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Partial hyperbolicity and foliations in $\mathbb{T}^3$
1. | Centro de Matemática, Facultad de Ciencias, Universidad de la República, Igua 4225, Montevideo, 11400, Uruguay |
References:
[1] |
P. Berger, Persistence of laminations, Bull. Braz. Math. Soc. (N.S.), 41 (2010), 259-319.
doi: 10.1007/s00574-010-0013-0. |
[2] |
C. Bonatti, L. Díaz and E. Pujals, A $C^1$ generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. (2), 158 (2003), 355-418.
doi: 10.4007/annals.2003.158.355. |
[3] |
C. Bonatti, L. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Sciences, 102, Mathematical Physics, III, Springer-Verlag, Berlin, 2005. |
[4] |
C. Bonatti and J. Franks, A Hölder continuous vector field tangent to many foliations, in Modern Dynamical Systems and Applications, Cambridge Univ. Press, Cambridge, 2004, 299-306. |
[5] |
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. |
[6] |
M. Brin, On dynamical coherence, Ergodic Theory Dynam. Systems, 23 (2003), 395-401.
doi: 10.1017/S0143385702001499. |
[7] |
M. Brin, D. Burago and S. 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. |
[8] |
M. Brin, D. Burago and S. 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. |
[9] |
M. Brin and A. Manning, Anosov diffeomorphisms with pinched spectrum, in Dynamical Systems and Turbulence, Warwick 1980 (Coventry, 1979/1980), Lecture Notes in Mathematics, 898, Springer, Berlin-New York, 1981, 48-53. |
[10] |
M. Brin and Ja. B. Pesin, Partially hyperbolic dynamical systems, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212. |
[11] |
D. Burago and S. Ivanov, Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups, J. Mod. Dyn., 2 (2008), 541-580.
doi: 10.3934/jmd.2008.2.541. |
[12] |
K. Burns, M. A. Rodriguez Hertz, F. Rodriguez Hertz, A. Talitskaya and R. Ures, Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center, Discrete Contin. Dynam. Syst., 22 (2008), 75-88.
doi: 10.3934/dcds.2008.22.75. |
[13] |
K. Burns and A. Wilkinson, Dynamical coherence and center bunching, Discrete Contin. Dynam. Syst., 22 (2008), 89-100.
doi: 10.3934/dcds.2008.22.89. |
[14] |
K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math. (2), 171 (2010), 451-489.
doi: 10.4007/annals.2010.171.451. |
[15] |
A. Candel and L. Conlon, Foliations I and II, Graduate Studies in Mathematics, {60}, American Math. Society, Providence, RI, 2003.
doi: 10.1090/gsm/060. |
[16] |
S. Crovisier, Perturbation de la dynamique de difféomorphismes en topologie $C^1$, Astérisque, 354 (2013). |
[17] |
L. J. Díaz, E. R. Pujals and R. Ures, Partial hyperbolicity and robust transitivity, Acta Math., 183 (1999), 1-43.
doi: 10.1007/BF02392945. |
[18] |
D. Dolgopyat and A. Wilkinson, Stable accesibility is $C^1$-dense. Geometric methods in dynamics. II, Astérisque, 287 (2003), 33-60. |
[19] |
J. Franks, Anosov diffeomorphisms, in 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, RI, 1970, 61-93. |
[20] |
A. Hammerlindl, Leaf conjugacies in the torus, Ergodic Th. and Dynam. Sys., 33 (2013), 896-933.
doi: 10.1017/etds.2012.171. |
[21] |
A. Hammerlindl and R. Potrie, Pointwise partial hyperbolicity in three-dimensional nilmanifolds, J. London Math. Soc. (2), 89 (2014), 853-875.
doi: 10.1112/jlms/jdu013. |
[22] |
G. Hector and U. Hirsch, Introduction to the Geometry of Foliations. Part B. Foliations of Codimension One, Second Edition, Aspects of Mathematics, E3, Friedr. Vieweg & Sohn, Braunschweig, 1987.
doi: 10.1007/978-3-322-90161-3. |
[23] |
M. A. Rodriguez Hertz, F. Rodriguez Hertz and R. Ures, Tori with hyperbolic dynamics in 3-manifolds, J. Mod. Dyn., 5 (2011), 185-202.
doi: 10.3934/jmd.2011.5.185. |
[24] |
M. A. Rodriguez Hertz, F. Rodriguez Hertz and R. Ures, A non dynamically coherent example in $\mathbbT^3$,, in preparation., ().
|
[25] |
M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math., 583, Springer-Verlag, Berlin-New York, 1977. |
[26] |
R. Mañe, An ergodic closing lemma, Ann. of Math. (2), 116 (1982), 503-540.
doi: 10.2307/2007021. |
[27] |
K. Parwani, On $3$-manifolds that support partially hyperbolic diffeomorphisms, Nonlinearity, 23 (2010), 589-606.
doi: 10.1088/0951-7715/23/3/009. |
[28] |
J. F. Plante, Foliations of $3$-manifolds with solvable fundamental group, Invent. Math., 51 (1979), 219-230.
doi: 10.1007/BF01389915. |
[29] |
R. Potrie, Partial Hyperbolicity and Attracting Regions in 3-Dimensional Manifolds, Ph.D. Thesis, arXiv:1207.1822, 2012. |
[30] |
I. Richards, On the classification of noncompact surfaces, Trans. Amer. Math. Soc., 106 (1963), 259-269.
doi: 10.1090/S0002-9947-1963-0143186-0. |
[31] |
V. V. Solodov, Components of topological foliations, Math. USSR-Sbornik, 47 (1984), 329-343.
doi: 10.1070/SM1984v047n02ABEH002645. |
[32] |
P. Walters, Anosov diffeomorphisms are topologically stable, Topology, 9 (1970), 71-78.
doi: 10.1016/0040-9383(70)90051-0. |
show all references
References:
[1] |
P. Berger, Persistence of laminations, Bull. Braz. Math. Soc. (N.S.), 41 (2010), 259-319.
doi: 10.1007/s00574-010-0013-0. |
[2] |
C. Bonatti, L. Díaz and E. Pujals, A $C^1$ generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. (2), 158 (2003), 355-418.
doi: 10.4007/annals.2003.158.355. |
[3] |
C. Bonatti, L. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Sciences, 102, Mathematical Physics, III, Springer-Verlag, Berlin, 2005. |
[4] |
C. Bonatti and J. Franks, A Hölder continuous vector field tangent to many foliations, in Modern Dynamical Systems and Applications, Cambridge Univ. Press, Cambridge, 2004, 299-306. |
[5] |
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. |
[6] |
M. Brin, On dynamical coherence, Ergodic Theory Dynam. Systems, 23 (2003), 395-401.
doi: 10.1017/S0143385702001499. |
[7] |
M. Brin, D. Burago and S. 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. |
[8] |
M. Brin, D. Burago and S. 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. |
[9] |
M. Brin and A. Manning, Anosov diffeomorphisms with pinched spectrum, in Dynamical Systems and Turbulence, Warwick 1980 (Coventry, 1979/1980), Lecture Notes in Mathematics, 898, Springer, Berlin-New York, 1981, 48-53. |
[10] |
M. Brin and Ja. B. Pesin, Partially hyperbolic dynamical systems, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212. |
[11] |
D. Burago and S. Ivanov, Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups, J. Mod. Dyn., 2 (2008), 541-580.
doi: 10.3934/jmd.2008.2.541. |
[12] |
K. Burns, M. A. Rodriguez Hertz, F. Rodriguez Hertz, A. Talitskaya and R. Ures, Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center, Discrete Contin. Dynam. Syst., 22 (2008), 75-88.
doi: 10.3934/dcds.2008.22.75. |
[13] |
K. Burns and A. Wilkinson, Dynamical coherence and center bunching, Discrete Contin. Dynam. Syst., 22 (2008), 89-100.
doi: 10.3934/dcds.2008.22.89. |
[14] |
K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math. (2), 171 (2010), 451-489.
doi: 10.4007/annals.2010.171.451. |
[15] |
A. Candel and L. Conlon, Foliations I and II, Graduate Studies in Mathematics, {60}, American Math. Society, Providence, RI, 2003.
doi: 10.1090/gsm/060. |
[16] |
S. Crovisier, Perturbation de la dynamique de difféomorphismes en topologie $C^1$, Astérisque, 354 (2013). |
[17] |
L. J. Díaz, E. R. Pujals and R. Ures, Partial hyperbolicity and robust transitivity, Acta Math., 183 (1999), 1-43.
doi: 10.1007/BF02392945. |
[18] |
D. Dolgopyat and A. Wilkinson, Stable accesibility is $C^1$-dense. Geometric methods in dynamics. II, Astérisque, 287 (2003), 33-60. |
[19] |
J. Franks, Anosov diffeomorphisms, in 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, RI, 1970, 61-93. |
[20] |
A. Hammerlindl, Leaf conjugacies in the torus, Ergodic Th. and Dynam. Sys., 33 (2013), 896-933.
doi: 10.1017/etds.2012.171. |
[21] |
A. Hammerlindl and R. Potrie, Pointwise partial hyperbolicity in three-dimensional nilmanifolds, J. London Math. Soc. (2), 89 (2014), 853-875.
doi: 10.1112/jlms/jdu013. |
[22] |
G. Hector and U. Hirsch, Introduction to the Geometry of Foliations. Part B. Foliations of Codimension One, Second Edition, Aspects of Mathematics, E3, Friedr. Vieweg & Sohn, Braunschweig, 1987.
doi: 10.1007/978-3-322-90161-3. |
[23] |
M. A. Rodriguez Hertz, F. Rodriguez Hertz and R. Ures, Tori with hyperbolic dynamics in 3-manifolds, J. Mod. Dyn., 5 (2011), 185-202.
doi: 10.3934/jmd.2011.5.185. |
[24] |
M. A. Rodriguez Hertz, F. Rodriguez Hertz and R. Ures, A non dynamically coherent example in $\mathbbT^3$,, in preparation., ().
|
[25] |
M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math., 583, Springer-Verlag, Berlin-New York, 1977. |
[26] |
R. Mañe, An ergodic closing lemma, Ann. of Math. (2), 116 (1982), 503-540.
doi: 10.2307/2007021. |
[27] |
K. Parwani, On $3$-manifolds that support partially hyperbolic diffeomorphisms, Nonlinearity, 23 (2010), 589-606.
doi: 10.1088/0951-7715/23/3/009. |
[28] |
J. F. Plante, Foliations of $3$-manifolds with solvable fundamental group, Invent. Math., 51 (1979), 219-230.
doi: 10.1007/BF01389915. |
[29] |
R. Potrie, Partial Hyperbolicity and Attracting Regions in 3-Dimensional Manifolds, Ph.D. Thesis, arXiv:1207.1822, 2012. |
[30] |
I. Richards, On the classification of noncompact surfaces, Trans. Amer. Math. Soc., 106 (1963), 259-269.
doi: 10.1090/S0002-9947-1963-0143186-0. |
[31] |
V. V. Solodov, Components of topological foliations, Math. USSR-Sbornik, 47 (1984), 329-343.
doi: 10.1070/SM1984v047n02ABEH002645. |
[32] |
P. Walters, Anosov diffeomorphisms are topologically stable, Topology, 9 (1970), 71-78.
doi: 10.1016/0040-9383(70)90051-0. |
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