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.
Yiming DING
,
Yun SUN
. α-LIMIT SETS AND LYAPUNOV FUNCTION FOR MAPS WITH ONE TOPOLOGICAL ATTRACTOR[J]. Acta mathematica scientia, Series B, 2022
, 42(2)
: 813
-824
.
DOI: 10.1007/s10473-022-0225-6
[1] Auslander J, Bhatia N P, Seibert P. Attractors in dynamical systems. Bol Soc Mat Mex, 1964, 9:55-66
[2] Batko B, Mrozek M. Weak index pairs and the Conley index for discrete multivalued dynamical systems. SIAM J Appl Dyn Syst, 2016, 15(2):1143-1162
[3] Balibrea F, Guirao J, Lampart M. A note on the definition of α-limit set. Appl Math Inf Sci, 2013, 7(5):1929-1932
[4] Bruin H, Keller G, Nowicki T, van Strien S. Wild Cantor attractors exist. Ann Math, 1996, 143(1):97-130
[5] Conley C. Isolated invariant sets and the Morse index//Regional Conference Series in Mathematics Vol 38. Am Math Soc, 1978
[6] Coddington E A, Levinson N. Theory of ordinary differential equations. Tata Mcgraw-Hill Education, 1955
[7] Collet P, Eckmann J P. Concepts and results in chaotic dynamics:a short course//Theoretical and Mathematical Physics. Berlin:Springer-Verlag, 2006
[8] Cui H F, Ding Y M. The α-limit sets of a unimodal map without homtervals. Topology Appl, 2010, 157(1):22-28
[9] Cui H F, Ding Y M. Renormalizaiton and conjugacy of piecewise linear Lorenz maps. Adv Math, 2015, 271:235-272
[10] Ding Y M. Renormalziation and α-limit set of expanding Lorenz map. Discrete Contin Dyn Syst, 2011, 29(3):979-999
[11] Ding Y M, Fan W T. The asymptotic periodicity of Lorenz maps. Acta Math Sci, 1999, 19B(1):114-120
[12] Franks J, Richeson D. Shift equivalence and the Conley index. Trans Amer Math Soc, 2000, 352(7):3305- 3322
[13] Good C, Meddaugh J, Mitchell J. Shadowing, internal chain transitivity and α-limit sets. J Math Anal Appl, 2020, 491(1):124291, 19 pp
[14] Guckenheimer J, Holmes P. Nonlinear oscillations, dyanmcial systems, and bifurcations of vector fields//Appl Math Sci Vol 42. New York:Springer-Verlag, 1983
[15] Graczyk J, Kozlovski O. On Hausdorff dimension of unimodal atttractors. Commun Math Phys, 2006, 264(3):565-581
[16] Glendinning P, Sparrow C. Prime and renormalizable kneading invariants and the dynamics of expanding Lorenz maps. Phys D, 1993, 62(1/4):22-50
[17] Hubbard J, Sparrow C. The classification of topologically expansive Lorenz maps. Comm Pure Appl Math, 1990, 43(4):431-443
[18] Kaczynski T, Mrozek M. Conley index for discrete multi-valued dynamical systems. Topology Appl, 1995, 65(1):83-96
[19] Li S M, Shen W X. Hausdorff dimension of Cantor attractors in one-dimensional dynamics. Invent Math, 2008, 171(2):345-387
[20] Liu Z X. Conley index for random dynamical systems. J Diffrential Equations, 2008, 244(7):1603-1628
[21] Milnor J. On the concepts of attractor//The Theory of Chaotic Attractors. New York:Springer, 1985:243-264
[22] Mischaikow K. The conley index theory:a brief introduction. Banach Center Publ, 1999, 47(1):9-19
[23] Mendelosn P. On unstable attractors. Bol Soc Mat Mex, 1960, 5:270-276
[24] Mane R. Hyperbolicity, sinks and measure in one-dimensional dynamics. Comm Math Phys, 1985, 100(4):495-524
[25] Smale S. Differential dynamical systems. Bull Amer Math Soc, 1967, 73:747-817
[26] Wang J T. On the theory of Conley index and shape Conley index in general metric spaces[D]. Tianjin:Tianjin University, 2016
[27] Williams R F. The zeta function of an atrractor//Conference on the Topology of Manifolds. Boston:Prindle Weber and Schmidt, 1968:155-161
