Chaotic behaviour on invariant sets of linear operators

TL;DR

This paper investigates chaotic properties like hypercyclicity and mixing in linear operators on topological vector spaces, focusing on invariant sets and their closed spans, establishing key links and providing illustrative examples.

Contribution

It establishes new connections between chaos properties on invariant sets and their closed spans in linear operators, with illustrative examples.

Findings

Links between chaos properties on invariant sets and their closed spans.

Conditions under which hypercyclicity and mixing are transferred.

Examples demonstrating the theoretical results.

Abstract

We study hypercyclicity, Devaney chaos, topological mixing properties and strong mixing in the measure-theoretic sense for operators on topological vector spaces with invariant sets. More precisely, our purpose is to establish links between the fact of satisfying any of these properties on certain invariant sets, and the analogous property on the closed span of the invariant set. We also give examples that illustrate these results.