The Variational Principle Of V-Statistics And A Conditional Variational Principle In Dynamical Systems

This article is divided into three parts. In the first part, we will review the definition of topological pressure in Pesin and Pitskel [24] and give a definition of saturated with respect to φ, which is a continuous function on X. Then we study the topological pressure of the limit of V-statistics taking Φ as kernel and get the variational principle. In the second part, we give the variational principle for discontinuous potentials. In the third part, we are devoted to study a topological dynamical system (X, f) with almost specification and give a conditional variational principle. Moreover, our result is applicable for β-shifts.The paper is organized as follows:In Chapter 1, the backgrounds of topological entropy, topological pressure, multifractal analysis and V-statistics are introduced and the main results are given.In Chapter 2, we review the definition of topological pressure and give a defi-nition of saturated with respect to φ. At last a variational principle is given.In Chapter 3, we introduce some important results and give the variational principle for discontinuous potentials.In Chapter 4, we give a conditional variational principle and an application for β-shifts.