Advanced Search
Article Contents
Article Contents

Minimal yet measurable foliations

Abstract Related Papers Cited by
  • In this paper we mainly address the problem of disintegration of Lebesgue measure along the central foliation of volume-preserving diffeomorphisms isotopic to hyperbolic automorphisms of 3-torus. We prove that atomic disintegration of the Lebesgue measure (ergodic case) along the central foliation has the peculiarity of being mono-atomic (one atom per leaf). This implies the measurability of the central foliation. As a corollary we provide open and nonempty subset of partially hyperbolic diffeomorphisms with minimal yet measurable central foliation.
    Mathematics Subject Classification: Primary: 37D30, 37D25; Secondary: 28D99, 37D10.


    \begin{equation} \\ \end{equation}
  • [1]

    A. Avila, M. Viana and A. Wilkinson, Absolute continuity, Lyapunov exponents and rigidity I: Geodesic flows, arXiv:1110.2365v2, 2011.


    A. Baraviera and C. Bonatti, Removing zero Lyapunov exponents, Ergodic Theory and Dynamical Systems, 23 (2003), 1655-1670.doi: 10.1017/S0143385702001773.


    L. Barreira and Y. Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents, Encyclopedia of Mathematics and its Applications, 115, Cambridge University Press, Cambridge, 2007.doi: 10.1017/CBO9781107326026.


    C. Bonatti and A. Wilkinson, Transitive partially hyperbolic diffeomorphisms on 3-manifolds, Topology, 44 (2005), 475-508.doi: 10.1016/j.top.2004.10.009.


    M. Brin, D. Burago and D. Ivanov, On partially hyperbolic diffeomorphisms of 3-manifolds with commutative fundamental group, in Modern Dynamical Systems and Applications, Cambridge Univ. Press, Cambridge, 2004, 307-312.


    M. Brin, D. Burago and D. Ivanov, Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus, J. Mod. Dyn., 3 (2009), 1-11.doi: 10.3934/jmd.2009.3.1.


    M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, 259, Springer-Verlag London, Ltd., London, 2011.doi: 10.1007/978-0-85729-021-2.


    J. Franks, Anosov diffeomorphisms, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, RI, 1970, 61-93.


    A. Gogolev, How typical are pathological foliations in partially hyperbolic dynamics: An example, Israel J. Math., 187 (2012), 493-507.doi: 10.1007/s11856-011-0088-3.


    A. Hammerlindl, Leaf conjugacies on the torus, to appear in Ergodic Theory and Dynamical Systems, 2009.


    A. Hammerlindl, Leaf Conjugacies on the Torus, Ph.D. Thesis, University of Toronto, Canada, 2009.


    A. Hammerlindl and R. Potrie, Pointwise partial hyperbolicity in 3-dimensional nilmanifolds, preprint, arXiv:1302.0543, 2013.


    A. Hammerlindl and R. Ures, Ergodicity and partial hyperbolicity on the 3-torus, Commun. Contemp. Math., 2013.doi: 10.1142/S0219199713500387.


    M. Hirayama and Y. Pesin, Non-absolutely continuous foliations, Israel J. Math., 160 (2007), 173-187.doi: 10.1007/s11856-007-0060-4.


    M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math., 583, Springer-Verlag, Berlin-New York, 1977.


    F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula, Ann. of Math. (2), 122 (1985), 509-539.doi: 10.2307/1971328.


    G. Ponce and A. Tahzibi, Central Lyapunov exponents of partially hyperbolic diffeomorphisms on $\mathbbT^3$, to appear in Proceedings of AMS, 2013.


    V. A. Rohlin, Lectures on the entropy theory of transformations with invariant measure, Uspehi Mat. Nauk, 22 (1967), 3-56.


    D. Ruelle and A. Wilkinson, Absolutely singular dynamical foliations, Comm. Math. Phys., 219 (2001), 481-487.doi: 10.1007/s002200100420.


    R. Saghin and Z. Xia, Geometric expansion, Lyapunov exponents and foliations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 689-704.doi: 10.1016/j.anihpc.2008.07.001.


    M. Shub and A. Wilkinson, Pathological foliations and removable zero exponents, Invent. Math., 139 (2000), 495-508.doi: 10.1007/s002229900035.


    D. Sullivan, A counterexample to the periodic orbit conjecture, Inst. Hautes Études Sci. Publ. Math., 46 (1976), 5-14.


    R. Ures, Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part, Proc. Amer. Math. Soc., 140 (2012), 1973-1985.doi: 10.1090/S0002-9939-2011-11040-2.


    R. Varão, Center foliation: Absolute continuity, disintegration and rigidity, to appear in Ergodic Theory and Dynamical Systems, 2014


    Y. Pesin, Characteristic Lyapunov exponents, and smooth ergodic theory, Russ. Math. Surv., 32 (1977), 55-114.doi: 10.1070/RM1977v032n04ABEH001639.

  • 加载中

Article Metrics

HTML views() PDF downloads(124) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint