Asymptotically Stable Sets And Topological Sequence Entropy Of Graph Maps

Dynamic systems generated by graphs maps study mainly the topological structures and asymptotic properties of orbits of graph maps, etc. In recent years dynamical properties generated by graph maps have attracted extreme attention since there is a close; connection between dynamical properties of auto-homeomorphisms on surfaces and these of graph maps . In the field of graph maps, one has done many researches and obtained a series of remarkable results. Our basic objective in this paper is to study asymptotically stable sets and topological sequence entropy of graph maps. Three necessary and sufficient conditions for a fixed point to be asymptotically stable are obtained and the commutativity of topological sequence entropy is studied.In section 1, some notions about topological dynamical systems and graph maps are introduced.In section 2, asymptotically stable sets of tree maps are studied, and two necessary and sufficient conditions for a fixed point to be asymptotically stable are obtained. By discussing the relations between the unstable manifold of fixed point z of graph maps and asymptotically stable properties, we show a simple characterization of asymptotically stable fixed point of graph maps. It is proven that a fixed point z of a graph map is asymptotically stable if and only if W(z, f) = {z} and z is an isolate point of P(f).In section 3, the commutativity of topological sequence entropy of graph maps is studied. It trims out that hA(f o g)=hA(go f) for any sequence A = (ai)i=1 and any two graph maps f, g.