Chaos on compact metric spaces generated from symbolic dynamical systems

TL;DR

This paper explores chaos in compact metric spaces through symbolic dynamics, demonstrating how chaotic maps like the tent and baker maps can be understood via decomposition spaces and extending chaos to finite graphs.

Contribution

It introduces a method to analyze chaos on compact metric spaces using decomposition spaces linked to symbolic dynamics, providing new insights into classical chaotic maps.

Findings

Chaotic maps on [0,1] can be characterized via decomposition dynamics.

The chaotic character of tent and baker maps is reinterpreted through symbolic dynamics.

A chaotic map is constructed on any finite graph.

Abstract

We discuss Devaney chaos on compact metric spaces using a decomposition space characterized by topological nature of symbolic dynamics. A chaotic map obtained here is defined as a topologically conjugate of the chaotic map on a decomposition space which is induced by a chaotic map of symbolic dynamics. In particular, the chaotic character of the tent map and the baker map on [0,1] are reconsidered based on decomposition dynamics involving symbolic dynamics with different two chaotic maps. As an example of compact metric space we exhibit a chaotic map existing on any given finite graph.