Katok's Formulae And Box Dimensions In Some Dynamical Systems

The notions of measure-theoretic entropy and topological entropy constitute im-portant invariants in the characterization of complexity of a dynamical system.The re-lationship between these two quantities is the well-known variational principle.Brin-Katok formula and Katok's entropy formula are two important entropy formulae in the study of entropy theory.Topological pressure is a generalization of topological entropy.The dimension of invariant sets is also among the most important character-istics of dynamical systems.This thesis is devoted to studying the Katok's formulae and box dimensions in dynamical systems.Particularly,we establish some versions of Brin-Katok formula and Katok's entropy formula in a topological dynamical system as well as a partially hyperbolic system,and for inhomogeneous self-conformal sets with overlaps,we give the upper box dimension formula and lower box dimension formula.The organization of this thesis is as follows:In Chapter 1,we give a conditional version of Brin-Katok formula and as well as a conditional version of Katok's entropy formula in mean metrics over a topological dynamical system.In Chapter 2,using spanning sets in mean metrics,we introduce a definition of measure-theoretic pressure of additive potentials for ergodic measures over a topo-logical dynamical system.We also give a new definition of topological pressure by replacing the Bowen metrics with the corresponding mean metrics.And we establish a pressure version of Katok's entropy formula in mean metrics.Using this formula,we prove that the topological pressure defined in mean metrics is equivalent to the classical topological pressure defined in Bowen metrics.Finally,we obtain that the topological pressure defined in mean metrics satisfies the variational principle.In Chapter 3,we devote to establishing the Katok's entorpy formula of unstable metric entropy for partially hyperbolic diffeomorphisms.In Chapter 4,we construct the Katok's entropy formula of unstable metric en-tropy in mean-u metrics which are the mean metrics on the unstable manifold.And we also give the definition of unstable topological entropy in mean-u metrics for partial-ly hyperbolic diffomorphisms and prove that the newly defined unstable topological entropy coincides with the unstable topological entropy defined in[24].Finally,we establish the variational principle relating the unstable metric entropy and the unstable topological entropy defined in mean-u metrics.In Chapter 5,we investigate the box dimensions of inhomogeneous self-conformal sets with overlaps.Firstly,we significantly generalize the result of[18]concerning the upper box dimension under some mild overlap conditions.Second-ly,we give the corresponding formula of lower box dimension for inhomogeneous attractor under some conditions.These results extend the Bowen's equation for inho-mogeneous iterated function system.