Topological Conjugacy

Abstract

For convenience, let us define a dynamical endomorphism to be a piecewise continuous self-map f : X → X of a complete separable metric space. Sometimes we use the term endomorphism for short, although we do not wish to confuse this with other uses of the term (e.g. as in group endomorphism, etc.) Let f : X → X and g : Y → Y be dynamical endomorphisms. We say that f is topologically conjugate to g if there is a homeomorphism h from X onto Y such that gh = hf (or h −1 gh = f). We sometimes call h a topological conjugacy between f and g or from f to g. We also say that f and g are topologically conjugate. Note that topological conjugacy is an equivalence relation on any given collection of dynamical endomorphisms. We write f ∼ g to denote that f is topologically conjugate to g. A dynamical property of a system is one which is preserved under topo-logical conjugacy. The following are just a few examples of dynamical properties of a given endomorphism f. 1. f has a bounded orbit 2. f has a fixed point 3. f has a dense orbit 4. f has infinitely many periodic orbits 5. the set of periodic orbits of f is dense in the set of bounded orbits To understand the dynamical properties of a given endomorphism f , one tries to find an understandable topological model for f. This is another en-domorphism g whose orbit structure is easily describable (or at least many dynamical properties are easily describable) and is such that g ∼ f. We now proceed to construct a useful class of topological models for many systems. These are called Symbolic Systems.