Entropy And Weak Horseshoe With Bounded-gap-hitting Times

In this thesis,we mainly study weak horseshoe with bounded-gap-hitting times.The thesis is organized as follows:In the introduction,we introduce some background and main results of our study.In chapter 2,some basic knowledge of topological dynamical system and ergodic theory are presented,including topological entropy and independent set etc.In chapter 3,we discuss the main results and their proofs in detail.For a flow(M,?),it is shown that if the time one map ?1 has weak horseshoe with bounded-gap-hitting times,so does ?? for all ??0.In chapter 4,we prove that for an affine homeomorphsim of a compact metric abelian group,it has positive topological entropy if and only if it has weak horseshoe with bounded-gap-hitting times.We also prove that if a topological dynamical sys-tem has a weak horseshoe with bounded-gap-hitting times,then it has infinitely many minimal sets.Finally,we introduce some related open problems.