The Dynamics Of Tree Maps

Dynamical systems generated by tree maps study mainly the asymptotic properties and the topological structures of orbits of tree maps. In recent years, dynamical properties generated by tree maps have been attracted extreme attention. In the field of tree maps, people have done many researches and abtained a long series of remarkable results. In this paper, we study mainly the ω-limit sets of tree map and the topological entropy of commuting tree maps.In Chapter One, we introduce briefly historic background and basic notions of topo-logicl dynamical system and the background of this paper.In Chapter Two, we investigate mainly the properties of the ω-limit sets for the tree map g and the non-autonomous discrete-time dynamical system (f_n,T). We have that(1) If {f_n}_n=1^∞ converges uniformly to f and P(f) = F(f), then ω(x,f_n) is a connected closed subset of T containing only fixed points of f for any point x ∈T.(2)Λ(g)=∩_n=0^∞gⁿ(Ω(g)).(3) If x ∈ Λ(g) — P(g), then ω(x, g) is an infinite minimal set.In Chapter Three, we discuss mainly the topological entropy of commuting tree maps. We show thatIf f and g are commuting tree maps and there are a periodic point x ∈ T of g and some natural number m ≥ 2 with (m, l) = 1 for any 2 ≤l ≤ s, such that x is a km-periodic point of f for any k ∈ N, where s is the number of the end points of T. Then f o g has positive topological entropy.