Homeomorphisms group of normed vector space: Conjugacy problems and the Koopman operator

TL;DR

This paper explores conjugacy problems in homeomorphism groups of normed vector spaces, linking them to Koopman operators and eigenfunctions, and introduces a topology where conjugacy can be characterized as a limit process.

Contribution

It establishes a connection between conjugacy in homeomorphism groups and Koopman operator eigenfunctions, providing a new approach to analyze conjugacy via a specialized topology.

Findings

Conjugacy relates to common eigenfunctions of Koopman operators.

A topology on homeomorphism groups is constructed for conjugacy analysis.

Main theorem applies to generalized Lipeomorphisms.

Abstract

This article is concerned with conjugacy problems arising in homeomorphisms group, Hom($F$), of unbounded subsets $F$ of normed vector spaces $E$. Given two homeomorphisms $f$ and $g$ in Hom($F$), it is shown how the existence of a conjugacy may be related to the existence of a common generalized eigenfunction of the associated Koopman operators. This common eigenfunction serves to build a topology on Hom($F$), where the conjugacy is obtained as limit of a sequence generated by the conjugacy operator, when this limit exists. The main conjugacy theorem is presented in a class of generalized Lipeomorphisms.