α-limit sets and Lyapunov function for maps with one topological attractor

Abstract

We consider the topological behaviors of continuous maps with one topological attractor on compact metric space X. This kind of map is a generalization of maps such as topologically expansive Lorenz map, unimodal map without homtervals and so on. We provide a leveled A-R pair decomposition for such maps, and characterize α-limit set of each point. Based on weak Morse decomposition of X, we construct a bounded Lyapunov function V (x), which give a clear description of orbit behavior of each point in X except a meager set.