Study Of Some Dynamical Properties On Topological Systems In One Dimension

In this paper we study mainly the topological structures of some invariant sets and entropy of continuous self-maps on Warsaw circle and unstable mainifolds and entropy of continuous self-maps on a tree.In Chapter One we introduce briefly the historic background and notions of topological system and some known results about topological system in one dimension.Warsaw circle often appears as an example of circle-like but not arc-like in the theory of continuum .There are many differences between the topological properties of it and that of interval or circle and there is essentially difference between dynamical properties of it and that of interval or circle.So,it is worthy of discussing the dynamical properties of a continuous self-map on Warsaw circle.In chapter two we discus som topological structures of som invariant sets of a continuous self-map on Warsaw circle,we obtain thatAnd proved that when R (/) or P (/) is closed, h (/) = 0.In recent years dynamical properties generated by tree maps have attracted extreme attention since there is a close connection between dynamical properties of auto-homeomorphisms on surfaces and these of tree maps.In the field of tree maps ,one has done many researches and obtained a series of remarkable results.In chapter three we discus some properties of unstable mainifold of continuous self-maps on a tree.we mainly prove that a necessary and sufficient condition that the topological entropy of a continuous self-map on a tree is positive is that it has a homoclinic point.