The Variational Principle Of The Billingsley Dimension And The Localized Topological Pressure Of The Saturated Set

In this thesis, we investigate some problems of the Billingsley di-mension of noncompact sets, the localized topological pressure and Katok formula for weighted topological pressure in topological dy-namical systems. In the first part of the thesis, we define some kinds of topological pressure for countable state Markov shifts. In the sec-ond part, we define the localized pressure using spanning set and give the relative variational principle for it. In the third part, we recall the notion of the weighted pressure and prove the Katok formula for weighted pressure. The paper is organized as follows:In Chapter 1, we introduce the backgrounds of topological entropy, topological pressure and dynamic systems.In Chapter 2, we recall some classical definitions in topological dynamical systems, ergodic theory and symbolic dynamics.In Chapter 3, we define the saturated set for symbolic space and estimate the lower bound of the Billingsley dimension for the satu-rated set.In Chapter 4. we recall the definition of the localized pressure and obtain the variational principle for the localized pressure.In Chapter 5, we recall the definition of weighted pressure and give the Katok formula of weighted version.