Research On Some Point Sets Of Graph Maps And Non-Autonomous Dynamical Systems

Recently, research on graph maps and non-autonomous dynamical systems has attracted the attention of a considerable number of scholars. Some properties of graph maps on the chain recurrent sets, the special α-limit sets and the α-limit sets in the non-autonomous dynamical systems are studied in this thesis.In chapter3, some properties on the chain recurrent sets of graph maps are discussed, and two conclusions are given as follows:(1) Let G be a graph, f:Gâ†'G be a continuous map. If P(f) is a nonempty closed set and for every x∈G-P(f), there is a point in ω(x,f) which has a generalized attracting neighborhood containing no circle, then CR(f)=P(f).(2) Let G be a graph, f:Gâ†'G be a continuous map. If P(f) is a nonempty closed set and for every y∈P(f), y has a generalized attracting neighborhood containing no circle, then CR(f)=AP(f).In chapter4, the relationships among the topological entropy, the recurrent sets and the special α-limit sets of graph maps are explored, an equivalent characterization of graph maps with positive topological entropy is given:(3) Let G be a graph and f:Gâ†'G be a continuous map, then h(f)>0if and only if SA(f)-R(f)≠0.In chapter5, the related definitions of non-autonomous dynamical systems are introducted, and some properties of the autonomous dynamical systems are extended to non-autonomous dynamical systems, then the following conclusion is given:(4)Let f,g∈C∞([0,1]),<x-n>∞n=0be a negative trajectory in [0,1] with[f,g], then a(<x-n>∞n=0,[f,g) is locally expanding with [f,g].