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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 and 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.

[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.

[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.

[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.

[5]

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

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[16]

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

[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.

[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.

[19]

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

[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.

[21]

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

[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.

[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.

[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.

[25]

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

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.

[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.

[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.

[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.

[5]

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

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[16]

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

[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.

[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.

[19]

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

[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.

[21]

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

[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.

[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.

[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.

[25]

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

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