Thermodynamic Formalism for Group-Extended Markov Systems with Applications to Fuchsian Groups

In this thesis we develop a thermodynamic formalism for group-extended Markov systems. We show that previous results of Kesten, Grigorchuk, Cohen and Brooks, which relate amenability with probabilistic, combinatorial and geometric quantities in different settings, find a common framework within this thermodynamic formalism. We partially recover these results and extend them in various directions for group-extended Markov systems. In this way we obtain generalisations of certain deep results of Brooks, which relate amenability of Fuchsian groups with the Hausdorff dimension of certain associated limit sets, to the setting of graph directed Markov systems. We introduce the new notion of induced topological pressure, which is suitable for giving a thermodynamic description of arbitrary subsystems of a dynamical system. Moreover, it enables us to study exhaustion principles for induced topological pressure, which are strongly linked with amenability in the context of group-extended Markov systems. A further main result of this thesis is to obtain a new multifractal formalism for infinite conformal iterated function systems.