Journal of Symbolic Logic 77 (1):308-328 (2012)

Abstract
Dynamic Topological Logic ( $\mathcal{DTL}$ ) is a modal framework for reasoning about dynamical systems, that is, pairs 〈X, f〉 where X is a topological space and f: X → X a continuous function. In this paper we consider the case where X is a metric space. We first show that any formula which can be satisfied on an arbitrary dynamic topological system can be satisfied on one based on a metric space; in fact, this space can be taken to be countable and have no isolated points. Since any metric space with these properties is homeomorphic to the set of rational numbers, it follows that any satisfiable formula can be satisfied on a system based on $\mathbb{Q}$ . We then show that the situation changes when considering complete metric spaces, by exhibiting a formula which is not valid in general but is valid on the class of systems based on a complete metric space. While we do not attempt to give a full characterization of the set of valid formulas on this class we do give a relative completeness result; any formula which is satisfiable on a dynamical system based on a complete metric space is also satisfied on one based on the Cantor space
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DOI 10.2178/jsl/1327068705
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References found in this work BETA

Dynamic Topological Logic.Philip Kremer & Grigori Mints - 2005 - Annals of Pure and Applied Logic 131 (1-3):133-158.
The Modal Logic of Continuous Functions on the Rational Numbers.Philip Kremer - 2010 - Archive for Mathematical Logic 49 (4):519-527.
Dynamic Topological Logic Interpreted Over Minimal Systems.David Fernández-Duque - 2011 - Journal of Philosophical Logic 40 (6):767-804.
Dynamic Topological Completeness For.David Fernandez Duque - 2007 - Logic Journal of the IGPL 15 (1):77-107.

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