# American Institute of Mathematical Sciences

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

## Pathological center foliation with dimension greater than one

 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.

Citation: José Santana Campos Costa, Fernando Micena. Pathological center foliation with dimension greater than one. Discrete & Continuous Dynamical Systems - A, 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.Google Scholar [2] D. V. Anosov and Y. G. Sinai, Some smooth ergodic systems, Russian Mathematical Surveys, 22 (1967), 107-172, Turpion Ltd. Google Scholar [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. Google Scholar [4] A. Baraviera and C. Bonatti, Removing zero Lyapunov Exponents, Ergodic Theory of Dynamical Systems, 23 (2003), 1655-1670. doi: 10.1017/S0143385702001773. Google Scholar [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. Google Scholar [6] M. Brin, On dynamical coherence, Ergodic Theory and Dynamical Systems, 23 (2003), 395-401, Cambridge Univ Press. doi: 10.1017/S0143385702001499. Google Scholar [7] M. Brin and Y. Pesin, On Partially hyperbolic dynamical systems, Mathematics of the USSR-Izvestiya, 8 (1974), 177.Google Scholar [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. Google Scholar [9] J. Franks, Anosov diffeomorphisms on tori, Transactions of the American Mathematical Society, 145 (1969), 117-124. doi: 10.2307/1995062. Google Scholar [10] J. Franks, Necessary conditions for stability of diffeomorphisms, Transactions of the American Mathematical Society, 158 (1971), 301-308. doi: 10.2307/1995906. Google Scholar [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. Google Scholar [12] A. Hammerlindl, Leaf conjugacies on the torus, Ergodic Theory and Dynamical Systems, 33 (2013), 896-933. doi: 10.1017/etds.2012.171. Google Scholar [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. 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, New York, 1977. Google Scholar [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. Google Scholar [17] A. Manning, There are no new Anosov diffeomorphisms on tori, American Journal of Mathematics, 96 (1974), 422-429. doi: 10.2307/2373551. Google Scholar [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. Google Scholar [19] F. Micena and A. Tahzibi, Regularity of foliations and Lyapunov exponents for partially hyperbolic Dynamics, Nonlinearity, (2013), 1071-1082. Google Scholar [20] J. Milnor, Fubini foiled: Katok's paradoxical example in measure theory, The Mathematical Intelligencer, 19 (1997), 30-32. doi: 10.1007/BF03024428. Google Scholar [21] V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc, 19 (1968), 197-231. Google Scholar [22] Y. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, European Mathematical Society, 2004. doi: 10.4171/003. Google Scholar [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. Google Scholar [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. Google Scholar [25] C. Pugh, M. Viana and A. Wilkinson, Absolute continuity of foliations. In preparation.Google Scholar [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. Google Scholar [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. Google Scholar [28] D. Ruelle and A. Wilkinson, Absolutely singular dynamical foliations, Comm. Math. Phys., 219 (2001), 481-487. doi: 10.1007/s002200100420. Google Scholar [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. Google Scholar [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. Google Scholar [31] M. Viana and J. Yang, Measure-theoretical properties of center foliations, Contemporary Mathematics, 692 (2017), 291-320. Google Scholar

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##### 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.Google Scholar [2] D. V. Anosov and Y. G. Sinai, Some smooth ergodic systems, Russian Mathematical Surveys, 22 (1967), 107-172, Turpion Ltd. Google Scholar [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. Google Scholar [4] A. Baraviera and C. Bonatti, Removing zero Lyapunov Exponents, Ergodic Theory of Dynamical Systems, 23 (2003), 1655-1670. doi: 10.1017/S0143385702001773. Google Scholar [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. Google Scholar [6] M. Brin, On dynamical coherence, Ergodic Theory and Dynamical Systems, 23 (2003), 395-401, Cambridge Univ Press. doi: 10.1017/S0143385702001499. Google Scholar [7] M. Brin and Y. Pesin, On Partially hyperbolic dynamical systems, Mathematics of the USSR-Izvestiya, 8 (1974), 177.Google Scholar [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. Google Scholar [9] J. Franks, Anosov diffeomorphisms on tori, Transactions of the American Mathematical Society, 145 (1969), 117-124. doi: 10.2307/1995062. Google Scholar [10] J. Franks, Necessary conditions for stability of diffeomorphisms, Transactions of the American Mathematical Society, 158 (1971), 301-308. doi: 10.2307/1995906. Google Scholar [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. Google Scholar [12] A. Hammerlindl, Leaf conjugacies on the torus, Ergodic Theory and Dynamical Systems, 33 (2013), 896-933. doi: 10.1017/etds.2012.171. Google Scholar [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. 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, New York, 1977. Google Scholar [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. Google Scholar [17] A. Manning, There are no new Anosov diffeomorphisms on tori, American Journal of Mathematics, 96 (1974), 422-429. doi: 10.2307/2373551. Google Scholar [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. Google Scholar [19] F. Micena and A. Tahzibi, Regularity of foliations and Lyapunov exponents for partially hyperbolic Dynamics, Nonlinearity, (2013), 1071-1082. Google Scholar [20] J. Milnor, Fubini foiled: Katok's paradoxical example in measure theory, The Mathematical Intelligencer, 19 (1997), 30-32. doi: 10.1007/BF03024428. Google Scholar [21] V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc, 19 (1968), 197-231. Google Scholar [22] Y. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, European Mathematical Society, 2004. doi: 10.4171/003. Google Scholar [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. Google Scholar [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. Google Scholar [25] C. Pugh, M. Viana and A. Wilkinson, Absolute continuity of foliations. In preparation.Google Scholar [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. Google Scholar [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. Google Scholar [28] D. Ruelle and A. Wilkinson, Absolutely singular dynamical foliations, Comm. Math. Phys., 219 (2001), 481-487. doi: 10.1007/s002200100420. Google Scholar [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. Google Scholar [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. Google Scholar [31] M. Viana and J. Yang, Measure-theoretical properties of center foliations, Contemporary Mathematics, 692 (2017), 291-320. Google Scholar
The fixed point $p_n$ of $g_n$ and $g_{n,j}$, respectively
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