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January  2014, 8(1): 93-107. doi: 10.3934/jmd.2014.8.93

Minimal yet measurable foliations

Gabriel Ponce 1, , Ali Tahzibi 2, and  Régis Varão 1,

1. 

Departamento de Matemática, ICMC-USP São Carlos- SP, Brazil, Brazil
2. 

Departamento de Matematica, ICMC-USP São Carlos, Caixa Postal 668, 13560-970 São Carlos-SP

Received  August 2013 Published  July 2014

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.
Keywords: disintegration, minimal foliations, Partial hyperbolicity, measurable partitions, Lyapunov exponents., derived from Anosov.
Mathematics Subject Classification: Primary: 37D30, 37D25; Secondary: 28D99, 37D1.
Citation: Gabriel Ponce, Ali Tahzibi, Régis Varão. Minimal yet measurable foliations. Journal of Modern Dynamics, 2014, 8 (1) : 93-107. doi: 10.3934/jmd.2014.8.93
References:
[1]

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

[2]

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

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

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

[9]

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.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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.  Google Scholar

[21]

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

[22]

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

[23]

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.  Google Scholar

[24]

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

[25]

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

show all references

References:
[1]

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

[2]

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

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

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

[9]

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.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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.  Google Scholar

[21]

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

[22]

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

[23]

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.  Google Scholar

[24]

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

[25]

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

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