Ergodic theory for conformal dynamics with applications to fractal geometry

从遍历理论的角度研究了共形动力系统的动力学性质、几何性质和随机性质的相互作用。我们主要关注具有偶数角的Dirichlet基域的Fuchsian群的动力学。对于相关的双曲曲面，我们得到了与取向测地线相关的同生长速率的Hausdorff维谱的新结果。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示维度。作为随机对应物，我们研究了观察到某种同源增长率的概率。结果表明，增长率满足较大偏差，其速率函数与Hausdorff维谱密切相关。我们将基于测地线流的一些几何性质的畸变论证与可数马尔可夫串的符号热力学形式相结合。这是与高桥博树（Keio U.）合作的作品。我们还在具有反射边界的实线上的瞬态动力学方面取得了一些进展。特别是，我们得到了拓扑压力函数的公式，该公式表明可能从反射边界产生相变。

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.