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

Quasi-shadowing for partially hyperbolic flows

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.

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

[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

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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. HuY. 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. SunT. 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

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. HuY. 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. SunT. 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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