Dynamic topological logic

Dynamic Topological Logic provides a context for studying the confluence of the topological semantics for S4, based on topological spaces rather than Kripke frames; topological dynamics; and temporal logic. In the topological semantics for S4, 2 is interpreted as topological interior: thus S4 can be understood as the logic of topological spaces. Topological dynamics studies the asymptotic properties of continuous maps on topological spaces. Thus, we define a dynamic topological system to be a topological space X together with a continuous function f that can be thought of in temporal terms, moving the points of the topological space from one moment to the next. Dynamic topological logics are the logics of dynamic topological systems, defined for a trimodal language with an S4 topological modality, 2 (interior), and two temporal modalities, © (next) and ∗ (henceforth). One potential area of study is the expressive power of this language: for example, in it one can express the Poincaré Recurrence Theorem.