Chaos in Topological Spaces
Dynamical Systems
2013-08-14 v1
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.
Keywords
Cite
@article{arxiv.1308.2810,
title = {Chaos in Topological Spaces},
author = {John Taylor},
journal= {arXiv preprint arXiv:1308.2810},
year = {2013}
}