Pathological center foliation with dimension greater than one

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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 & 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

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.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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