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May  2017, 37(5): 2861-2879. doi: 10.3934/dcds.2017123

## On the limit quasi-shadowing property

 1 Chongqing College of Humanities Science and Technology, Chongqing, 401524, China 2 College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

* Corresponding author

Received  November 2014 Revised  December 2016 Published  February 2017

Fund Project: The second author is supported by NSFC (11471056) and Foundation and Frontier Research Program of Chongqing (cstc2016jcyjA0312).

In this paper, we study the limit quasi-shadowing property for diffeomorphisms. We prove that any quasi-partially hyperbolic pseudoorbit $\{x_{i},n_{i}\}_{i∈ \mathbb{Z}}$ can be $\mathcal{L}^p$-, limit and asymptotic quasi-shadowed by a points sequence $\{y_{k}\}_{k∈ \mathbb{Z}}$. We also investigate the $\mathcal{L}^p$-, limit and asymptotic quasi-shadowing properties for partially hyperbolic diffeomorphisms which are dynamically coherent.

Citation: Fang Zhang, Yunhua Zhou. On the limit quasi-shadowing property. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2861-2879. doi: 10.3934/dcds.2017123
##### References:
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##### References:
 [1] M. Benaim and M. Hirsch, Asymptotic pseudotrajectories and chain recurrent flows, with applications, J. Dynam. Diff. Equat., 8 (1996), 141-176.  doi: 10.1007/BF02218617.  Google Scholar [2] D. Bohnet and C. Bonatti, Partially hyperbolic diffeomorphisms with uniformly center foliation: The quotient dynamics, Ergodic Theory and Dynamical Systems, 36 (2015), 1067-1105.  doi: 10.1017/etds.2014.102.  Google Scholar [3] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms Lect. Notes in Math. 470, Springer, 1975. doi: 10.1007/BFb0081279.  Google Scholar [4] T. Eirola, O. Nevanlinna and S. Pilyugin, Limit shadowing property, Numer. Funct. Anal. Optim, 18 (1997), 75-92.  doi: 10.1080/01630569708816748.  Google Scholar [5] S. Gan, A generalized shadowing lemma, Discrete Contin. Dyn. Syst., 8 (2002), 627-632.  doi: 10.3934/dcds.2002.8.627.  Google Scholar [6] S. Gan, The star systems $\mathcal{X}^*$ and a proof of the $C^1$ Ω-stability conjecture for flows, J. Differential Equations, 163 (2000), 1-17.   Google Scholar [7] M. Hirsch, Asymptotic phase, shadowing and reaction-diffusion systems, In: Differential Equations, Dynamical Systems, and Control Science. Lect. Notes in Pure and Applied Math. , Marcel Dekker Inc. New York, Basel, Hong Kong, 152 (1994), 87–99. Google Scholar [8] H. Hu, Y. Zhou and Y. Zhu, Quasi-shadowing for partially hyperbolic diffeomorphisms, Ergodic Theory Dynam. Systems, 35 (2015), 412-430.  doi: 10.1017/etds.2014.126.  Google Scholar [9] H. Hu and Y. Zhu, Quasi-stability of partially hyperbolic diffeomorphisms, Trans. Amer. Math. Soc., 366 (2014), 3787-3804.  doi: 10.1090/S0002-9947-2014-06037-6.  Google Scholar [10] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Pub. Math. IHES, 51 (1980), 137-173.  doi: 10.1007/BF02684777.  Google Scholar [11] S. Kryzhevich and S. Tikhomirov, Partial hyperbolicity and central shadowing, Discrete Contin. Dyn. Syst., 33 (2013), 2901-2909.  doi: 10.3934/dcds.2013.33.2901.  Google Scholar [12] C. Liang, W. Sun and X. Tian, Ergodic properties of invariant measures for $C^{1+α}$ non-uniformly hyperbolic systems, Ergodic Theory Dynam. Systems, 33 (2013), 560-584.   Google Scholar [13] S. Liao, An existence theorem for periodic orbits, Acta Sci. Natur. Univ. Pekinensis, 1 (1979), 1-20.   Google Scholar [14] Ya. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity Zurich Lectures in Advanced Mathematics, 2006. doi: 10.4171/003.  Google Scholar [15] S. Pilyugin, Shadowing in dynamical systems, Lec. Notes in Math. , 1706, Springer-Verlag, 1999. Google Scholar [16] X. Wen, S. Gan and L. Wen, $C^1$-stably shadowable chain components are hyperbolic, J. Differential Equations, 246 (2009), 340-357.   Google Scholar
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