Sensitive actions in non-compact spaces

TL;DR

This paper generalizes a known result linking chaos conditions in dynamical systems to non-compact spaces, offering a simpler proof and extending previous theorems to broader settings.

Contribution

It extends the generalization of chaos conditions to non-compact spaces with a simplified proof, broadening the applicability of prior results.

Findings

Unified proof for chaos conditions in non-compact spaces

Extended previous theorems to more general settings

Simplified the understanding of chaos in dynamical systems

Abstract

Devaney defines a function as chaotic if it satisfies the following three conditions: transitivity, having a dense set of periodic points, and sensitive dependence on initial conditions. In \cite{3}, it was demonstrated that the first two conditions imply the third. This result was generalized in \cite{aak} by replacing the density of periodic points with the density of minimal points. The result was further generalized in \cite{g} for group actions, in \cite{km} for $C$-semigroups actions, and in \cite{d} for a continuous semi-flow with $X$ being a Polish space. Subsequently, in \cite{ip1} and \cite{ip2}, it was generalized for compact spaces and for non-compact spaces in \cite{z}. The objective of this work is to generalize the result in \cite{z}, providing a simple proof.