Topological Entropy Of Z~d-action And Some Related Problems

In this thesis,we mainly study topological entropy of Z^d actions.More precisely:In the introduction,the history and some recently progresses are introduced,and we give a brief introduction about what we have done in this thesis.In Chapter 2,some related notions and properties on topological dynamical system and dimension theory are introduced.In Chapter 3,for Z-action,some basic notions and properties of some classical types of topological entropy are summarized.In Chapter 4,we study some types of topological entropy under Z^d-actions: Bowen dimensional entropy,Packing dimensional entropy and Bowen set entropy.For Bowen dimensional entropy,we proved that the Bowen dimension entropy of subset can be estimated by the lower measure local entropy of the points in this set: Let m be a Borel probability on X,E be a Borel subset of X and 0?s??.?1?If h_??x??s for all x?E,then h_top^B?E??s.?2?If h_??x??s for all x?E and ??E??0,then h_top^B?s.For Packing dimensional entropy,we have a similarly result,we proved that the packing dimension entropy of subset can be estimated by the upper measure local entropy of the points in this set: Let m be a Borel probability on X,E be a Borel subset of X and 0For Packing dimensional entropy,we have a similarly result,we proved that the packing dimension entropy of subset can be estimated by the upper measure local entropy of the points in this set: Let m be a Borel probability on X,E be a Borel subset of X and 0?s??,then If h*?x??s for all x?E,then h_top^B?E??s.Moreover,the relationship of Bowen dimension entropy,Packing dimension entropy and Bowen set entropy are studied.In the last Chapter,we give a summary for this paper and list several questions for further study.