Abstract
We prove the absolute continuity of stable foliations for mappings of Banach spaces satisfying conditions consistent with time-t maps of certain classes of dissipative PDEs. This property is crucial for passing information from submanifolds transversal to the stable foliation to the rest of the phase space; it is also used in proofs of ergodicity. Absolute continuity of stable foliations is well known in finite dimensional hyperbolic theory. On Banach spaces, the absence of nice geometric properties poses some additional difficulties.
Similar content being viewed by others
References
Anosov D.V.: Geodesic flows on closed Riemannian manifolds of negative curvature. Trudy Matematicheskogo Instituta im. VA Steklova 90, 3–210 (1967)
Benyamini, Y., Lindenstrauss, J.: Geometric nonlinear functional analysis, vol. 48. American Mathematical Society (1998)
Blumenthal A., Young L.-S.: Entropy, volume growth and SRB measures for Banach space mappings. Invent. Math. 207(2), 833–893 (2017)
Eckmann J.-P., Ruelle D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57(3), 617 (1985)
Henry D.: Geometric Theory of Semilinear Parabolic Equations, vol. 840. Springer, Berlin (1981)
Hopf, E.: Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung (1939)
Hunt B.R., Sauer T., Yorke J.A.: Prevalence: a translation-invariant ‘almost every’ on infinite-dimensional spaces. Bull. Am. Math. Soc. 27(2), 217–238 (1992)
Kato T.: Perturbation Theory for Linear Operators. Springer, Berlin (1995)
Katok, A., Strelcyn, J.-M., Ledrappier, F., Przytycki, F.: Invariant manifolds, entropy and billiards; smooth maps with singularities (1986)
Lian Z., Liu P., Lu K.: SRB measures for a class of partially hyperbolic attractors in hilbert spaces. J. Diff. Equ. 261(7), 1532–1603 (2016)
Lian Z., Young L.-S., Zeng C.: Absolute continuity of stable foliations for systems on Banach spaces. J. Differ. Equ. 254(1), 283–308 (2013)
Lian, Z., Young, L.-S.: Lyapunov exponents, periodic orbits and horseshoes for mappings of Hilbert spaces. In: Annales Henri Poincaré, vol. 12, pp. 1081–1108. Springer (2011)
Liverani, C., Wojtkowski, M.P.: Ergodicity in Hamiltonian systems. In: Dynamics reported, pp. 130–202. Springer (1995)
Nussbaum, R.D. et al.: The radius of the essential spectrum. Duke Math. J. 37, 473–478 (1970)
Pesin Y.B.: Characteristic Lyapunov exponents and smooth ergodic theory. Russ. Math. Surv. 32(4), 55–114 (1977)
Pugh C., Shub M.: Ergodic attractors. Trans. Am. Math. Soc. 312(1), 1–54 (1989)
Rokhlin V.A.: On the fundamental ideas of measure theory. Matematicheskii Sbornik 67(1), 107–150 (1949)
Sinai Y.G.: Dynamical systems with elastic reflections. Russ. Math. Surv. 25(2), 137–189 (1970)
Thieullen, P.: Fibrés dynamiques asymptotiquement compacts exposants de Lyapounov. Entropie. Dimension. In: Annales de l’institut Henri Poincaré (C) Analyse non linéaire, vol. 4, pp. 49–97. Gauthier-Villars (1987)
Young, L.-S.: Ergodic theory of differentiable dynamical systems. In: Real and complex dynamical systems, pp. 293–336. Springer (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Liverani
Lai-Sang Young was supported in part by NSF grant DMS-1363161. This work was done while Alex Blumenthal was a doctoral student at the Courant Institute of Mathematical Sciences, New York University.
Rights and permissions
About this article
Cite this article
Blumenthal, A., Young, LS. Absolute Continuity of Stable Foliations for Mappings of Banach Spaces. Commun. Math. Phys. 354, 591–619 (2017). https://doi.org/10.1007/s00220-017-2912-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-017-2912-z