Partial hyperbolicity and foliations in \mathbb{T}^3

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2015, 9: 81-121. doi: 10.3934/jmd.2015.9.81

Partial hyperbolicity and foliations in $\mathbb{T}^3$

1.	Centro de Matemática, Facultad de Ciencias, Universidad de la República, Igua 4225, Montevideo, 11400, Uruguay

Received January 2013 Revised June 2014 Published June 2015

Abstract
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We prove that dynamical coherence is an open and closed property in the space of partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ isotopic to Anosov. Moreover, we prove that strong partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ are either dynamically coherent or have an invariant two-dimensional torus which is either contracting or repelling. We develop for this end some general results on codimension one foliations which may be of independent interest.

Keywords: codimension-one foliations., dynamical coherence, Partial hyperbolicity (pointwise), global product structure.

Mathematics Subject Classification: Primary: 37C05, 37C20; Secondary: 37C25, 37C29, 37D30, 57R3.

Citation: Rafael Potrie. Partial hyperbolicity and foliations in $\mathbb{T}^3$. Journal of Modern Dynamics, 2015, 9: 81-121. doi: 10.3934/jmd.2015.9.81

References:

[1]	P. Berger, Persistence of laminations,, Bull. Braz. Math. Soc. (N.S.), 41 (2010), 259. 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. 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, (2005).
[4]	C. Bonatti and J. Franks, A Hölder continuous vector field tangent to many foliations,, in Modern Dynamical Systems and Applications, (2004), 299.
[5]	C. Bonatti and A. Wilkinson, Transitive partially hyperbolic diffeomorphisms on $3$-manifolds,, Topology, 44 (2005), 475. doi: 10.1016/j.top.2004.10.009.
[6]	M. Brin, On dynamical coherence,, Ergodic Theory Dynam. Systems, 23 (2003), 395. 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, (2004), 307.
[8]	M. Brin, D. Burago and S. Ivanov, Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus,, J. Mod. Dyn., 3 (2009), 1. doi: 10.3934/jmd.2009.3.1.
[9]	M. Brin and A. Manning, Anosov diffeomorphisms with pinched spectrum,, in Dynamical Systems and Turbulence, 898* (1980), 48.*
[10]	M. Brin and Ja. B. Pesin, Partially hyperbolic dynamical systems,, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170.
[11]	D. Burago and S. Ivanov, Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups,, J. Mod. Dyn., 2 (2008), 541. 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. 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. 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. doi: 10.4007/annals.2010.171.451.
[15]	A. Candel and L. Conlon, Foliations I and II,, Graduate Studies in Mathematics, (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. 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.
[19]	J. Franks, Anosov diffeomorphisms,, in 1970 Global Analysis* (Proc. Sympos. Pure Math.*, (1968), 61.
[20]	A. Hammerlindl, Leaf conjugacies in the torus,, Ergodic Th. and Dynam. Sys., 33 (2013), 896. 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. 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, (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. 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., (1977).
[26]	R. Mañe, An ergodic closing lemma,, Ann. of Math. (2), 116 (1982), 503. doi: 10.2307/2007021.
[27]	K. Parwani, On $3$-manifolds that support partially hyperbolic diffeomorphisms,, Nonlinearity, 23 (2010), 589. 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. doi: 10.1007/BF01389915.
[29]	R. Potrie, Partial Hyperbolicity and Attracting Regions in 3-Dimensional Manifolds,, Ph.D. Thesis, (2012).
[30]	I. Richards, On the classification of noncompact surfaces,, Trans. Amer. Math. Soc., 106 (1963), 259. doi: 10.1090/S0002-9947-1963-0143186-0.
[31]	V. V. Solodov, Components of topological foliations,, Math. USSR-Sbornik, 47 (1984), 329. doi: 10.1070/SM1984v047n02ABEH002645.
[32]	P. Walters, Anosov diffeomorphisms are topologically stable,, Topology, 9 (1970), 71. doi: 10.1016/0040-9383(70)90051-0.

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References:

[1]	P. Berger, Persistence of laminations,, Bull. Braz. Math. Soc. (N.S.), 41 (2010), 259. 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. 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, (2005).
[4]	C. Bonatti and J. Franks, A Hölder continuous vector field tangent to many foliations,, in Modern Dynamical Systems and Applications, (2004), 299.
[5]	C. Bonatti and A. Wilkinson, Transitive partially hyperbolic diffeomorphisms on $3$-manifolds,, Topology, 44 (2005), 475. doi: 10.1016/j.top.2004.10.009.
[6]	M. Brin, On dynamical coherence,, Ergodic Theory Dynam. Systems, 23 (2003), 395. 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, (2004), 307.
[8]	M. Brin, D. Burago and S. Ivanov, Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus,, J. Mod. Dyn., 3 (2009), 1. doi: 10.3934/jmd.2009.3.1.
[9]	M. Brin and A. Manning, Anosov diffeomorphisms with pinched spectrum,, in Dynamical Systems and Turbulence, 898* (1980), 48.*
[10]	M. Brin and Ja. B. Pesin, Partially hyperbolic dynamical systems,, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170.
[11]	D. Burago and S. Ivanov, Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups,, J. Mod. Dyn., 2 (2008), 541. 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. 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. 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. doi: 10.4007/annals.2010.171.451.
[15]	A. Candel and L. Conlon, Foliations I and II,, Graduate Studies in Mathematics, (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. 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.
[19]	J. Franks, Anosov diffeomorphisms,, in 1970 Global Analysis* (Proc. Sympos. Pure Math.*, (1968), 61.
[20]	A. Hammerlindl, Leaf conjugacies in the torus,, Ergodic Th. and Dynam. Sys., 33 (2013), 896. 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. 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, (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. 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., (1977).
[26]	R. Mañe, An ergodic closing lemma,, Ann. of Math. (2), 116 (1982), 503. doi: 10.2307/2007021.
[27]	K. Parwani, On $3$-manifolds that support partially hyperbolic diffeomorphisms,, Nonlinearity, 23 (2010), 589. 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. doi: 10.1007/BF01389915.
[29]	R. Potrie, Partial Hyperbolicity and Attracting Regions in 3-Dimensional Manifolds,, Ph.D. Thesis, (2012).
[30]	I. Richards, On the classification of noncompact surfaces,, Trans. Amer. Math. Soc., 106 (1963), 259. doi: 10.1090/S0002-9947-1963-0143186-0.
[31]	V. V. Solodov, Components of topological foliations,, Math. USSR-Sbornik, 47 (1984), 329. doi: 10.1070/SM1984v047n02ABEH002645.
[32]	P. Walters, Anosov diffeomorphisms are topologically stable,, Topology, 9 (1970), 71. doi: 10.1016/0040-9383(70)90051-0.

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