Multifractal Analysis Of Birkhoff Averages And Bowen Topological Entropy For Actions Of Countable Groups

This thesis is devoted to the study of multifractal analysis of Birkhoff averages in dynamical systems and Bowen topological entropy for actions of countable dis-crete groups. By a topological dynamical system(X,d,f), we mean that (X,d) is a compact metric space and f is a continuous map from X to itself. From Chapter 1 to Chapter 3, we study the maps with the almost weak specification property and apply the results to ergodic automorphisms of the compact metric abelian groups. We study the pressure of the historic set, the generic property of the historic set and a conditional variational principle of the topological pressure of the set E(I), where.In Chapter 4, we prove a variational principle of the topological entropy of the level sets of the form for maps with the gluing orbit property, which can be applied to the transitive subshift of finite type and the geodesic flow on geodesic metric space.In Chapter 5, we define the Bowen topological entropy, measure-theoretical en-tropy and establish a variational principle related to the Bowen topological entropy and the Brin-Katok's measure-theoretical entropy for actions of countable discrete groups.The method in this chapter can be also applied to sequence entropy, the semigroup of finitely generated, discrete non-autonomous dynamical systems and so on.