-
Previous Article
Persistence and stationary distribution of a stochastic predator-prey model under regime switching
- DCDS Home
- This Issue
-
Next Article
Global solutions to the Chemotaxis-Navier-Stokes equations with some large initial data
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.
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.
|
| [1] |
Zhiping Li, Yunhua Zhou. Quasi-shadowing for partially hyperbolic flows. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2089-2103. doi: 10.3934/dcds.2020107 |
| [2] |
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 |
| [3] |
Sergey Kryzhevich, Sergey Tikhomirov. Partial hyperbolicity and central shadowing. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2901-2909. doi: 10.3934/dcds.2013.33.2901 |
| [4] |
Yujun Zhu. Topological quasi-stability of partially hyperbolic diffeomorphisms under random perturbations. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 869-882. doi: 10.3934/dcds.2014.34.869 |
| [5] |
Keith Burns, Amie Wilkinson. Dynamical coherence and center bunching. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 89-100. doi: 10.3934/dcds.2008.22.89 |
| [6] |
Antonio Algaba, Estanislao Gamero, Cristóbal García. The reversibility problem for quasi-homogeneous dynamical systems. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3225-3236. doi: 10.3934/dcds.2013.33.3225 |
| [7] |
Tuhin Ghosh, Karthik Iyer. Cloaking for a quasi-linear elliptic partial differential equation. Inverse Problems and Imaging, 2018, 12 (2) : 461-491. doi: 10.3934/ipi.2018020 |
| [8] |
Laurent Boudin, Francesco Salvarani. The quasi-invariant limit for a kinetic model of sociological collective behavior. Kinetic and Related Models, 2009, 2 (3) : 433-449. doi: 10.3934/krm.2009.2.433 |
| [9] |
Harry Crimmins. Stability of hyperbolic Oseledets splittings for quasi-compact operator cocycles. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2795-2857. doi: 10.3934/dcds.2022001 |
| [10] |
Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355 |
| [11] |
Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 0: 331-348. doi: 10.3934/jmd.2020012 |
| [12] |
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 |
| [13] |
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 |
| [14] |
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 |
| [15] |
David Burguet, Todd Fisher. Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2253-2270. doi: 10.3934/dcds.2013.33.2253 |
| [16] |
Xinsheng Wang, Weisheng Wu, Yujun Zhu. Local unstable entropy and local unstable pressure for random partially hyperbolic dynamical systems. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 81-105. doi: 10.3934/dcds.2020004 |
| [17] |
Raquel Ribeiro. Hyperbolicity and types of shadowing for $C^1$ generic vector fields. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2963-2982. doi: 10.3934/dcds.2014.34.2963 |
| [18] |
Qiangchang Ju, Hailiang Li, Yong Li, Song Jiang. Quasi-neutral limit of the two-fluid Euler-Poisson system. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1577-1590. doi: 10.3934/cpaa.2010.9.1577 |
| [19] |
Jianfeng Huang, Haihua Liang. Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 861-873. doi: 10.3934/dcdsb.2020145 |
| [20] |
Zecen He, Haihua Liang, Xiang Zhang. Limit cycles and global dynamic of planar cubic semi-quasi-homogeneous systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 421-441. doi: 10.3934/dcdsb.2021049 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]