On Lipschitz equivalence of finite-dimensional linear flows

TL;DR

This paper provides a comprehensive classification of finite-dimensional linear flows up to Lipschitz equivalence, revealing fundamental roles of linearity and finite-dimensionality through an elementary yet intricate analysis.

Contribution

It establishes a complete classification of linear flows under Lipschitz equivalence using basic linear algebra, extending previous results by considering a broader equivalence relation.

Findings

Classification based on linear algebra properties

Extension of known results to Lipschitz equivalence

Elementary analysis highlighting linearity and finite-dimensionality

Abstract

Two flows on a finite-dimensional normed space $X$ are Lipschitz equivalent if some homeomorphism $h$ of $X$ that is bi-Lipschitz near the origin preserves all orbits, i.e., $h$ maps each orbit onto an orbit. A complete classification by Lipschitz equivalence is established for all linear flows on $X$, in terms of basic linear algebra properties of their generators. Utilizing equivalence instead of the much more restrictive conjugacy, the classification theorem significantly extends known results. The analysis is entirely elementary though somewhat intricate. It highlights, more clearly than does the existing literature, the fundamental roles played by linearity and finite-dimensionality.