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.
Higher dimensional Frobenius problem and Lipschitz equivalence of Cantor sets
高维Frobenius问题由先前的论文[Fan，Rao和Zhang，Higher dimensional Frobenius problem：maximum saturated cones，growth function and rigidity，Preprint 2014]引入。引入了高维Frobenius问题，研究了一个方向增长函数.本文研究了$\mathbb R^d$中类尘自相似集的Lipschitz等价性.对于任何自相似集，我们与它相关联的高维Frobenius问题，我们表明，相关的高维Frobenius问题的方向增长函数是Lipschitz不变。作为应用，我们解决了当两个尘埃自相似集$E$和$F$具有共面比时的Lipschitz等价问题，证明了它们是Lipschitz等价的当且仅当对于某些$p, q\geq 1$，$E$的第$p$次迭代的收缩向量是$F$的第$q$次迭代的收缩向量的置换.这部分地回答了Falconer和Marsh提出的一个问题[On the Lipschitz equivalence of Cantor sets，\emph{Mathematika,} \textbf{39}（1992），223- 233]。
Lipschitz Equivalence of Self-Similar Sets: Algebraic and Geometric Properties
本文对自相似集的Lipschitz等价性的研究进行了综述。Lipschitz等价是分形几何中的一个重要性质，因为它保持了分形集的许多重要性质。Falconer和Marsh [On the Lipschitz equivalence of Cantor sets，\textit{Mathematika}，\textbf{39}（1992），223- 233]的一个基本结果建立了基于自相似集的压缩比的代数性质的Lipschitz等价的条件。最近在这一领域取得了其他重大进展。本文是对这一领域的一次全面考察。它提供了一个重要的和有趣的结果在该领域的总结。此外，我们提供了几个重要的技术，已被用来证明一些关键结果的详细讨论。这是我们的希望，该文件将提供一个很好的概述的主要成果和技术，和一个友好的切入点，任何人谁是有兴趣在这一领域的研究问题。
Exponential ergodicity in the bounded-Lipschitz distance for a subclass of piecewise-deterministic Markov processes with random switching between flows
