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August  2017, 37(7): 3905-3920. doi: 10.3934/dcds.2017164

Existence of SRB measures for a class of partially hyperbolic attractors in banach spaces

Zeng Lian 1, , Peidong Liu 2, and  Kening Lu 1,3,

1. 

School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2. 

School of Mathematical Sciences, Peking University, Beijing 100871, China
3. 

Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA

Received  August 2016 Revised  February 2017 Published  April 2017

Fund Project: This work was partially supported by NSFC (11331007, 11541003, and 11671279), and NSF (1413603).

In this paper, we study the existence of SRB measures for infinite dimensional dynamical systems in a Banach space. We show that if the system has a partially hyperbolic attractor with nontrivial finite dimensional unstable directions, then it has an SRB measure.

Keywords: SRB measure, infinite dimensional dynamical systems, partial hyperbolicity, chaotic behavior.
Mathematics Subject Classification: Primary: 37D30, 37C40.
Citation: Zeng Lian, Peidong Liu, Kening Lu. Existence of SRB measures for a class of partially hyperbolic attractors in banach spaces. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3905-3920. doi: 10.3934/dcds.2017164
References:
[1]

A. Blumenthal and L.-S. Young, Entropy, volume growth and SRB measures for Banach space mappings, Invent. Math., 207 (2017), 833-893, arXiv:1510.04312v1. doi: 10.1007/s00222-016-0678-0.  Google Scholar

[2]

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[4]

J. K. Hale, Attractors and dynamics in partial differential equations. From finite to infinite dimensional dynamical systems, (Cambridge, 1995,), NATO Sci. Ser. Ⅱ Math. Phys. Chem., Kluwer Acad. Publ., Dordrecht, 19 (2001), 85-112.  doi: 10.1007/978-94-010-0732-0_4.  Google Scholar

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D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, New York, 1981.  Google Scholar

[6]

W. Huang and K. Lu, Entropy, Chaos and weak horseshoe for infinite dimensional random dynamical systems, XVIIth International Congress on Mathematical Physics, (2012), 281-281, arXiv: 1504.05275. doi: 10.1142/9789814449243_0017.  Google Scholar

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F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms, Ann. Math., 122 (1985), 509-574.  doi: 10.2307/1971329.  Google Scholar

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Z. Li and L. Shu, The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula, Discrete Contin. Dyn. Syst., 33 (2013), 4123-4155.  doi: 10.3934/dcds.2013.33.4123.  Google Scholar

[9]

Z. Lian, P. Liu and K. Lu, SRB measures for a class of partially hyperbolic attractors in Hilbert spaces, J. Differential Equations, 261 (2016), 1532-1603, arXiv: 1508.03301. doi: 10.1016/j.jde.2016.04.006.  Google Scholar

[10]

Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space Memoirs of AMS., 206 (2010), vi+106 pp. doi: 10.1090/S0065-9266-10-00574-0.  Google Scholar

[11]

Z. Lian and L.-S. Young, Lyapunov exponents, periodic orbits and horseshoes for mappings of Hilbert spaces, Annales Henri Poincaré, 12 (2011), 1081-1108.  doi: 10.1007/s00023-011-0100-9.  Google Scholar

[12]

K. Lu, Q. Wang and L. -S. Young, Strange attractors for periodically forced parabolic equations Mem. Amer. Math. Soc., 224 (2013), vi+85 pp. doi: 10.1090/S0065-9266-2012-00669-1.  Google Scholar

[13]

R. Mañé, Lyapunov exponents and stable manifolds for compact transformations, Lecture Notes in Mathematics, Springer, 1007 (1983), 522-577.  doi: 10.1007/BFb0061433.  Google Scholar

[14]

J. C. Álvarez Paiva and A. C. Thompson, Volumes on normed and Finsler spaces, Riemann-Finsler Geometry, MSRI Publications, 50 (2004), 1-48.  doi: 10.4171/PRIMS/123.  Google Scholar

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J. Palis, A global perspective for non-conservative dynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 485-507.  doi: 10.1016/j.anihpc.2005.01.001.  Google Scholar

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P. Pesin, Characteristic Lyapunov exponents, and smooth ergodic theory, Russian Math. Surveys, 32 (1977), 55-112.   Google Scholar

[17]

Ya. B. Pesin and Ya. G. Sinai, Gibbs measures for partially hyperbolic attractors, Ergodic Theory Dynam. Systems, 2 (1982), 417-438.  doi: 10.1017/S014338570000170X.  Google Scholar

[18]

M. Qian, J. -S. Xie and S. Zhu, Smooth Ergodic Theory for Endomorphisms, Lecture Notes in Mathematics, 1978, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-01954-8.  Google Scholar

[19]

V. A. Rokhlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation, 71 (1952), 55 pp.  Google Scholar

[20]

D. Ruelle, Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics, J. Statist. Phys., 95 (1999), 393-468.  doi: 10.1023/A:1004593915069.  Google Scholar

[21]

D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. Math., 115 (1982), 243-290.  doi: 10.2307/1971392.  Google Scholar

[22]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[23]

P. Thieullen, Asymptotically compact dynamic bundles, Lyapunov exponents, entropy, dimension, Ann. Inst. H. Poincaré, Anal. Non linéaire, 4 (1987), 49-97.  doi: 10.1016/S0294-1449(16)30373-0.  Google Scholar

[24]

L.-S. Young, What are SRB measures, and which dynamical systems have them? Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays, J. Statist. Phys., 108 (2002), 733-754.  doi: 10.1023/A:1019762724717.  Google Scholar

[25]

L.-S. Young, Stochastic stability of hyperbolic attractors, Ergodic Theory Dynam. Systems, 6 (1986), 311-319.  doi: 10.1017/S0143385700003473.  Google Scholar

show all references

References:
[1]

A. Blumenthal and L.-S. Young, Entropy, volume growth and SRB measures for Banach space mappings, Invent. Math., 207 (2017), 833-893, arXiv:1510.04312v1. doi: 10.1007/s00222-016-0678-0.  Google Scholar

[2]

C. Bonatti, L. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, Ⅲ. Springer-Verlag, Berlin, 2005.  Google Scholar

[3]

J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys., 57 (1985), 617-656.  doi: 10.1103/RevModPhys.57.617.  Google Scholar

[4]

J. K. Hale, Attractors and dynamics in partial differential equations. From finite to infinite dimensional dynamical systems, (Cambridge, 1995,), NATO Sci. Ser. Ⅱ Math. Phys. Chem., Kluwer Acad. Publ., Dordrecht, 19 (2001), 85-112.  doi: 10.1007/978-94-010-0732-0_4.  Google Scholar

[5]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, New York, 1981.  Google Scholar

[6]

W. Huang and K. Lu, Entropy, Chaos and weak horseshoe for infinite dimensional random dynamical systems, XVIIth International Congress on Mathematical Physics, (2012), 281-281, arXiv: 1504.05275. doi: 10.1142/9789814449243_0017.  Google Scholar

[7]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms, Ann. Math., 122 (1985), 509-574.  doi: 10.2307/1971329.  Google Scholar

[8]

Z. Li and L. Shu, The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula, Discrete Contin. Dyn. Syst., 33 (2013), 4123-4155.  doi: 10.3934/dcds.2013.33.4123.  Google Scholar

[9]

Z. Lian, P. Liu and K. Lu, SRB measures for a class of partially hyperbolic attractors in Hilbert spaces, J. Differential Equations, 261 (2016), 1532-1603, arXiv: 1508.03301. doi: 10.1016/j.jde.2016.04.006.  Google Scholar

[10]

Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space Memoirs of AMS., 206 (2010), vi+106 pp. doi: 10.1090/S0065-9266-10-00574-0.  Google Scholar

[11]

Z. Lian and L.-S. Young, Lyapunov exponents, periodic orbits and horseshoes for mappings of Hilbert spaces, Annales Henri Poincaré, 12 (2011), 1081-1108.  doi: 10.1007/s00023-011-0100-9.  Google Scholar

[12]

K. Lu, Q. Wang and L. -S. Young, Strange attractors for periodically forced parabolic equations Mem. Amer. Math. Soc., 224 (2013), vi+85 pp. doi: 10.1090/S0065-9266-2012-00669-1.  Google Scholar

[13]

R. Mañé, Lyapunov exponents and stable manifolds for compact transformations, Lecture Notes in Mathematics, Springer, 1007 (1983), 522-577.  doi: 10.1007/BFb0061433.  Google Scholar

[14]

J. C. Álvarez Paiva and A. C. Thompson, Volumes on normed and Finsler spaces, Riemann-Finsler Geometry, MSRI Publications, 50 (2004), 1-48.  doi: 10.4171/PRIMS/123.  Google Scholar

[15]

J. Palis, A global perspective for non-conservative dynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 485-507.  doi: 10.1016/j.anihpc.2005.01.001.  Google Scholar

[16]

P. Pesin, Characteristic Lyapunov exponents, and smooth ergodic theory, Russian Math. Surveys, 32 (1977), 55-112.   Google Scholar

[17]

Ya. B. Pesin and Ya. G. Sinai, Gibbs measures for partially hyperbolic attractors, Ergodic Theory Dynam. Systems, 2 (1982), 417-438.  doi: 10.1017/S014338570000170X.  Google Scholar

[18]

M. Qian, J. -S. Xie and S. Zhu, Smooth Ergodic Theory for Endomorphisms, Lecture Notes in Mathematics, 1978, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-01954-8.  Google Scholar

[19]

V. A. Rokhlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation, 71 (1952), 55 pp.  Google Scholar

[20]

D. Ruelle, Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics, J. Statist. Phys., 95 (1999), 393-468.  doi: 10.1023/A:1004593915069.  Google Scholar

[21]

D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. Math., 115 (1982), 243-290.  doi: 10.2307/1971392.  Google Scholar

[22]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[23]

P. Thieullen, Asymptotically compact dynamic bundles, Lyapunov exponents, entropy, dimension, Ann. Inst. H. Poincaré, Anal. Non linéaire, 4 (1987), 49-97.  doi: 10.1016/S0294-1449(16)30373-0.  Google Scholar

[24]

L.-S. Young, What are SRB measures, and which dynamical systems have them? Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays, J. Statist. Phys., 108 (2002), 733-754.  doi: 10.1023/A:1019762724717.  Google Scholar

[25]

L.-S. Young, Stochastic stability of hyperbolic attractors, Ergodic Theory Dynam. Systems, 6 (1986), 311-319.  doi: 10.1017/S0143385700003473.  Google Scholar

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