Dimension Theory and Fuchsian Groups

In this paper we review some concepts of Dimension Theory in Dynamical Systems and we show how to apply them for studying growth rates of Kleinian groups acting on the hyperbolic plane H ². The mainly focus on: multifractal analysis, additive and nonadditive thermodynamic formalisms and Gibbs states. In order to connect these concepts with groups we define a family of potentials _n ():=d _h (O,e ₀ e ₁...e _n (O)), (the limit set of ), where d _h is the hyperbolic metric in H ² and e ₀ e ₁... is a sequence in the generators of assigned to . These sequences are obtained from the method by C. Series for coding hyperbolic geodesics. Next, a decomposition in level sets K :={:lim _n =} is considered and a variational multifractal analysis of the entropy spectrum of K , by means of the formalism developed by Barreira, is done.