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Quasi-shadowing for partially hyperbolic flows
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China |
In this paper, we study the quasi-shadowing property for partially hyperbolic flows. A partially hyperbolic flow $ \varphi_{t} $ has the quasi-shadowing property if for any $ (\delta,T) $-pseudoorbit $ g(t) $ of $ \varphi_{t} $ there exist a sequence of points $ \{y_{k}\}_{k\in\mathbb{Z}} $ and a reparametrization $ \alpha $ such that $ \varphi_{\alpha(t)-\alpha(kT)}(y_k) $ trace $ g(t) $ in which $ y_{k} $ is obtained from $ \varphi_{\alpha(kT)-\alpha((k-1)T)}(y_{k-1}) $ by a motion along the central direction. We prove that any partially hyperbolic flow $ \varphi_{t} $ has the quasi-shadowing property. We also investigate the limit quasi-shadowing properties for flows. That is, a partially hyperbolic flow has the $ \mathcal{L}^p $, limit and asymptotic quasi-shadowing properties.
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A generalized shadowing lemma, Discrete Contin. Dyn. Syst., 8 (2002), 627-632.
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Partial hyperbolicity and central shadowing, Discrete Contin. Dyn. Syst., 33 (2013), 2901-2909.
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S. Y. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Mathematics, 1706. Springer-Verlag, Berlin, 1999. |
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Shadowing in structurally stable flows, J. Differential Equations, 140 (1997), 238-265.
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J. G. Sinai,
Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64.
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W. X. Sun and Y. Yang,
Hyperbolic periodic points for chain hyperbolic homoclinic classes, Discrete Contin. Dyn. Syst., 36 (2016), 3911-3925.
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W. X. Sun, T. Young and Y. H. Zhou,
Topological entropies of equivalent smooth flows, Trans. Amer. Math. Soc., 361 (2009), 3071-3082.
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doi: 10.3934/dcds.2017123. |
show all references
References:
| [1] |
D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Pro. Steklov Inst. Math., 90 (1967), 209 pp. |
| [2] |
D. Bohnet and C. Bonatti,
Partially hyperbolic diffeomorphisms with uniformly center foliation: The quotient dynamics, Ergod. Theory Dyn. Syst., 36 (20165), 1067-1105.
doi: 10.1017/etds.2014.102. |
| [3] |
R. Bowen,
Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.
doi: 10.2307/1995452. |
| [4] |
R. Bowen,
Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1-30.
doi: 10.2307/2373590. |
| [5] |
S. B. Gan,
A generalized shadowing lemma, Discrete Contin. Dyn. Syst., 8 (2002), 627-632.
doi: 10.3934/dcds.2002.8.627. |
| [6] |
S. B. Gan and L. Wen,
Nonsingular star flows satisfy Axiom A and the no-cycle condition, Invent. Math., 164 (2006), 279-315.
doi: 10.1007/s00222-005-0479-3. |
| [7] |
S. B. Gan and D. W. Yang,
Morse-Smale systems and horseshoes for three dimensional singular flows, Ann. Sci. Éc. Norm. Supér., 51 (2018), 39-112.
doi: 10.24033/asens.2351. |
| [8] |
B. Han and X. Wen,
A shadowing lemma for quasi-hyperbolic strings of flows, J. Differential Equations, 264 (2018), 1-29.
doi: 10.1016/j.jde.2017.08.065. |
| [9] |
S. Hayashi,
Connecting invariant manifolds and the solution of the $C^1$ stability conjecture and $\Omega$-stability conjecture for flows, Ann. Math., 145 (1997), 81-137.
doi: 10.2307/2951824. |
| [10] |
H. Y. Hu, Y. H. Zhou and Y. J. Zhu,
Quasi-shadowing for partially hyperbolic diffeomorphisms, Ergod. Theory Dyn. Syst., 35 (2015), 412-430.
doi: 10.1017/etds.2014.126. |
| [11] |
A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., (1980), 137–173. |
| [12] |
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. |
| [13] |
K. Palmer, Shadowing in Dynamical Systems, Theory and applications. Mathematics and its Applications, 501. Kluwer Academic Publishers, Dordrecht, 2000.
doi: 10.1007/978-1-4757-3210-8. |
| [14] |
S. Y. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Mathematics, 1706. Springer-Verlag, Berlin, 1999. |
| [15] |
S. Y. Pilyugin,
Shadowing in structurally stable flows, J. Differential Equations, 140 (1997), 238-265.
doi: 10.1006/jdeq.1997.3295. |
| [16] |
J. G. Sinai,
Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64.
|
| [17] |
W. X. Sun and Y. Yang,
Hyperbolic periodic points for chain hyperbolic homoclinic classes, Discrete Contin. Dyn. Syst., 36 (2016), 3911-3925.
doi: 10.3934/dcds.2016.36.3911. |
| [18] |
W. X. Sun, T. Young and Y. H. Zhou,
Topological entropies of equivalent smooth flows, Trans. Amer. Math. Soc., 361 (2009), 3071-3082.
doi: 10.1090/S0002-9947-08-04743-0. |
| [19] |
L. Wang and Y. J. Zhu,
Center specification property and entropy for partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst., 36 (2016), 469-479.
doi: 10.3934/dcds.2016.36.469. |
| [20] |
F. Zhang and Y. H. Zhou,
On the limit quasi-shadowing property, Discrete Contin. Dyn. Syst., 37 (2017), 2861-2879.
doi: 10.3934/dcds.2017123. |
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