Ergodic theory for conformal dynamics with applications to fractal geometry

共形动力学的遍历理论及其在分形几何中的应用

基本信息

批准号：
21K03269
负责人：
イェーリッシュヨハネス
金额：
$ 2.58万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2021
资助国家：
日本
起止时间：
2021-04-01 至 2024-03-31
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-21K03269/
关键词：
Fuchsian groups Hausdorff dimension Large deviations Ergodic theory Fractal geometry Multifractal analysis Non-uniformly hyperbolic Ergodic Theory Fractal Geometry

项目摘要

We studied the interplay of dynamical, geometric and stochastic properties of conformal dynamical systems from the viewpoint of ergodic theory. We focused mainly on the dynamics of Fuchsian groups admitting a Dirichlet fundamental domain with even corners. For the associated hyperbolic surface, we obtained new results on the Hausdorff dimension spectrum of homological growth rates associated with oriented geodesics. In particular, we are able to express the dimension in terms of a generalized Poincare exponent associated with a given inverse temperature.As a stochastic counterpart we studied the probability to observe a certain homological growth rate. It turns out that the growth rate satisfies large deviations with a rate function closely related to the Hausdorff dimension spectrum. We have combined distortion arguments based on some geometric properties of the geodesic flow with the symbolic thermodynamic formalism for countable Markov shits. This is a joint work with Hiroki Takahasi (Keio U.).We also had some progress on transient dynamics on the real line with a reflective boundary. In particular, we obtained formulas for the topological pressure function which indicate a possible phase transition which arises from the reflective boundary.

我们从遍历理论的角度研究了共形动力系统的动力学、几何和随机特性的相互作用。我们主要关注具有偶数角的狄利克雷基本域的 Fuchsian 群的动力学。对于相关的双曲曲面，我们获得了与定向测地线相关的同调增长率的豪斯多夫维数谱的新结果。特别是，我们能够用与给定逆温度相关的广义庞加莱指数来表达维度。作为随机对应物，我们研究了观察到特定同调增长率的概率。事实证明，增长率满足与豪斯多夫维数谱密切相关的速率函数的大偏差。我们将基于测地线流的某些几何特性的畸变论证与可数马尔可夫狗屎的符号热力学形式主义结合起来。这是与 Hiroki Takahasi (Keio U.) 的合作成果。我们在具有反射边界的实线上的瞬态动力学方面也取得了一些进展。特别是，我们获得了拓扑压力函数的公式，该公式表明了反射边界可能出现的相变。