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.
Citation: |
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. | |
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. | |
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. | |
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. | |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, New York, 1981. | |
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. | |
F. Ledrappier and L.-S. Young , The metric entropy of diffeomorphisms, Ann. Math., 122 (1985) , 509-574. doi: 10.2307/1971329. | |
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. | |
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. | |
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. | |
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. | |
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. | |
R. Mañé , Lyapunov exponents and stable manifolds for compact transformations, Lecture Notes in Mathematics, Springer, 1007 (1983) , 522-577. doi: 10.1007/BFb0061433. | |
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. | |
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. | |
P. Pesin , Characteristic Lyapunov exponents, and smooth ergodic theory, Russian Math. Surveys, 32 (1977) , 55-112. | |
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. | |
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. | |
V. A. Rokhlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation, 71 (1952), 55 pp. | |
D. Ruelle , Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics, J. Statist. Phys., 95 (1999) , 393-468. doi: 10.1023/A:1004593915069. | |
D. Ruelle , Characteristic exponents and invariant manifolds in Hilbert space, Ann. Math., 115 (1982) , 243-290. doi: 10.2307/1971392. | |
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. | |
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. | |
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. | |
L.-S. Young , Stochastic stability of hyperbolic attractors, Ergodic Theory Dynam. Systems, 6 (1986) , 311-319. doi: 10.1017/S0143385700003473. |