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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 |
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.
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Asymptotic pseudotrajectories and chain recurrent flows, with applications, J. Dynam. Diff. Equat., 8 (1996), 141-176.
doi: 10.1007/BF02218617. |
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D. Bohnet and C. Bonatti,
Partially hyperbolic diffeomorphisms with uniformly center foliation: The quotient dynamics, Ergodic Theory and Dynamical Systems, 36 (2015), 1067-1105.
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Limit shadowing property, Numer. Funct. Anal. Optim, 18 (1997), 75-92.
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S. Gan,
A generalized shadowing lemma, Discrete Contin. Dyn. Syst., 8 (2002), 627-632.
doi: 10.3934/dcds.2002.8.627. |
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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. |
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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. |
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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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