• Previous Article
    Global solutions to the Chemotaxis-Navier-Stokes equations with some large initial data
  • DCDS Home
  • This Issue
  • Next Article
    Persistence and stationary distribution of a stochastic predator-prey model under regime switching
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 - A, 2017, 37 (5) : 2861-2879. doi: 10.3934/dcds.2017123
References:
[1]

M. Benaim, 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, 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, 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, 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, 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, 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, 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, L. Wen, $C^1$-stably shadowable chain components are hyperbolic, J. Differential Equations, 246 (2009), 340-357.

show all references

References:
[1]

M. Benaim, 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, 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, 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, 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, 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, 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, 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, L. Wen, $C^1$-stably shadowable chain components are hyperbolic, J. Differential Equations, 246 (2009), 340-357.

[1]

Michael Brin, Dmitri Burago, Sergey Ivanov. Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus. Journal of Modern Dynamics, 2009, 3 (1) : 1-11. doi: 10.3934/jmd.2009.3.1

[2]

Sergey Kryzhevich, Sergey Tikhomirov. Partial hyperbolicity and central shadowing. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2901-2909. doi: 10.3934/dcds.2013.33.2901

[3]

Yujun Zhu. Topological quasi-stability of partially hyperbolic diffeomorphisms under random perturbations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 869-882. doi: 10.3934/dcds.2014.34.869

[4]

Keith Burns, Amie Wilkinson. Dynamical coherence and center bunching. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1/2) : 89-100. doi: 10.3934/dcds.2008.22.89

[5]

Antonio Algaba, Estanislao Gamero, Cristóbal García. The reversibility problem for quasi-homogeneous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3225-3236. doi: 10.3934/dcds.2013.33.3225

[6]

Laurent Boudin, Francesco Salvarani. The quasi-invariant limit for a kinetic model of sociological collective behavior. Kinetic & Related Models, 2009, 2 (3) : 433-449. doi: 10.3934/krm.2009.2.433

[7]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355

[8]

Yakov Pesin. On the work of Dolgopyat on partial and nonuniform hyperbolicity. Journal of Modern Dynamics, 2010, 4 (2) : 227-241. doi: 10.3934/jmd.2010.4.227

[9]

Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Raúl Ures. Partial hyperbolicity and ergodicity in dimension three. Journal of Modern Dynamics, 2008, 2 (2) : 187-208. doi: 10.3934/jmd.2008.2.187

[10]

Jérôme Buzzi, Todd Fisher. Entropic stability beyond partial hyperbolicity. Journal of Modern Dynamics, 2013, 7 (4) : 527-552. doi: 10.3934/jmd.2013.7.527

[11]

David Burguet, Todd Fisher. Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2253-2270. doi: 10.3934/dcds.2013.33.2253

[12]

Qiangchang Ju, Hailiang Li, Yong Li, Song Jiang. Quasi-neutral limit of the two-fluid Euler-Poisson system. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1577-1590. doi: 10.3934/cpaa.2010.9.1577

[13]

Lu Yang, Meihua Yang, Peter E. Kloeden. Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2635-2651. doi: 10.3934/dcdsb.2012.17.2635

[14]

Ugo Locatelli, Letizia Stefanelli. Quasi-periodic motions in a special class of dynamical equations with dissipative effects: A pair of detection methods. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1155-1187. doi: 10.3934/dcdsb.2015.20.1155

[15]

Mikhail B. Sevryuk. Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2/3) : 569-595. doi: 10.3934/dcds.2007.18.569

[16]

Junxiang Xu. On quasi-periodic perturbations of hyperbolic-type degenerate equilibrium point of a class of planar systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2593-2619. doi: 10.3934/dcds.2013.33.2593

[17]

Wen-Rong Dai. Formation of singularities to quasi-linear hyperbolic systems with initial data of small BV norm. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3501-3524. doi: 10.3934/dcds.2012.32.3501

[18]

Priyanjana M. N. Dharmawardane. Decay property of regularity-loss type for quasi-linear hyperbolic systems of viscoelasticity. Conference Publications, 2013, 2013 (special) : 197-206. doi: 10.3934/proc.2013.2013.197

[19]

Raquel Ribeiro. Hyperbolicity and types of shadowing for $C^1$ generic vector fields. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2963-2982. doi: 10.3934/dcds.2014.34.2963

[20]

Andy Hammerlindl. Partial hyperbolicity on 3-dimensional nilmanifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3641-3669. doi: 10.3934/dcds.2013.33.3641

2016 Impact Factor: 1.099

Metrics

  • PDF downloads (1)
  • HTML views (2)
  • Cited by (0)

Other articles
by authors

[Back to Top]