Research Of Equicontinuity And Measure's Dimension

Equicontinuity is a much strong form of stability in topological dynamics. It plays a very important role in studying maps' sensitivety to initial conditions, topological transitivity, and minimal set etc. The property of the invariant set and measure's dimension of the IFS are main objects in the studying of fractal geometry. At the beginning of this paper we are devoted to studying the equicontinuity on the set of periodic points. Later we are researching on the property of the invariant set and measure's dimension of the self-similiar IFS. There are three parts:In chapter one, we discuss the equicontinuity on the set of periodic points of self-maps denned on ordinary compact metric space. We generalize the corresponding results of Zhou [1] and Yang [2]. Later we give a necessary condition and a sufficient condition for the self-maps which have the property of equicontinuity on their sets of periodic points, i.e. theorem 1.3.1 and theorem 1.4.1.In chapter two, by studying the properties of self-maps on trees, we obtain five necessary and sufficient conditions for the product self-maps on trees which have the property of equicontinuity on their sets of periodic points, i.e. theorem 2.3.1, theorem 2.3.2, theorem 2.3.3 and theorem 2.3.4. During the proofs we find easily that all conclusions in this chapter can be generalized to finite product self-maps on trees.In chapter three, For the similar interated function system {/j, /2, ??? fm} satis-fying open set condition, a continuous map on a full measure subset of the similar set A is given and a necessary and sufficient condition for the maximal entropy measure's equality to the maximal dimension measure is proved. Therefore, we generalize the corresponding conclusion in paper [3].