Ergodic theory of low-dimensional dynamical systems

本计画的目标是利用遍历理论的方法来研究低维系统的动力学性质。* 特别是，我将发展多项式的核心熵理论，最近由W。瑟斯顿这个函数的性质将与Mandelbrot集的几何和拓扑有关，并将用于更好地理解高阶多项式和有理映射的参数空间的几何。* 此外，我将研究作用于具有双曲性质的空间上的群上的随机游动的遍历理论，并导出几何群论，Teichmueller动力学和低维拓扑的应用。这将扩展李群上的随机游动理论，* 由Furstenberg，Margulis，Zimmer等人开发，到几何和拓扑中感兴趣的群体，如映射类群和自由群的外自同构群。*****

The goal of this project is to investigate dynamical properties of low-dimensional systems using methods from ergodic theory. ***In particular, I will develop the theory of core entropy of polynomials as recently introduced by W. Thurston. Properties of this function will be related to the geometry and topology of the Mandelbrot set, and will be used to better understand the geometry of the parameter space of polynomials of higher degree and rational maps. ***Moreover, I will investigate the ergodic theory of random walks on groups acting on spaces with hyperbolic properties, and derive applications to geometric group theory, Teichmueller dynamics and low-dimensional topology. This will extend the theory of random walks on Lie groups, ***as developed by Furstenberg, Margulis, Zimmer, and others, to groups of interest in geometry and topology such as the mapping class group and the group of outer automorphisms of the free group. *****