Rescaled-Expansive Flows: Unstable Sets and Topological Entropy

TL;DR

This paper develops a rescaled-theory for local stable and unstable sets in rescaled-expansive flows, providing conditions for their structure and applying these to establish positive topological entropy in certain dynamical systems.

Contribution

It introduces a novel rescaled framework for analyzing local unstable sets and connects this to entropy properties in rescaled-expansive flows.

Findings

Conditions for non-trivial unstable sets

Positive topological entropy for specific flows

New insights into local stable and unstable structures

Abstract

In this work we introduce and explore a rescaled-theory of local stable and unstable sets for rescaled-expansive flows and its applications to topological entropy. We introduce a rescaled version of the local unstable sets and the unstable points. We find conditions for points of the phase space to exhibit non-trivial connected pieces of such unstable sets. We apply these results to the problem of proving positive topological entropy for rescaled-expansive flows with non-singular Lyapunov stable sets.