Analyzing Topological Mixing and Chaos on Continua with Symbolic Dynamics

Abstract

This work describes the way that topological mixing and chaos in continua, as induced by discrete dynamical systems, can or can't be understood through topological conjugacy with symbolic dynamical systems. For example, there is no symbolic dynamical system that is topologically conjugate to any discrete dynamical system on an entire continuum, and there is no finer topology that can be given to a continuum which simultaneously makes the continuum homeomorphic to a symbolic dynamical system and contains its original topology. However, this paper demonstrates an analytical mechanism by which the existence of topological mixing and/or chaos can be shown through conjugacy with qualitative dynamical systems outside the usual purview of symbolic dynamics. Two examples of this mechanism are demonstrated on classic textbook models of chaotic dynamics; the first proving the existence of topological mixing everywhere in the dyadic map on the interval by showing that there exists a qualitative system that is topologically conjugate to the dyadic map on the interval with a finer topology than the usual Euclidean topology, and the other following a similar approach to demonstrate the existence of Devaney chaos everywhere in the $2$-tent map on the interval. The content is presented in a somewhat self-contained fashion, reiterating some standard results in the field, to aid new learners of topological mixing/chaos.