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February  2019, 39(2): 1049-1070. doi: 10.3934/dcds.2019044

Pathological center foliation with dimension greater than one

José Santana Campos Costa 1, and  Fernando Micena 2,

1. 

DEMAT-UFMA, São Luís-MA, Brazil
2. 

IMC-UNIFEI, Itajubá-MG, Brazil

Received  March 2018 Published  November 2018

We are considering partially hyperbolic diffeomorphims of the torus, with ${\rm{dim}}(E^c) > 1.$ We prove, under some conditions, that if the all center Lyapunov exponents of the linearization $A,$ of a DA-diffeomorphism $f,$ are positive and the center foliation of $f$ is absolutely continuous, then the sum of the center Lyapunov exponents of $f$ is bounded by the sum of the center Lyapunov exponents of $A.$ After, we construct a $C^1-$open class of volume preserving DA-diffeomorphisms, far from Anosov diffeomorphisms, with non compact pathological two dimensional center foliation. Indeed, each $f$ in this open set satisfies the previously established hypothesis, but the sum of the center Lyapunov exponents of $f$ is greater than the corresponding sum with respect to its linearization. It allows to conclude that the center foliation of $f$ is non absolutely continuous. We still build an example of a DA-diffeomorphism, such that the disintegration of volume along the two dimensional, non compact center foliation is neither Lebesgue nor atomic.

Keywords: Partial hyperbolic diffeomorphisms, center foliation, absolutely continuous foliation, volume growth, Lyapunov exponents.
Mathematics Subject Classification: Primary: 37D10, 37D20, 37D25, 37D30.
Citation: José Santana Campos Costa, Fernando Micena. Pathological center foliation with dimension greater than one. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1049-1070. doi: 10.3934/dcds.2019044
References:
[1]

D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Matematicheskogo Instituta Imeni VA Steklova, 90 (1967), 3-210, Russian Academy of Sciences, Steklov Mathematical Institute of Russian Academy of Sciences.

[2]

D. V. Anosov and Y. G. Sinai, Some smooth ergodic systems, Russian Mathematical Surveys, 22 (1967), 107-172, Turpion Ltd.

[3]

A. Avila, M. Viana and A. Wilkinson, Absolute continuity, Lyapunov exponents and rigidity Ⅰ: Geodesic flows, Journal of the European Mathematical Society, 17 (2015), 1435-1462.  doi: 10.4171/JEMS/534.

[4]

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

[5]

C. Bonatti, L. J. Díaz and E. Pujals, A C1 generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Annals of Mathematics, 158 (2003), 355-418.  doi: 10.4007/annals.2003.158.355.

[6]

M. Brin, On dynamical coherence, Ergodic Theory and Dynamical Systems, 23 (2003), 395-401, Cambridge Univ Press.  doi: 10.1017/S0143385702001499.

[7]

M. Brin and Y. Pesin, On Partially hyperbolic dynamical systems, Mathematics of the USSR-Izvestiya, 8 (1974), 177.

[8]

T. Fisher, R. Potrie and M. Sambarino, Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov, Mathematische Zeitschrift, 278 (2014), 149-168.  doi: 10.1007/s00209-014-1310-x.

[9]

J. Franks, Anosov diffeomorphisms on tori, Transactions of the American Mathematical Society, 145 (1969), 117-124.  doi: 10.2307/1995062.

[10]

J. Franks, Necessary conditions for stability of diffeomorphisms, Transactions of the American Mathematical Society, 158 (1971), 301-308.  doi: 10.2307/1995906.

[11]

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.

[12]

A. Hammerlindl, Leaf conjugacies on the torus, Ergodic Theory and Dynamical Systems, 33 (2013), 896-933.  doi: 10.1017/etds.2012.171.

[13]

B. Hasselblatt and Y. Pesin, Partially hyperbolic dynamical systems, Handbook of Dynamical Systems, Elsevier B. V., Amsterdam, 1 (2006), 1-55. doi: 10.1016/S1874-575X(06)80026-3.

[14]

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

[15]

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

[16]

C. Liang, G. Liu and W. X. Sun, Equivalent conditions of dominated splitting. for volume-preserving diffeomorphisms, Acta Mathematica Sinica, English Series, 23 (2007), 1563-1576.  doi: 10.1007/s10114-005-0889-6.

[17]

A. Manning, There are no new Anosov diffeomorphisms on tori, American Journal of Mathematics, 96 (1974), 422-429.  doi: 10.2307/2373551.

[18]

F. Micena, New Derived from Anosov Diffeomorphisms with pathological center foliation, Journal of Dynamics and Differential Equations, 29 (2017), 1150-1172.  doi: 10.1007/s10884-016-9523-9.

[19]

F. Micena and A. Tahzibi, Regularity of foliations and Lyapunov exponents for partially hyperbolic Dynamics, Nonlinearity, (2013), 1071-1082. 

[20]

J. Milnor, Fubini foiled: Katok's paradoxical example in measure theory, The Mathematical Intelligencer, 19 (1997), 30-32.  doi: 10.1007/BF03024428.

[21]

V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc, 19 (1968), 197-231. 

[22]

Y. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, European Mathematical Society, 2004. doi: 10.4171/003.

[23]

G. Ponce and A. Tahzibi, Central Lyapunov exponents of partially hyperbolic diffeomorphisms of ${\mathbb{T}^3}$, Proceedings of American Math. Society, 142 (2014), 3193-3205.  doi: 10.1090/S0002-9939-2014-12063-6.

[24]

G. Ponce, A. Tahzibi and R. Varão, Minimal yet measurable foliations, Journal of Modern Dynamics, 8 (2014), 93-107.  doi: 10.3934/jmd.2014.8.93.

[25]

C. Pugh, M. Viana and A. Wilkinson, Absolute continuity of foliations. In preparation.

[26]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent.Math., 172 (2008), 353-381.  doi: 10.1007/s00222-007-0100-z.

[27]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, A non-dynamically coherent example ${\mathbb{T}^3}$, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 33 (2016), 1023-1032.  doi: 10.1016/j.anihpc.2015.03.003.

[28]

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

[29]

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

[30]

R. Varão, Center foliation: absolute continuity, disintegration and rigidity, Ergod. Th. and Dynam. Sys., 36 (2016), 256-275.  doi: 10.1017/etds.2014.53.

[31]

M. Viana and J. Yang, Measure-theoretical properties of center foliations, Contemporary Mathematics, 692 (2017), 291-320. 

show all references

References:
[1]

D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Matematicheskogo Instituta Imeni VA Steklova, 90 (1967), 3-210, Russian Academy of Sciences, Steklov Mathematical Institute of Russian Academy of Sciences.

[2]

D. V. Anosov and Y. G. Sinai, Some smooth ergodic systems, Russian Mathematical Surveys, 22 (1967), 107-172, Turpion Ltd.

[3]

A. Avila, M. Viana and A. Wilkinson, Absolute continuity, Lyapunov exponents and rigidity Ⅰ: Geodesic flows, Journal of the European Mathematical Society, 17 (2015), 1435-1462.  doi: 10.4171/JEMS/534.

[4]

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

[5]

C. Bonatti, L. J. Díaz and E. Pujals, A C1 generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Annals of Mathematics, 158 (2003), 355-418.  doi: 10.4007/annals.2003.158.355.

[6]

M. Brin, On dynamical coherence, Ergodic Theory and Dynamical Systems, 23 (2003), 395-401, Cambridge Univ Press.  doi: 10.1017/S0143385702001499.

[7]

M. Brin and Y. Pesin, On Partially hyperbolic dynamical systems, Mathematics of the USSR-Izvestiya, 8 (1974), 177.

[8]

T. Fisher, R. Potrie and M. Sambarino, Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov, Mathematische Zeitschrift, 278 (2014), 149-168.  doi: 10.1007/s00209-014-1310-x.

[9]

J. Franks, Anosov diffeomorphisms on tori, Transactions of the American Mathematical Society, 145 (1969), 117-124.  doi: 10.2307/1995062.

[10]

J. Franks, Necessary conditions for stability of diffeomorphisms, Transactions of the American Mathematical Society, 158 (1971), 301-308.  doi: 10.2307/1995906.

[11]

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.

[12]

A. Hammerlindl, Leaf conjugacies on the torus, Ergodic Theory and Dynamical Systems, 33 (2013), 896-933.  doi: 10.1017/etds.2012.171.

[13]

B. Hasselblatt and Y. Pesin, Partially hyperbolic dynamical systems, Handbook of Dynamical Systems, Elsevier B. V., Amsterdam, 1 (2006), 1-55. doi: 10.1016/S1874-575X(06)80026-3.

[14]

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

[15]

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

[16]

C. Liang, G. Liu and W. X. Sun, Equivalent conditions of dominated splitting. for volume-preserving diffeomorphisms, Acta Mathematica Sinica, English Series, 23 (2007), 1563-1576.  doi: 10.1007/s10114-005-0889-6.

[17]

A. Manning, There are no new Anosov diffeomorphisms on tori, American Journal of Mathematics, 96 (1974), 422-429.  doi: 10.2307/2373551.

[18]

F. Micena, New Derived from Anosov Diffeomorphisms with pathological center foliation, Journal of Dynamics and Differential Equations, 29 (2017), 1150-1172.  doi: 10.1007/s10884-016-9523-9.

[19]

F. Micena and A. Tahzibi, Regularity of foliations and Lyapunov exponents for partially hyperbolic Dynamics, Nonlinearity, (2013), 1071-1082. 

[20]

J. Milnor, Fubini foiled: Katok's paradoxical example in measure theory, The Mathematical Intelligencer, 19 (1997), 30-32.  doi: 10.1007/BF03024428.

[21]

V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc, 19 (1968), 197-231. 

[22]

Y. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, European Mathematical Society, 2004. doi: 10.4171/003.

[23]

G. Ponce and A. Tahzibi, Central Lyapunov exponents of partially hyperbolic diffeomorphisms of ${\mathbb{T}^3}$, Proceedings of American Math. Society, 142 (2014), 3193-3205.  doi: 10.1090/S0002-9939-2014-12063-6.

[24]

G. Ponce, A. Tahzibi and R. Varão, Minimal yet measurable foliations, Journal of Modern Dynamics, 8 (2014), 93-107.  doi: 10.3934/jmd.2014.8.93.

[25]

C. Pugh, M. Viana and A. Wilkinson, Absolute continuity of foliations. In preparation.

[26]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent.Math., 172 (2008), 353-381.  doi: 10.1007/s00222-007-0100-z.

[27]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, A non-dynamically coherent example ${\mathbb{T}^3}$, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 33 (2016), 1023-1032.  doi: 10.1016/j.anihpc.2015.03.003.

[28]

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

[29]

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

[30]

R. Varão, Center foliation: absolute continuity, disintegration and rigidity, Ergod. Th. and Dynam. Sys., 36 (2016), 256-275.  doi: 10.1017/etds.2014.53.

[31]

M. Viana and J. Yang, Measure-theoretical properties of center foliations, Contemporary Mathematics, 692 (2017), 291-320. 

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