Conditional Entropy And Non-additive Thermodynamic Formalism For Group Actions

In this thesis, we mainly study the relative topological entropy for amenable groups and the non-additive thermodynamic formalism for Zd-actions. The thesis is organized as follows.In the introduction, the origin, developments and main contents of topological dynamics and ergodic theory, are briefly recalled.In Chapter2, we mainly recall some basic notions and properties in topological dynamics and ergodic theory which will be used in this thesis.In Chapter3, we study the relative version of entropy for actions of countable amenable groups. Firstly, we introduce several notions of topological conditional en-tropy and fiber entropy for a given factor map between two amenable group actions. In addition, we connect the notions of topological fiber entropy, topological conditional entropy and measure-theoretic conditional entropy, namely, we prove three variational principles. Finally, as an application of our variational principle we prove that both of the countable-to-one extension and the distal extension have zero conditional topolog-ical entropy. In particular, we give an alternative proof of Corollary8.5in [129](Kerr and Li), that is, we will use ergodic method to prove the topological entropy of a distal system is zero.The topological pressure and thermodynamic formalism for nonadditive sequences has become a valuable tool in the study of the multifractal formalism of dimension the-ory, especially for nonconformal dynamical systems. Nevertheless, despite these and many other significant developments, only the case of Z-actions is understood. So it would be desirable to continue studying the nonadditive thermodynamic formalis-m, dimension theory and multifractal analysis for larger group actions. In Chapter4, we introduce the topological pressure of sub-additive potentials for Zd(d≥1)-actions and establish the local and global variational principle for it. These results general-ize Cao, Feng and Huang’s results to general continuous Zd-actions. We note that the sub-additive potential for amenable group, which was first defined by us in [142], is different from the classical sub-additive notion for Zd-actions. For comparison, we briefly review the definition of sub-additive potential and the sub-additive thermody-namic formalism for amenable group actions in the last section. In Chapter5, we consider the more general class of asymptotically sub-additive potentials for Zd-actions. Firstly, we give a definition of asymptotically sub-additive potentials for Zd-actions, and establish a variational principle for asymptotically sub-additive topological pressure. Then we give several nontrivial examples of sub-additive and asymptotically sub-additive potentials. Secondly, we obtain two characterizations of the topological pressure when the entropy function is upper semi-continuous. Final-ly, we establish the existence of equilibrium measures for arbitrary asymptotically sub-additive potentials. One hand, these results extend Ruelle and Misiurewicz’s results to asymptotically sub-additive case. On the other hand, we also generalize Feng-Huang’s results to general continuous Zd-actions.