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Continuous shift commuting maps between ultragraph shift spaces
Pathological center foliation with dimension greater than one
1. | DEMAT-UFMA, São Luís-MA, Brazil |
2. | IMC-UNIFEI, Itajubá-MG, Brazil |
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.
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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