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Existence of SRB measures for a class of partially hyperbolic attractors in banach spaces
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 |
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.
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. Ma |
[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. Ma |
[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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