Induced Topological Pressure For Topological Dynamical Systems

In this thesis, we define the induced topological pressure, the induced measure-theoretic entropy for compact dynamical systems and study their properties.We organize the thesis as follows.In Preface, we introduce some background about the study of the induced topo-logical pressure.In Chapter 1, some preliminary notions and results in ergodic theory and topo-logical dynamics, which will be used in the thesis, are reviewed.In Chapter 2, we define the induced topological pressure for compact dynamical systems and consider the relation between it and the topological pressure. Basing on this, we set up a variational principle for the induced topological pressure. As its application, we point out that the BS dimension is a special case of the induced topological pressure. We also consider the problem of the existence of equilibrium measures for the induced topological pressure.In Chapter 3, we study the relation for two induced topological pressures un-der a factor map. Specifically speaking, let Ï€:(X, f) â†' (Y, g) be a factor map between dynamical systems, i.e., a continuous surjective mapping from X onto Y compatible with the actions on X and on Y. We study the relation for two induced topological pressures between dynamical systems. As an application, we consider the zero-dimensional extension for BS dimensions.In Chapter 4, we define a topological version of the induced measure-theoretic entropy and obtain its Katok entropy formula. As its application, we show the in-duced measure-theoretic entropy coincides with Hausdorff dimension of measure in a symbolic space.In Chapter 5, we define the induced Gurevich pressure of almost-additive poten-tials for countable state Markov shifts and obtain its variational principle.