Some Properties About Topological Entropy Of Free Semigroup Action

In this thesis, we are concerned with some properties of the topological entropy for free semigroup action. Bufetov (1997, J. Dynam. Control Systems) provided a definition for this entropy using spanning sets and separated sets, we gave another equivalent defini-tion of the topological entropy by open covers. Next, we got some basic properties such as topological conjugacy and power rule of entropy and a quantitative analogue of Bowen’s theorem for semiconjugacy. This paper also compared the entropies presented by Bufetov and Bis (2004, Discrete Contin. Dyn. Syst.). Finally, we studied the relationship between the entropy and skew-product.This thesis is organized as follows:In Chapter 1, the history and the development of a free semigroup action and dy-namical system of topological entropy were presented.In Chapter 2, we introduced some definitions and properties of semigroup action, symbolic space, and topological entropy.In Chapter 3, by studying the topological entropy of free semigroup defined by span-ning sets and separated sets, we showed that such entropy can be defined by open covers. Next, we proved that such entropy equals the topological entropy defined by spanning sets and separated sets. Finally, we presented an example for computing topological entropy of free semigroup action which is generated by symbolic space{0,1}N.In Chapter 4, we gave an inequalityh(Fk)≤kh(F) concerning the power rule of the topological entropy of free semigroup action, where h(F) is topological entropy of free semigroup generated by m continuous maps and Fk={g1(?)g1(?)…(?)gk:gi∈F,1≤i≤k}, and we gave a counterexample indicating the equality is not always true. Moreover, we proved the topological entropy of free semigroup is conjugacy invariant. A relationship between two semiconjugate systems’entropies analogue to Bowen’s theorem was obtained. Bufetov defined entropy by summing up the complexities of the iterations of every possible path in finite time and taking the average and Bis defined it by calculating the complexity of all possible paths in finite time. By comparing the entropies presented by them, the entropy defined by Bis is bigger than another. Finally, we studied the relationships among entropy, an invariant set of symbolic space and skew-product transformation.