The Mixing Properties On Maps Of Warsaw Circle And Dense Chaos Of Tree Maps

In this paper we mainly study the mixing properties on maps of warsaw circle and dense chaos of tree maps.In Chapter One, we introduce simply the development about topological dynamical system and the background of this paper.In Chapter Two, we study mainly the mixing properties on maps of Warsaw circle W. For a coutinuous map f: W→W, we prove the following results:(1) f is topological transitive if and only if f is chaotic in the sense Devaney; (2) f is topological transitive if and only if f is mixing; (3) f is topological transitive implies f has a horseshoe; (4) f is topological transitive implies f have periodic points of n period for each integer n.In Chapter Three, we study mainly dense chaos of tree maps. Let T be a tree and f : T→T be a continuous map, we show the following four conditions are equivalent:(1) f is generically chaotic, (2) f is genericallyδ—chaotic for someδ> 0, (3) f is denselyδ- chaotic for someδ> 0, (4) either there exists a unique transitive closed non degenerate connected set or there exist k(k≥2) transitive closed non degenerate connected components having a common endpoint; moreover, if J is a non degenerate connected set then f(J) is non degenerate, and there exist a transitive connected setâ… _o and an integer n such that fⁿ(J) n Int(â… ₀)≠φ.