Chaos in Topological Spaces

TL;DR

This paper extends the concept of chaos to general topological spaces, establishing conditions for sensitive dependence and providing a new example of chaos in a non-metrizable space.

Contribution

It introduces a generalized definition of chaos for topological spaces, relates it to sensitive dependence in uniform Hausdorff spaces, and constructs a chaotic map in a non-metrizable space.

Findings

Chaos definition coincides with Devanney's in metric spaces

Chaotic maps in uniform Hausdorff spaces exhibit sensitive dependence

Constructed a chaotic map on a non-metrizable space

Abstract

We give a definition of chaos for a continuous self-map of a general topological space. This definition coincides with the Devanney definition for chaos when the topological space happens to be a metric space. We show that in a uniform Hausdorff space, there is a meaningful definition of sensitive dependence on initial conditions, and prove that if a map is chaotic on a such a space, then it necessarily has sensitive dependence on initial conditions. The proof is interesting in that it explains very clearly what causes a chaotic process to have sensitive dependence. Finally, we construct a chaotic map on a non-metrizable topological space.