On the phenomenon of topological chaos and statistical triviality

TL;DR

This paper explores the coexistence of topological chaos and statistical triviality in certain vector fields on a compact manifold, revealing a complex structure with an infinitely dimensional connected subset.

Contribution

It demonstrates the existence of a large, infinitely dimensional set of vector fields exhibiting both topological chaos and statistical triviality on a compact manifold.

Findings

Existence of an infinitely dimensional connected subset of vector fields

Presence of topological chaos with statistical triviality

Complex scale of such vector fields in terms of dimension

Abstract

There exists a compact manifold so that the set of topologically chaotic but statistically trivial $C^{r} (1\leq r \leq \infty)$ vector fields on this manifold displays considerable scale in the view of dimension. More specifically, it contains an infinitely dimensional connected subset.