# American Institute of Mathematical Sciences

May  2018, 38(5): 2487-2503. doi: 10.3934/dcds.2018103

## Topological stability and spectral decomposition for homeomorphisms on noncompact spaces

 Department of Mathematics, Chungnam National University, Daejeon 305-764, Korea

* Corresponding author (yangyinong1201@gmail.com)

Received  August 2017 Published  March 2018

In this paper, we introduce the notions of expansiveness, shadowing property and topological stability for homeomorphisms on noncompact metric spaces which are dynamical properties and equivalent to the classical definitions in case of compact metric spaces. Then we extend the Walters's stability theorem and Smale's spectral decomposition theorem to homeomorphisms on locally compact metric spaces.

Citation: Keonhee Lee, Ngoc-Thach Nguyen, Yinong Yang. Topological stability and spectral decomposition for homeomorphisms on noncompact spaces. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2487-2503. doi: 10.3934/dcds.2018103
##### References:
 [1] N. Aoki, On homeomorphisms with pseudo-orbit tracing property, Tokyo J. Math., 6 (1983), 329-334.  doi: 10.3836/tjm/1270213874. [2] B. Carvalho and W. Cordeiro, N-expansive homeomorphisms with the shadowing property, J. Differential Equations, 261 (2016), 3734-3755.  doi: 10.1016/j.jde.2016.06.003. [3] N.-P. Chung and K. Lee, Topological stability and pseudo-orbit tracing property of group actions, Proc. Amer. Math. Soc., 146 (2018), 1047-1057. [4] W. Cordeiro, M. Denker and X. Zhang, On specification and measure expansiveness, Discrete Contin. Dyn. Syst., 37 (2017), 1941-1957. [5] T. Das, K. Lee, D. Richeson and J. Wiseman, Spectral decomposition for topologically Anosov homemorphisms on noncompact and non-metrizable spaces, Topology Appl., 160 (2013), 149-158.  doi: 10.1016/j.topol.2012.10.010. [6] M. Hurley, Chain recurrence, semiflows, and gradients, J. Dynam. Differential Equations, 7 (1995), 437-456.  doi: 10.1007/BF02219371. [7] K. Lee and C. A. Morales, Topological stability and pseudo-orbit tracing property for expansive measures, J. Differential Equations, 262 (2017), 3467-3487.  doi: 10.1016/j.jde.2016.04.029. [8] P. Oprocha, Chain recurrence in multidimensional time discrete dynamical systems, Discrete Conti. Dyn. Syst., 20 (2008), 1039-1056.  doi: 10.3934/dcds.2008.20.1039. [9] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1. [10] P. Walters, On the pseudo-orbit tracing property and its relationship to stability, Lecture Notes in Math., Springer, 668 (1978), 231-244.

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##### References:
 [1] N. Aoki, On homeomorphisms with pseudo-orbit tracing property, Tokyo J. Math., 6 (1983), 329-334.  doi: 10.3836/tjm/1270213874. [2] B. Carvalho and W. Cordeiro, N-expansive homeomorphisms with the shadowing property, J. Differential Equations, 261 (2016), 3734-3755.  doi: 10.1016/j.jde.2016.06.003. [3] N.-P. Chung and K. Lee, Topological stability and pseudo-orbit tracing property of group actions, Proc. Amer. Math. Soc., 146 (2018), 1047-1057. [4] W. Cordeiro, M. Denker and X. Zhang, On specification and measure expansiveness, Discrete Contin. Dyn. Syst., 37 (2017), 1941-1957. [5] T. Das, K. Lee, D. Richeson and J. Wiseman, Spectral decomposition for topologically Anosov homemorphisms on noncompact and non-metrizable spaces, Topology Appl., 160 (2013), 149-158.  doi: 10.1016/j.topol.2012.10.010. [6] M. Hurley, Chain recurrence, semiflows, and gradients, J. Dynam. Differential Equations, 7 (1995), 437-456.  doi: 10.1007/BF02219371. [7] K. Lee and C. A. Morales, Topological stability and pseudo-orbit tracing property for expansive measures, J. Differential Equations, 262 (2017), 3467-3487.  doi: 10.1016/j.jde.2016.04.029. [8] P. Oprocha, Chain recurrence in multidimensional time discrete dynamical systems, Discrete Conti. Dyn. Syst., 20 (2008), 1039-1056.  doi: 10.3934/dcds.2008.20.1039. [9] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1. [10] P. Walters, On the pseudo-orbit tracing property and its relationship to stability, Lecture Notes in Math., Springer, 668 (1978), 231-244.
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