H\"{o}lder classifications of finite-dimensional linear flows

TL;DR

This paper provides a comprehensive classification of finite-dimensional linear flows based on Hölder equivalence, revealing how linear algebra properties determine flow equivalence under Hölder continuous homeomorphisms.

Contribution

It introduces a complete classification of linear flows using Hölder equivalence, extending and unifying previous results by focusing on orbit-preserving homeomorphisms.

Findings

Classification in terms of linear algebra properties

Extension of known results to Hölder equivalence

Clarification of the roles of linearity and finite-dimensionality

Abstract

Two flows on a finite-dimensional normed space $X$ are equivalent if some homeomorphism $h$ of $X$ preserves all orbits, i.e., $h$ maps each orbit onto an orbit. Under the assumption that $h$, $h^{-1}$ both are $\beta$-H\"{o}lder continuous near the origin for some (or all) $0<\beta< 1$, a complete classification with respect to some-H\"{o}lder (or all-H\"{o}lder) equivalence is established for linear flows on $X$, in terms of basic linear algebra properties of their generators. Consistently utilizing equivalence instead of the more restrictive conjugacy, the classification theorems extend and unify known results. Though entirely elementary, the analysis is somewhat intricate and highlights, more clearly than does the existing literature, the fundamental roles played by linearity and the finite-dimensionality of $X$.