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、几何和随机性质之间的相互作用。我们主要集中在动力学的Fuchsian群承认狄利克雷基本域，甚至角落。对于相关的双曲曲面，我们得到了与定向测地线相关的同调增长率的Hausdorff维数谱的新结果。特别地，我们能够用与给定的逆温度相关联的广义Poincare指数来表示维数。作为随机对应，我们研究了观察到某个同调增长率的概率。结果表明，增长率满足大的偏差与Hausdorff维数谱密切相关的速率函数。 我们结合了变形参数的基础上的一些几何性质的测地线流的符号热力学形式主义可数马尔可夫大便。这是与Hiroki Takahasi（Keio U.）的合作作品。我们在具有反射边界的真实的线上的瞬态动力学也有了一些进展。特别是，我们得到的拓扑压力函数，这表明可能的相变所产生的反射边界的公式。

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.