Equivalence of topological dynamics without well-posedness

TL;DR

This paper extends the concept of topological equivalence to dynamical systems lacking well-posedness by generalizing Yorke's axiomatic theory, including group actions, thus broadening the framework for analyzing such systems.

Contribution

It introduces a new notion of topological equivalence for ill-posed systems and generalizes Yorke's theory to topological group actions.

Findings

Established a generalized framework for topological equivalence without well-posedness.

Connected the new notion with classical definitions in well-posed cases.

Extended Yorke's axiomatic theory to include topological group actions.

Abstract

The notion of topological equivalence plays an essential role in the study of dynamical systems of flows. However, it is inherently difficult to generalize this concept to systems without well-posedness in the sense of Hadamard. In this study, we formulate a notion of "topological equivalence" between such systems based on the axiomatic theory of topological dynamics proposed by Yorke, and discuss its relation with the usual definition. During this process, we generalize Yorke's theory to the action of topological groups.