On the limit quasi-shadowing property

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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.

Keywords: limit quasi-shadowing, quasi-partially hyperbolic pseudoorbit, partial hyperbolicity, dynamical coherence.

Mathematics Subject Classification: Primary: 37C50; Secondary: 37D30.

Citation: Fang Zhang, Yunhua Zhou. On the limit quasi-shadowing property. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2861-2879. doi: 10.3934/dcds.2017123

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.
[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.
[3]	R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms Lect. Notes in Math. 470, Springer, 1975. doi: 10.1007/BFb0081279.
[4]	T. Eirola, O. Nevanlinna and S. Pilyugin, Limit shadowing property, Numer. Funct. Anal. Optim, 18 (1997), 75-92. doi: 10.1080/01630569708816748.
[5]	S. Gan, A generalized shadowing lemma, Discrete Contin. Dyn. Syst., 8 (2002), 627-632. doi: 10.3934/dcds.2002.8.627.
[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.
[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.
[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.
[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.
[10]	A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Pub. Math. IHES, 51 (1980), 137-173. doi: 10.1007/BF02684777.
[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.
[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.
[13]	S. Liao, An existence theorem for periodic orbits, Acta Sci. Natur. Univ. Pekinensis, 1 (1979), 1-20.
[14]	Ya. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity Zurich Lectures in Advanced Mathematics, 2006. doi: 10.4171/003.
[15]	S. Pilyugin, Shadowing in dynamical systems, Lec. Notes in Math. , 1706, Springer-Verlag, 1999.
[16]	X. Wen, S. Gan and L. Wen, $C^1$-stably shadowable chain components are hyperbolic, J. Differential Equations, 246 (2009), 340-357.

show all references

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.
[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.
[3]	R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms Lect. Notes in Math. 470, Springer, 1975. doi: 10.1007/BFb0081279.
[4]	T. Eirola, O. Nevanlinna and S. Pilyugin, Limit shadowing property, Numer. Funct. Anal. Optim, 18 (1997), 75-92. doi: 10.1080/01630569708816748.
[5]	S. Gan, A generalized shadowing lemma, Discrete Contin. Dyn. Syst., 8 (2002), 627-632. doi: 10.3934/dcds.2002.8.627.
[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.
[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.
[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.
[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.
[10]	A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Pub. Math. IHES, 51 (1980), 137-173. doi: 10.1007/BF02684777.
[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.
[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.
[13]	S. Liao, An existence theorem for periodic orbits, Acta Sci. Natur. Univ. Pekinensis, 1 (1979), 1-20.
[14]	Ya. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity Zurich Lectures in Advanced Mathematics, 2006. doi: 10.4171/003.
[15]	S. Pilyugin, Shadowing in dynamical systems, Lec. Notes in Math. , 1706, Springer-Verlag, 1999.
[16]	X. Wen, S. Gan and L. Wen, $C^1$-stably shadowable chain components are hyperbolic, J. Differential Equations, 246 (2009), 340-357.

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