does not hold. To get 10.2.1. we prove now explicit upper bounds on the Hausdorff dimension of Λθ. Proposition 10.2.2. If β is the reciprocal for a PV number, τ ∈ (0, 0.5) and θ = (β, β, τ, τ) we have: dimH Λθ ≤ log( ∑ P∈Πn,β(]P ) log β log τ ) n log β−1 ∀n ≥ 1 where Πn,β is the partition of Σ + defined in 9.2. and ]P denotes the number of cylinder sets of length n contained in an element of this partition. Proof Fix a reciprocal of a PV number β, τ ∈ (0, 0.5) and θ = (β, β, τ, τ). Let n ≥ 1 and set un = log( ∑ P∈Πn,β(]P ) log β log τ ) n log β−1 . 76 Consider the set of cylinders in Σ+ given by Cn = {[s1s2 . . . sm]0 | si ∈ {−1, 1}n i = 1 . . .m}. Define a set function η on Cn by η([s]0) = ]P (s)log β/ log τ ]P (s) βnun and η([s1s2 . . . sm]0) = η([s1]0) * η([s2]0) * . . . * η([sm]0) where s, s1, . . . sm are elements of {−1, 1}n and P (s) denotes the element of the partition Πn,β containing the cylinder [s]0. Note the facts that Cn is a basis of the metric topology of Σ + and that ∑ s∈{−1,1}n η([s]0) = 1 by the definition of un. Thus we can extend η to a Borel probability measure on Σ+. Now recall that the map πθ = πβ,β,τ,τ given by πθ(s) = ( ∞∑ i=0 si(1− β)βi, ∞∑ i=0 si(1− τ)τ i) is a homeomorphism from Σ+ onto Λθ. Thus ηβ,τ := η ◦ π−1θ defines a Borel probability measure on Λθ. Given m ≥ 1 we set q(m) = dm(log β/ log τ)e. Given a sequence si ∈ {−1, 1}n for i = 1 . . .m we define a subset of Λθ by Rs1...sn = {( ∞∑ i=0 si(1− β)βi, ∞∑ i=0 ti(1− τ)τ i) | si, ti ∈ {−1, 1} (s(i−1)n, . . . , sin−1) = si i = 1 . . .m and (t(i−1)n, . . . , tin−1) = si i = 1 . . . q(m)}. We see that Rs1...sm is "almost" a square in Λθ of side length β mn. We have: c1β mn ≤ diamRs1...sm ≤ c2βmn (1) where the constants c1, c2 are independent of the choice of si. Now let as examine the ηβ,τ measure of the sets Rs1...sm . Assume that ti ∼n,β si for i = q(m) + 1 . . .m where ∼n,β is the equivalence relation introduced in 9.2. . The rectangles πθ([s1 . . . sq(m)tq(m)+1 . . . tm]0) are all disjoint and lie above each other in the set Rs1...sm . Hence we have ηβ,τ (Rs1...sm) ≥ η( ⋃ ti∼n,β si i=q(m)+1...m πθ([s1 . . . sq(m)tq(m)+1 . . . tm]0) = = ∑ ti∼n,β si i=q(m)+1...m η([s1 . . . sq(m)tq(m)+1 . . . tm]0). 77 Using the fact s ∼n,β t⇒ ]P (s) = ]P (t)⇒ η([s]0) = η([t]0) this equals m∏ i=1 η([si]0) ∑ ti∼n,β si i=q(m)+1...m 1 = m∏ i=1 ]P (si) log β/ log τ ]P (si) βmnun ∑ ti∼n,β si i=q(m)+1...m 1 = = ∏m i=1 ]P (si) log β/ log τ ∏q(m) i=1 ]P (si) βmnun = (φs1...smβ nun)m where φs1...sm = ( ∏m i=1 ]P (si) log β/ log τ ∏q(m) i=1 ]P (si) )1/m. Now fix an ε > 0 We use the sets Rs1...sm to construct a good cover of Λθ in the sense for Hausdorff dimension. To this end set Rm := {Rs1...sm |φs1...sm ≥ βnε}. We have an upper bound on the cardinality of Rm. If R ∈ Rm then ηβ,τ (R) ≥ βmn(un+ε) and since ηβ,τ is a probability measure we see: card(Rm) ≤ β−mn(un+ε) (2). Now let R(M) = ⋃ m≥M Rm. We want to prove that R(M) is a cover of Λθ for all M ≥ 1. For s = (sk) ∈ Σ+ we define the function φm by φm(s) = φs0...smn−1 . In addition we need two auxiliary functions on Σ+: fm(s) = ∏m i=0 ]P ((s(i−1)n, . . . , sin−1)) 1/m ∏q(m) i=0 ]P ((s(i−1)n, . . . , sin−1)) 1/q(m) , gm(s) = ( q(m) ∏ i=1 ]P ((s(i−1)n, . . . , sin−1))) 1/q(m)(log β log τ−q(m)/m). Since 1 ≤ ]P (s) ≤ 2n we have 1 ≤ gm(s) ≤ 2n(log β/ log τ−q(m)/m). Thus by the definition of q(m) we have gm(s) −→ 1. Moreover we have limm−→∞fm(s) ≥ 1 because ∏t i=0 ]P ((si−1n, . . . , sin−1)) 1/t ≥ 1 ∀t ≥ 1. A simple calculation shows φm(s) = (fm(s)) log β/ log τgm(s). The properties of f and g thus imply: limm−→∞φm(s) ≥ 1 ∀ s ∈ Σ+. This will help us to show that R(M) is a cover of Λθ. For all s = (sk) ∈ Σ+ there is an m ≥ M such that φm(s) ≥ βnε and thus πθ(s) ∈ Rs0,...,smn−1 ∈ R(M). Since πθ 78 is onto Λθ we see that R(M) is indeed a cover of Λθ. We are now able to complete the proof. For every ε > 0 and every M ∈ IN we have: ∑ R∈R(M) (diamR)un+2ε = ∑ m≥M ∑ R∈Rm (diamR)un+2ε ≤(1) ∑ m≥M ∑ R∈Rm (c2β mn)un+2ε = ∑ m≥M card(Rm)(c2β mn)un+2ε ≤(2) cun+2ε2 ∑ m≥M βmnε. The last expression goes to zero with M −→ 0. By the definition for Hausdorff dimension we thus get dimH Λθ ≤ un+2ε and since ε is arbitrary, we have dimH Λθ ≤ un. 2 Some ideas we have used here are to due the prove of McMullen's theorem (2.1.2.) by Pesin in [PE2]. Now we use strategies developed in the proof of 9.2.3. to get: Proposition 10.2.3. If β is the reciprocal for a PV number, τ ∈ (0, 0.5) and θ = (β, β, τ, τ) we have: ∃ N ∀ n > N log( ∑ P∈Πn,β(]P ) log β log τ ) n log β−1 < log(2β/τ) log(1/τ) . Proof Fix a reciprocal of a PV number β. Consider the proof of 9.2.3. for the equal weighted Bernoulli measure b. Recall that we denote by xni i = 1 . . . ](n) the distinct points of the form ∑n−1 k=0 ±(1−β)βk and by mni the b measure of corresponding element P in from the partition Πn,β. By the singularity of bβ we have more than we used in 9.2.3. : ∀C ∈ (0, 1) ∀ε > 0 ∃ disjoint intervals (a1, b1), . . . , (au, bu) with u∑ l=1 (bl − al) < ε and bβ(O) > C where O := u⋃ l=0 (al, bl). 79 By the same arguments we used in the proof of 9.2.3., we conclude: ∃c > 0 ∀C ∈ (0, 1) ∀ε > 0 ∃N = N(ε, C) ∀n ≥ N : ∑ xni ∈Ō mni > C and ](n) := card{xni ∈ Ō} ≤ εcβ−n. Since mni = b(P i n) = ]P i n/2 n, where ]P denotes the number of cylinder sets of length n contained in P , it follows that there is a subset Πn,β of Πn,β with ](n) elements such that ∑ P∈Πn,β ]P ≥ C2n Now we estimate: ∑ P∈Πn,β (]P )log β/ log τ = ∑ P∈Πn,β (]P )log β/ log τ + ∑ P∈Πn,β\Pin,β (]P )log β/ log τ ≤ ](n)1−log β/ log τ ( ∑ P∈Πn,β ]P ) log β/ log τ (](n)− ](n))1−log β/ log τ+( ∑ P∈Πn,β\Pin,β ]P ) log β/ log τ ≤ (εcβ−n)1−log β/ log τ2n log β/ log τ + (cβ−n)1−log β/ log τ ((1− C)2)n log β/ log τ = βn(log β/ log τ−1)2n log β/ log τ ((εc)1−log β/ log τ + c1−log β/ log τ (1− C)log β/ log τ ). Now choose ε and C such that ((εc)1−log β/ log τ + c1−log β/ log τ (1− C)log β/ log τ ) < 1. For all n ≥ N(ε, C) we have: log( ∑ P∈Πn,β(]P ) log β log τ ) n log β−1 < log(2β/τ) log(1/τ) + log((εc)1−log β/ log τ + c1−log β/ log τ (1− C)log β/ log τ ) n log β−1 . The last term in this sum is negative and hence our proof is complete. 2 Now the proof of our theorem is obvious: Proof of 10.2.1. From 4.1. we know that the box-counting dimension of Λθ is given by log(2β/τ)/ log(1/τ) in the situation we study here. Thus 10.2.2. and 10.2.3. immediately imply dimH Λθ < dimB Λθ. The inequality dimH Λθ < dimB Λθ follows from this with the help of proposition A5. 2 80 We end this work with three problems concerning number theoretical peculiarities that we were not able to solve. Open problems (1) What is the Hausdorff dimension of Λθ if β is the reciprocal of a PV number, τ ∈ (0, 0.5) and θ = (β, β, τ, τ)? (2) Does the variational principle for Hausdorff dimension hold for the systems (Λθ, Tθ) in this situation? (3) Are there number theoretical peculiarities for the systems (Λθ, Tθ), (Λθ, fθ) and ([−1, 1], fβ1,β2) in the asymmetric situation, θ = (β1, β2, τ1, τ2) ∈ P 4all with β1 6= β2? 81 Appendix A: General facts in dimension theory We will here first define the most important quantities in dimension theory and then collect some basic facts. We refer to the book of Falconer [FA1] and the book of Pesin [PE2] for a more detailed discussion of dimension theory. Let Z ⊆ IRq. We define the s-dimensional Hausdorff measure Hs(Z) of Z by Hs(Z) = lim λ−→0 inf{ ∑ i∈I (diamUi) s|Z ⊆ ⋃ i∈I Ui and diam(Ui) ≤ λ}. The Hausdorff dimension dimH Z of Z is given by dimH Z = sup{s|Hs(Z) =∞} = inf{s|Hs(Z) = 0}. Let Nε(Z) be the minimal number of balls of radius ε that are needed to cover Z. We define the upper box-counting dimension dimB resp. lower box-counting dimension dimB of Z by dimBZ = limε−→0 logNε(Z) − log ε dimBZ = limε−→0 logNε(Z) − log ε . We remark that these quantities are not changed if we replace Nε(Z) by the minimal number of squares parallel to the axis with side length ε that are needed to cover Z. Furthermore we note that limit in the definition exists, if it exists for some exponential decreasing sequence. Now let μ be a Borel probability measure on IRq. We define the dimensional theoretical quantities for μ by dimH μ = inf{dimH Z|μ(Z) = 1} and dimBμ = lim ρ−→0 inf{dimBZ|μ(Z) ≥ 1− ρ}. We introduce one more notion of dimension for a measure μ. Let hμ(ε) = inf{Hμ(Π)|Π a partition with diamΠ ≤ ε} where Hμ(Π) is the usual entropy of Π. We define the upper Rényi dimension dimR resp. lower Rényi dimension dimR of Z by dimRZ = limε−→0 hμ(ε) − log ε dimRZ = limε−→0 hμ(ε) − log ε . The upper local dimension d(x, μ) resp. lower local dimension d(x, μ) of the measure μ in a point x is defined by d(x, μ) = limε−→0 μ(Bε(x)) log ε d(x, μ) = limε−→0 μ(Bε(x)) log ε . Basic relations of the dimensions defined here are stated in the following proposition 82 Proposition A1 (1) dimH Z ≤ dimBZ ≤ dimBZ holds for all Z ⊆ IRq. (2) dimH μ ≤ dimBμ ≤ dimBμ holds for all Borel probability measures μ on IRq. (3) dimRμ ≤ dimBμ holds for all Borel probability measures μ on IRq. The first two inequalities are obvious and third one is proved in [YO]. The relations between the local dimension and the other notion of dimension of measures are described in the following theorem: Theorem A2 (1) d(x, μ) ≤ c -a.e. ⇒ dimH μ ≤ c. (2) d(x, μ) ≥ c -a.e. ⇒ dimH μ ≥ c and dimRμ ≥ c. (3) d(x, μ) ≤ c -a.e. ⇒ dimBμ ≤ c. (4) d(x, μ) = d(x, μ) = c a.e. ⇒ dimHμ = dimB μ = dimR μ = c. A proof of this theorem is contained in the work of Young [YO]. If the condition in part (4) of the last theorem holds, the measure μ is called exact dimensional and the common value of the dimensions is denoted by dimμ. In particular absolute continuous measures are exact dimensional: Proposition A3 If μ is an absolutely continuous Borel probability measure on IRq then d(x, μ) = d(x, μ) = q μ-a.e. . One basic fact we have to mention is that dimensional theoretical quantities are not increased by Lipschitz maps and are hence bi-Lipschitz invariants. Proposition A4 Let f be a Lipschitz map from IRq into itself then we have: (1) dimB/H f(Z) ≤ dimB/H Z for all Z ⊆ IRq. (2) dimB/H μ ◦ f−1 ≤ dimB/H μ for all Borel probability measures μ on IRq. Here dimB can be both upper and lower box-counting dimension. The proof of this proposition is obvious from the definitions. Especially we see that a projection on a linear subspace of IRq does not increase Hausdorff and boxcounting dimension of a set or a measure. There is one other elemental fact we use in our work: 83 Proposition A5 If Z ⊆ IRq and I is an interval then dimH/B(Z × I) = dimH/B +1, where dimB can be both upper and lower box-counting dimension. The statement for Hausdorff dimension follows from proposition 7.4. of [FA1] and the statement for box-counting dimension is easy to see using 3.1. of [FA1]. At the end of this appendix we like to remark that the terminology in dimension theory is not unique. What we called box-counting dimension is also known as Minkowsky dimension or as capacity. The Rényi dimension is often called information dimension. 84 Appendix B: Pisot-Vijayarghavan numbers A Pisot-Vijayarghavan number (short: PV number) is by definition the root of an algebraic equitation whose conjugates lie all inside the unit circle in the complex plane. Salem [SA] showed that the set of PV numbers is a closed subset of the reals and that 1 is an isolated element. In our context we are interested in numbers β ∈ (0.5, 1) such that β−1 is a PV number. We list some examples including all reciprocals of PV numbers with minimal polynomial of degree two and three and a sequence of such numbers decreasing to 0.5. x2 + x− 1 ( √ 5− 1)/2 x3 + x2 + x− 1 0.5436898. . . x3 + x2 − 1 0.754877 . . . x3 + x− 1 0.6823278. . . x3 − x2 + 2x− 1 0.5698403. . . x4 − x3 − 1 0.7244918. . . xn + xn−1 . . .+ x− 1 rn −→ 0.5 Table 1: Reciprocals of PV numbers An important property of PV numbers is that their powers are near integers. More precise: Proposition B1 If α is a PV number then there is a constant 0 < θ < 1 such that ||αn||ZZ ≤ θn ∀n ≥ 0 where ||.|| denotes the distance to the nearest integer. This statement can be found in [ER1]. There is an another property of PV numbers that is of great importance for us. For β ∈ (0, 1) we denote by ]β(n) the number of distinct points of the for ∑n−1 k=0 ±βk and by ωβ(n) the minimal distance between two of those points. Proposition B2 If β ∈ (0.5, 1) is the reciprocal of a PV number then there are positive constants c > 0 and C > 0 such that ωβ(n) ≥ cβn and ]β(n) ≥ Cβ−1 holds for all n ≥ 0. For the first inequality we refer to [GA2] lemma 1.6. and for the second inequality see (15) of [PU]. Finally we like to mention that there is a whole book about Pisot and Salem numbers [BDGPS]. Certainly the reader will find much more information about the role of these numbers in algebraic number theory in this book than we provided here for our purposes. 85 General notations IN denotes the set of natural numbers {1, 2, 3, 4, 5, . . .} IN0 := IN ∪ {0} ZZ denotes the set of integers {. . . ,−2,−1, 0, 1, 2, . . .} ZZ− denotes the set of negative integers {−1,−2,−3,−4, . . .} ZZ−0 := ZZ − ∪ {0} IR denotes the set of real numbers sup(A) denotes the supremum of a set A ⊆ IR inf(A) denotes the infimum of a set A ⊆ IR lim denotes the limes superior lim denotes the limes inferior dxe denotes the smallest integer bigger then x ∈ IR bxc denotes the biggest integer smaller then x ∈ IR |x| denotes the absolute value of x ∈ IR d(x, y) denotes the distance between two points x and y in a metric space Bε(x) denotes the open ball of radius ε around x in a metric space diam(A) denotes the diameter of a subset A of a metric space := sup{d(x, y)|x ∈ A y ∈ A} card(A) denotes the cardinality of a set A closure(A) denotes the closure of the set A with respect to a given topology prX(A) denotes the projection of A ⊆ IRq onto the first component prY (A) denotes the projection of A ⊆ IRq onto the second component 86 prZ(A) denotes the projection of A ⊆ IRq onto the third component prXY (A) denotes the projection of A ⊆ IRq onto the first two components prXZ(A) denotes the projection of A ⊆ IRq onto the first and the third component prY Z(A) denotes the projection of A ⊆ IRq onto the second and the third component Σ := {−1, 1}ZZ Σ := Σ\{(sk)|∃k0∀k ≤ k0 : sk = 1}) ∪ {(1)} Σ+ := {−1, 1}IN0 ΣA denotes a Markov chain in Σ ; see chapter eight Σ+A denotes a Markov chain in Σ + ; see chapter eight pr+ denotes the projection from Σ onto Σ+ σ denotes the shift map; σ((sk)) = (sk+1) bp denotes the Bernoulli measure on Σ resp. Σ+ which is the product of the discrete measure giving 1 the probability p and −1 the probability (1− p) b := b0.5 ` denotes the normalized Lebesgue measure on the interval [−1, 1] mp denotes a Markov measure; see chapter eight htop(T ) denotes the topological entropy of a continuous transformation on a topological space; see [KH] for definition hμ(T ) denotes the metric entropy of a transformation T with respect to an invariant measure μ; see [KH] for definition dimB denotes the box-counting dimension; see appendix A dimH denotes the Hausdorff dimension; see appendix A 87 dimR denotes the Renyi dimension; see appendix A d(x, μ) denotes the local dimension of a measure μ in a point x; see appendix A Some notations and basic relations in our work system (Λθ, fθ) (Λθ, Tθ) ([−1, 1]2, fβ1,β2) type attractor repeller endomorphism parameters θ ∈ P 4all θ ∈ P 4all (β1, β2) ∈ P 2olapp projections prXY ◦ fθ = fβ1,β2 prXZΛθ = Λθ see page 14 page 9 page 12 coding system (Σ, σ−1) (Σ+, σ) (Σ, σ−1) coding map πθ πθ πβ1,β2 projections prXZ ◦ πθ = πθ ◦ pr+ prX ◦ πθ = πβ1,β2 prX ◦ πβ1,β2 = prXY ◦ πθ = πβ1,β2 πβ1,β2 ◦ pr+ see page 17 page 16 page 18 ergodic measures μθ μθ μβ1,β2 projections prXY μθ = μθ prXμθ = μβ1,β2 prX μβ1,β2 = (pr +μ)β1,β2 see page 23 page 21 page 21 88 References [AY] J.C. 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Wir untersuchen eine Klasse von selbstaffinen Repellern und eine Klasse von Attraktoren stückweise affiner Abbildungen, die jeweils von vier Parametern abhängen. Darüber hinaus betrachten wir verallgemeinerte Baker's Transformationen, ein Klasse von Endomorphismen abhängig von zwei Parametern. Bei unserer dimensionstheoretischen Analyse leiten uns im wesentlichen zwei Fragestellungen. Erstens fragen wir, ob die Hausdorff Dimension der betrachteten invarianten Mengen mit deren Box-Counting Dimension übereinstimmt. Zweitens fragen wir, ob auf den betrachteten invarianten Mengen ein ergodisches Mass voller Hausdorff Dimension existiert bzw. ob das Variationsprinzip der Hausdorff Dimension gilt, was bedeutete, dass sich die Dimension der betrachteten Menge durch die Dimension ergodischer Masse auf der Menge approximieren lässt. Im Rahmen dieser Arbeit konnten wir ein ganze Reihe neuer Ergebnisse erzielen, die interessante Phänomene im Bereich der Dimensionstheorie dynamischer Systeme, anhand der von uns gewählten Beispiele, aufzeigen. Wir denken, dass unsere Ergebnisse und Methoden auch bei der Entwicklung einer allgemeinen Theorie relevant sein könnten. Wir werden nun unsere Hauptergebnisse zusammenfassend darstellen. Die Berechnung der Box-Counting Dimension der Attraktoren und Repellern, die wir betrachten, ist mit elementaren Überdeckungs Argumenten möglich und wir erhalten eine allgemein gültige Formel. Weiterhin zeigen wir, dass die Box-Counting Dimension der Repeller und Attraktoren generisch (im Sinne des Lebesgue Masses auf Teilen des Parameterraums) mit deren Hausdorff Dimension übereinstimmt. Für die Repeller finden wir generisch ergodische Masse voller Hausdorff Dimension. Auf der anderen Seite zeigen wir, dass das Variationsprinzip für die Attraktoren nicht generisch gilt. Für die verallgemeinerte Baker's Transformation gibt es Parameterbereich in denen generisch ein ergodisches Mass voller Hausdorff Dimension existiert und Bereiche in denen das Variationsprinzip nicht gilt. Die Beweise dieser generischen Resultate basieren zum einen auf einer geeigneten Anwendung der allgemeinen Dimensionstheorie ergodischer Masse und zum anderen auf einem Studium bestimmter selbstähnlicher Masse. Weitere Hauptergebnisse unserer Arbeit beziehen sich auf zahlentheoretische Ausnahmen zu unseren generischen Resultaten in einer symmetrischen Situation. Wir zeigen, dass die Identität zwischen Hausdorff und Box-Counting Dimension der Attraktoren und der Repeller nicht gilt, wenn die Parameter bestimmte zahlentheoretische Eigenschaften besitzen. Weiterhin zeigen wir, dass für die symmetrische Attraktoren sowie für die Fat Baker's Transformationen das Variationsprinzip der Hausdorff Dimension unter bestimmten zahlentheoretischen Bedingungen nicht gilt, obwohl es in diesem symmetrischen Fall generisch gilt. Für die Reppeller konnten wir unter diesen Bedingungen nur zeigen, dass kein Bernoulli Mass voller Dimension existieren kann. 92 Lebenslauf von Jörg Neunhäuserer (Stand: Juni 1999) Persönliche Daten: Name: Jörg Neunhäuserer Geburtsdatum: 26.4.1969 Geburtsort: Wuppertal Staatsangehörigkeit: deutsch Familienstand: ledig Schule: 1976-1980: Grundschule Bornscheuerstrasse, Wuppertal 1980-1989: Märkisches Gymnasium, Schwelm Mai 1989: Abitur Studium: 1989-1992: Grundstudium der Mathematik mit Nebenfach Philosophie an der Freien Universität Berlin (FUB) M"arz 92: Vordiplom im Fach Mathematik 1992-1997: Hauptstudium der Mathematik mit Nebenfach Philosophie an der FUB Februar 97: Diplom im Fach Mathematik 1997-99: Promotionsstudium im Fach Mathematik an der FUB, gefördert durch ein Promotionsstipendium der Berliner Universitäten Juni 99: Abgabe der Dissertation vor. September 99: Disputation Berufliche Tätigkeit: 1990-1992: Verschiedene Jobs der studentischen Arbeitsvermittlung 1992-1995: Lehrtutor für Mathematik an der FUB 1998-1999: Freie Mitarbeit beim Test von Lernsoftware des Schulbuchverlags Cornelson in Berlin