发表机构
ICEx-UFMG; Penn State; Department of Mathematics and Shenzhen International Center for Mathematics - Southern University of Science and Technology（米纳斯吉拉斯联邦大学; 宾夕法尼亚州立大学; 南方科技大学数学系及深圳国际数学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对伪阿诺索夫同胚类中拓扑熵相同的微分同胚，本文修正了Handel关于半共轭原像连通性的结论，并证明了对应不变测度的性质及半共轭诱导的度量同构。
AI 中文摘要
设$g$是伪阿诺索夫同胚$f$的同痕类中的$C^\infty$微分同胚，且$g$与$f$具有相同的拓扑熵。1988年，Handel证明这一结论意味着存在从$g$到$f$的半共轭$\pi$，他指出一般情况下至少存在一个点$x$使得$\pi^{-1}(x)$不连通。本文证明该结论不成立：对任意$x$，集合$\pi^{-1}(x)$是一列嵌套闭拓扑圆盘的交集，因此是连通的。本文还证明存在唯一的$g$-不变概率测度，其投影到$f$的最大熵测度，该测度是熵最大化的、双曲的且为伯努利测度，且该半共轭诱导了对应保测系统间的度量同构。
英文摘要
Let $g$ be a $C^\infty$ diffeomorphism in the isotopy class of a pseudo-Anosov homeomorphism $f$ such that $g$ and $f$ have the same topological entropy. In 1988, Handel proved that this implies the existence of a semiconjugacy $π$ from $g$ to $f$. He stated that, in general, there is at least one point $x$ such that $π^{-1}(x)$ is disconnected. We show that this is not the case: for every $x$, the set $π^{-1}(x)$ is the intersection of a nested sequence of closed topological disks, and hence is connected. We also prove that there is a unique $g$-invariant probability measure projecting to the measure of maximal entropy of $f$. This measure is entropy-maximizing, hyperbolic, and Bernoulli, and the semiconjugacy induces a metric isomorphism between the corresponding measure-preserving systems.
