Acta mathematica scientia,Series B ›› 2022, Vol. 42 ›› Issue (2): 813-824.doi: 10.1007/s10473-022-0225-6

• Articles • Previous Articles    

α-LIMIT SETS AND LYAPUNOV FUNCTION FOR MAPS WITH ONE TOPOLOGICAL ATTRACTOR

Yiming DING, Yun SUN   

  1. Center for Mathematical Sciences, School of Science, Wuhan University of Technology, Wuhan 430070, China
  • Received:2020-12-28 Revised:2021-05-12 Online:2022-04-25 Published:2022-04-22
  • Supported by:
    This work was supported by the National Key Research and Development Program of China (2020YFA0714200), and also supported by the Excellent Dissertation Cultivation Funds of Wuhan University of Technology (2018-YS-077).

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. Under the finiteness and basin conditions, 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 gives a clear description of orbit behavior of each point in X except a meager set.

Key words: Attracting set, leveled A-R pair decomposition, α-limit set, Lyapunov function

CLC Number: 

  • 37B25
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