LAUSR

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. Under the finiteness and basin conditions, we provide a leveled A-R pair decomposition for such maps, and characterize β -limit set of each point. Based on weak Morse decomposition of X , we construct a bounded Lyapunov function V ( x ), which gives a clear description of orbit behavior of each point in X except a meager set.