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On Lipschitz equivalence of finite-dimensional linear flows

Arno Berger, Anthony Wynne

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.

On Lipschitz equivalence of finite-dimensional linear flows

Abstract

Two flows on a finite-dimensional normed space are Lipschitz equivalent if some homeomorphism of that is bi-Lipschitz near the origin preserves all orbits, i.e., maps each orbit onto an orbit. A complete classification by Lipschitz equivalence is established for all linear flows on , 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.
Paper Structure (6 sections, 24 theorems, 206 equations, 4 figures)

This paper contains 6 sections, 24 theorems, 206 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Analytic preliminaries
  3. Linear flows and Lipschitz similarity
  4. Distortion points for stable flows
  5. Characterizing Lipschitz equivalence
  6. Pointwise Lipschitz equivalence at $0$

Key Result

Theorem 1.1

Let $\Phi$, $\Psi$ be linear flows on $X$. Then each of the following four statements implies the other three: Moreover, $\Phi \stackrel{1}{\cong}\Psi$ if and only if $A^{\Phi}$, $A^{\Psi}$ are Lipschitz similar while $A^{\Phi_{\sf C}}$, $A^{\Psi_{\sf C}}$ are similar.

Figures (4)

  • Figure 1: Given $A\in \mathbb{R}^{2\times 2}$, the matrix $\alpha A$ is similar, Lipschitz similar, or Lyapunov similar for some $\alpha \in \mathbb{R} \setminus \{0\}$ to precisely one of the matrices shown in the respective row; the scalar $\alpha$ is uniquely determined if and only if $\sigma (A)\not \subset i\mathbb{R}$, visualized here as the cases to the right of the dotted line.
  • Figure 2: If $U_0$, $V_0$ are finite then $f|_{U\setminus U_0} = g\circ \widetilde{h}$ for some bijection $\widetilde{h}:U\setminus U_0 \to V\setminus V_0$; see Proposition \ref{['prop2zz']}.
  • Figure 3: The differentiable, Lipschitz, and Hölder classes of $A\in \mathbb{R}^{2\times 2}$ coincide for $A\in \mathcal{G}_2^{\dagger}$ yet differ for $A\in \mathcal{G}_2 \setminus \mathcal{G}_2^{\dagger}$. Whenever $A\in \mathcal{G}_2 \setminus \mathcal{G}_2^{\dagger}$, the class $[A]_1$ resembles $[A]_{1^-}$ much more (left) than it resembles $[A]_{\sf diff}$. Notice that all matrices in $\mathbb{R}^{2\times 2}\setminus \mathcal{G}_2$ correspond to points on the solid black curves. Circles indicate the six different classes $[A]_0$, with $O_2$, $J_2$ corresponding to the same point.
  • Figure 4: Constructing $h\in \mathcal{H}_{{\sf pw}1}(X)$ to prove (\ref{['eq6_p1']}) for hyperbolic $\Phi$.

Theorems & Definitions (44)

  • Theorem 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Remark 1.4
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 3.1
  • Remark 3.2
  • Proposition 3.3
  • ...and 34 more