Ergodic theory of low-dimensional dynamical systems

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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. *****