Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T14:04:57.451Z Has data issue: false hasContentIssue false
  • English
  • Français

Absolute continuity, Lyapunov exponents, and rigidity II: systems with compact center leaves

Part of: Smooth dynamical systems: general theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  31 May 2021

A. AVILA ,
MARCELO VIANA Open the ORCID record for MARCELO VIANA [Opens in a new window]  and
A. WILKINSON Open the ORCID record for A. WILKINSON [Opens in a new window]
A. AVILA
Affiliation:
Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057Zürich, Switzerland IMPA—Estrada D. Castorina 110, Jardim Botânico, 22460-320Rio de Janeiro, Brazil (e-mail: [email protected])
MARCELO VIANA*
Affiliation:
IMPA—Estrada D. Castorina 110, Jardim Botânico, 22460-320Rio de Janeiro, Brazil (e-mail: [email protected])
A. WILKINSON
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois60637, USA (e-mail: [email protected])
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We explore new connections between the dynamics of conservative partially hyperbolic systems and the geometric measure-theoretic properties of their invariant foliations. Our methods are applied to two main classes of volume-preserving diffeomorphisms: fibered partially hyperbolic diffeomorphisms and center-fixing partially hyperbolic systems. When the center is one-dimensional, assuming the diffeomorphism is accessible, we prove that the disintegration of the volume measure along the center foliation is either atomic or Lebesgue. Moreover, the latter case is rigid in dimension three (this does not require accessibility): the center foliation is actually smooth and the diffeomorphism is smoothly conjugate to an explicit rigid model. A partial extension to fibered partially hyperbolic systems with compact fibers of any dimension is also obtained. A common feature of these classes of diffeomorphisms is that the center leaves either are compact or can be made compact by taking an appropriate dynamically defined quotient. For volume-preserving partially hyperbolic diffeomorphisms whose center foliation is absolutely continuous, if the generic center leaf is a circle, then every center leaf is compact.

Keywords

Lyapunov exponentrigidityabsolute continuityinvariance principlepartial hyperbolicity

MSC classification

Primary: 37C40: Smooth ergodic theory, invariant measures 37D30: Partially hyperbolic systems and dominated splittings
Secondary: 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification 37C75: Stability theory
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 2: Anatole Katok Memorial Issue Part 1: Special Issue of Ergodic Theory and Dynamical Systems , February 2022 , pp. 437 - 490
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alves, J. F., Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140 (2000), 351–398.CrossRefGoogle Scholar
Anosov, D. V.. Geodesic flows on closed Riemannian manifolds of negative curvature. Proc. Steklov Inst. Math. 90 (1967), 1–235.Google Scholar
Anosov, D. V. and Sinai, Y. G.. Certain smooth ergodic systems. Russian Math. Surveys 22 (1967), 103–167.CrossRefGoogle Scholar
Avila, A., Santamaria, J. and Viana, M.. Holonomy invariance: rough regularity and applications to Lyapunov exponents. Astérisque 358 (2013), 13–74.Google Scholar
Avila, A. and Viana, M.. Extremal Lyapunov exponents: an invariance principle and applications. Invent. Math. 181 (2010), 115–189.CrossRefGoogle Scholar
Avila, A., Viana, M. and Wilkinson, A.. Absolute continuity, Lyapunov exponents and rigidity I: geodesic flows. J. Eur. Math. Soc. (JEMS) 17 (2015), 1435–1462.CrossRefGoogle Scholar
Bohnet, D.. Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation. J. Mod. Dyn. 7 (2013), 565–604.CrossRefGoogle Scholar
Bonatti, C., Díaz, L. J. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity (Encyclopaedia of Mathematical Sciences, 102) . Springer, Berlin, 2005.Google Scholar
Bonatti, C. and Wilkinson, A.. Transitive partially hyperbolic diffeomorphisms on 3-manifolds. Topology 44 (2005), 475–508.CrossRefGoogle Scholar
Brin, M. and Pesin, Y.. Partially hyperbolic dynamical systems. Izv. Acad. Nauk SSSR 1 (1974), 177–212.Google Scholar
Brin, M. and Stuck, G.. Introduction to Dynamical Systems . Cambridge: Cambridge University Press, 2002.CrossRefGoogle Scholar
Burns, K., Pugh, C. and Wilkinson, A.. Stable ergodicity and Anosov flows. Topology 39 (2000), 149–159.CrossRefGoogle Scholar
Burns, K. and Wilkinson, A.. On the ergodicity of partially hyperbolic systems. Ann. of Math. 171 (2010), 451–489.CrossRefGoogle Scholar
Carrasco, P.. Compact dynamical foliations. Ergod. Th. & Dynam. Sys. 35 (2015), 2474–2498.CrossRefGoogle Scholar
Damjanović, D., Wikinson, A. and Xu, D.. Pathology and asymmetry: centralizer rigidity for partially hyperbolic diffeomorphisms. Duke Math. J., to appear.Google Scholar
de la Llave, R.. Invariants for smooth conjugacy of hyperbolic dynamical systems. II. Comm. Math. Phys. 109 (1987), 369–378.CrossRefGoogle Scholar
Epstein, D.. Foliations with all leaves compact. Ann. Inst. Fourier 26 (1976), 265–282.CrossRefGoogle Scholar
Gogolev, A.. Partially hyperbolic diffeomorphisms with compact center foliations. J. Mod. Dyn. 5 (2011), 747–769.CrossRefGoogle Scholar
Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583) . Springer, Berlin, 1977.CrossRefGoogle Scholar
Journé, J.-L.. A regularity lemma for functions of several variables. Rev. Mat. Iberoam. 4 (1988), 187–193.CrossRefGoogle Scholar
Katok, A. and Spatzier, R.. Invariant measures for higher-rank hyperbolic abelian actions. Ergod. Th. & Dynam. Sys. 16 (1996), 751–778.CrossRefGoogle Scholar
Ledrappier, F.. Quelques propriétés des exposants caractéristiques. École d’Été de Probabilités de Saint-Flour XII—1982 (Lecture Notes in Mathematics, 1097) . Ed. Hennequin, P. L.. Springer, Berlin and Heidelberg, 1984, pp. 305–396.CrossRefGoogle Scholar
Ledrappier, F.. Positivity of the exponent for stationary sequences of matrices. Lyapunov Exponents (Bremen, 1984) (Lecture Notes in Mathematics, 1186) . Ed. Arnold, L. and Wihstutz, V.. Springer, Berlin, 1986, pp. 56–73.Google Scholar
Lindenstrauss, E.. Invariant measures and arithmetic quantum unique ergodicity. Ann. of Math. 163 (2006), 165–219.CrossRefGoogle Scholar
Milnor, J.. Fubini foiled: Katok’s paradoxical example in measure theory. Math. Intelligencer 19 (1997), 30–32.CrossRefGoogle Scholar
Oxtoby, J. C.. Measure and Category (Graduate Texts in Mathematics, 2). Springer, New York, 1980.CrossRefGoogle Scholar
Palis, J. and Yoccoz, J.-C.. Centralizers of Anosov diffeomorphisms on tori. Ann. Sci. Éc. Norm. Supér. (4) 22 (1989), 99–108.CrossRefGoogle Scholar
Pugh, C., Shub, M. and Wilkinson, A.. Hölder foliations. Duke Math. J. 86 (1997), 517–546.CrossRefGoogle Scholar
Pugh, C., Shub, M. and Wilkinson, A.. Hölder foliations, revisited. J. Mod. Dyn. 6 (2012), 79–120.CrossRefGoogle Scholar
Pugh, C., Viana, M. and Wilkinson, A.. Absolute continuity of foliations. In preparation.Google Scholar
Repovš, D., Skopenkov, A. and Ščepin, E.. ${C}^1$ -homogeneous compacta in ${\mathsf{R}}^n$ are ${C}^1$ -submanifolds of ${\mathsf{R}}^n$ . Proc. Amer. Math. Soc. 124 (1996), 1219–1226.CrossRefGoogle Scholar
Rokhlin, V. A.. On the fundamental ideas of measure theory. Amer. Math. Soc. Transl. 10 (1962), 1–54. Transl. from Mat. Sb. 25 (1949), 107–150. First published by the American Mathematical Society in 1952 as Translation Number 71.Google Scholar
Ruelle, D.. Perturbation theory for Lyapunov exponents of a toral map: extension of a result of Shub and Wilkinson. Israel J. Math. 134 (2003), 345–361.CrossRefGoogle Scholar
Ruelle, D. and Wilkinson, A.. Absolutely singular dynamical foliations. Comm. Math. Phys. 219 (2001), 481–487.CrossRefGoogle Scholar
Shanon, M.. Dehn surgeries and smooth structures on 3-dimensional transitive Anosov flows. Thesis, University of Bourgogne Franche-Comté 2020, http://www.theses.fr/2020UBFCK035##.Google Scholar
Shub, M. and Wilkinson, A.. Pathological foliations and removable zero exponents. Invent. Math. 139 (2000), 495–508.CrossRefGoogle Scholar
Wilkinson, A.. The cohomological equation for partially hyperbolic diffeomorphisms. Astérisque 358 (2013), 75–165.Google Scholar