On The Topological Pressure Of The Free Semigroup Action Generated By Proper Maps

In this paper,we introduce the topological pressure of a free semigroup action generated by proper maps on a metric space in the sense of Bufetov and Bi?s,which is not only the generalization of Patr?ao entropy,but also the extension of Bufetov entropy and Bi?s entropy.We obtain that the topological pressure of a free semigroup action generated by proper maps is equivalent to the topological pressure on its one-point compactification space.According to this relation,we obtain the partial variational principle of the topological pressure on the locally compact separable space.Then,considering the relationship between the topological pressure of a free semigroup action and the skew-product transformation,taking the topological pressure in the sense of Bufetov as an example,the properties of topological pressure of the skew-product transformation are discussed.Finally,the two definitions of topological pressure are compared.This thesis includes four parts as follows:In Chapter1,we introduce the development?research status and importance of topological pressure.In Chapter2,the preliminaries including some notions what we need are given.In Chapter3,we give the definition of topological pressure of a free semigroup action generated on proper maps on a metric space in the sense of Bufetov.Then we give equivalent notions by using spanning set and separated set.Furthermore,we obtain the topological pressure of a free semigroup action generated by proper maps is equivalent to the topological pressure on its one-point compactification space also we give the partial variational principle and some properties on locally compact separable metric space.Moreover,we extend the new definition of topological pressure to the skew-product transformation and get a new skew-product topological pressure.In Chapter4,we define a topological pressure of a free semigroup action generated on proper maps in the sense of Bi?s on a metric space,which is the extension of Bi?s entropy.Furthermore,we compare the size of the two topological pressure.