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

On the limit quasi-shadowing property

Fang Zhang 1, and  Yunhua Zhou 2,,

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, 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.  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

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.  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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