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

• 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; Fuchsian groups; Hausdorff dimension; Large deviations; Ergodic theory; Fractal geometry; Multifractal analysis; Non-uniformly hyperbolic; Ergodic Theory; Fractal Geometry

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.

我们研究了沿阵行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。我们主要关注紫红色群体的动态，承认一个奇特的基本领域，甚至弯道。对于相关的双曲线表面，我们在与定向的大地学相关的同源生长速率的Hausdorff尺寸光谱上获得了新的结​​果。特别是，我们能够以与给定反向温度相关的广义繁殖指数来表达维度。作为随机对应物，我们研究了观察一定同源生长速率的概率。事实证明，增长率与与Hausdorff尺寸频谱密切相关的速率函数满足了较大的偏差。 我们已经根据测量流的某些几何特性与可数的马尔可夫sh的符号热力学形式主义结合了失真参数。这是与高崎hiroki（keio u。）的联合作品。我们在真实线上的瞬态动力学方面也取得了一些进展，并具有反射性边界。特别是，我们获得了拓扑压力函数的公式，该公式表明可能是由反射边界产生的相变。