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

In this paper,we introduce the topological entropy of a free semigroup action gener-ated by proper maps on a topological space.We introduce two topological d-entropy of a free semigroup action generated by proper maps in Befetov and Bis sense on a metric space,which extends the notions of Patrao entropy,Bufetov entropy and Bis entropy.We give some properties of these notions and discuss the relation between them.We also give a partial variational principle for locally compact separable metric spaces.Moreover,the relationship between topological entropy of the free semigroup generated by proper maps and topological entropy of a skew-product transformation is given.These results extend the results obtained by Patrao[15],Bufetov[20],Bis and Lin,Ma and Wang[10].This paper is organized as follows:In Chapter 1,we introduce the development history,applications and the importance of entropy.Some notions what we need and definitions of entropy are given in preliminaries in Chapter2.In Chapter3,we introduce the topological entropy of a free semigroup action gener-ated by proper maps on a topological space and topological Bowen entropy and d-entropy of a free semigroup action generated by proper maps in Befutov sense on a metric space.We give some properties and the partial variational principle for locally compact separa-ble metric space.Moreover,we extend the new definition of topological entropy to the skew-product transformation and get a new skew-product topological entropy.In Chapter4,we introduce another d-entropy of a free semigroup action generated by proper maps in Bis sense on a metric space,which extend the definition of Bis entropy.The size of two topological d-entropy are also compared.