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April  2020, 40(4): 2089-2103. doi: 10.3934/dcds.2020107

Quasi-shadowing for partially hyperbolic flows

Zhiping Li and  Yunhua Zhou

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Received  March 2019 Published  January 2020

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.

Keywords: Quasi-shadowing, limit quasi-shadowing, partial hyperbolicity, flow.
Mathematics Subject Classification: Primary: 37C50, 37D30; Secondary: 37C10.
Citation: Zhiping Li, Yunhua Zhou. Quasi-shadowing for partially hyperbolic flows. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2089-2103. doi: 10.3934/dcds.2020107
References:
[1]

D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Pro. Steklov Inst. Math., 90 (1967), 209 pp.  Google Scholar

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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.  Google Scholar

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S. B. Gan, A generalized shadowing lemma, Discrete Contin. Dyn. Syst., 8 (2002), 627-632.  doi: 10.3934/dcds.2002.8.627.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., (1980), 137–173.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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S. Y. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Mathematics, 1706. Springer-Verlag, Berlin, 1999.  Google Scholar

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S. Y. Pilyugin, Shadowing in structurally stable flows, J. Differential Equations, 140 (1997), 238-265.  doi: 10.1006/jdeq.1997.3295.  Google Scholar

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J. G. Sinai, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64.   Google Scholar

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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.  doi: 10.3934/dcds.2016.36.3911.  Google Scholar

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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.  doi: 10.1090/S0002-9947-08-04743-0.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

R. Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.  doi: 10.2307/1995452.  Google Scholar

[4]

R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1-30.  doi: 10.2307/2373590.  Google Scholar

[5]

S. B. Gan, A generalized shadowing lemma, Discrete Contin. Dyn. Syst., 8 (2002), 627-632.  doi: 10.3934/dcds.2002.8.627.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., (1980), 137–173.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

S. Y. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Mathematics, 1706. Springer-Verlag, Berlin, 1999.  Google Scholar

[15]

S. Y. Pilyugin, Shadowing in structurally stable flows, J. Differential Equations, 140 (1997), 238-265.  doi: 10.1006/jdeq.1997.3295.  Google Scholar

[16]

J. G. Sinai, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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