# American Institute of Mathematical Sciences

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

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.
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. Google Scholar [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. Google Scholar [3] C. Bonatti, L. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective,, Encyclopaedia of Mathematical Sciences, (2005). Google Scholar [4] C. Bonatti and J. Franks, A Hölder continuous vector field tangent to many foliations,, in Modern Dynamical Systems and Applications, (2004), 299. Google Scholar [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. Google Scholar [6] M. Brin, On dynamical coherence,, Ergodic Theory Dynam. Systems, 23 (2003), 395. doi: 10.1017/S0143385702001499. Google Scholar [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. Google Scholar [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. Google Scholar [9] M. Brin and A. Manning, Anosov diffeomorphisms with pinched spectrum,, in Dynamical Systems and Turbulence, 898 (1980), 48. Google Scholar [10] M. Brin and Ja. B. Pesin, Partially hyperbolic dynamical systems,, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] A. Candel and L. Conlon, Foliations I and II,, Graduate Studies in Mathematics, (2003). doi: 10.1090/gsm/060. Google Scholar [16] S. Crovisier, Perturbation de la dynamique de difféomorphismes en topologie $C^1$,, Astérisque, 354 (2013). Google Scholar [17] L. J. Díaz, E. R. Pujals and R. Ures, Partial hyperbolicity and robust transitivity,, Acta Math., 183 (1999), 1. doi: 10.1007/BF02392945. Google Scholar [18] D. Dolgopyat and A. Wilkinson, Stable accesibility is $C^1$-dense. Geometric methods in dynamics. II,, Astérisque, 287 (2003), 33. Google Scholar [19] J. Franks, Anosov diffeomorphisms,, in 1970 Global Analysis (Proc. Sympos. Pure Math., (1968), 61. Google Scholar [20] A. Hammerlindl, Leaf conjugacies in the torus,, Ergodic Th. and Dynam. Sys., 33 (2013), 896. doi: 10.1017/etds.2012.171. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [24] M. A. Rodriguez Hertz, F. Rodriguez Hertz and R. Ures, A non dynamically coherent example in $\mathbbT^3$,, in preparation., (). Google Scholar [25] M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds,, Lecture Notes in Math., (1977). Google Scholar [26] R. Mañe, An ergodic closing lemma,, Ann. of Math. (2), 116 (1982), 503. doi: 10.2307/2007021. Google Scholar [27] K. Parwani, On $3$-manifolds that support partially hyperbolic diffeomorphisms,, Nonlinearity, 23 (2010), 589. doi: 10.1088/0951-7715/23/3/009. Google Scholar [28] J. F. Plante, Foliations of $3$-manifolds with solvable fundamental group,, Invent. Math., 51 (1979), 219. doi: 10.1007/BF01389915. Google Scholar [29] R. Potrie, Partial Hyperbolicity and Attracting Regions in 3-Dimensional Manifolds,, Ph.D. Thesis, (2012). Google Scholar [30] I. Richards, On the classification of noncompact surfaces,, Trans. Amer. Math. Soc., 106 (1963), 259. doi: 10.1090/S0002-9947-1963-0143186-0. Google Scholar [31] V. V. Solodov, Components of topological foliations,, Math. USSR-Sbornik, 47 (1984), 329. doi: 10.1070/SM1984v047n02ABEH002645. Google Scholar [32] P. Walters, Anosov diffeomorphisms are topologically stable,, Topology, 9 (1970), 71. doi: 10.1016/0040-9383(70)90051-0. Google Scholar

show all references

##### References:
 [1] P. Berger, Persistence of laminations,, Bull. Braz. Math. Soc. (N.S.), 41 (2010), 259. doi: 10.1007/s00574-010-0013-0. Google Scholar [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. Google Scholar [3] C. Bonatti, L. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective,, Encyclopaedia of Mathematical Sciences, (2005). Google Scholar [4] C. Bonatti and J. Franks, A Hölder continuous vector field tangent to many foliations,, in Modern Dynamical Systems and Applications, (2004), 299. Google Scholar [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. Google Scholar [6] M. Brin, On dynamical coherence,, Ergodic Theory Dynam. Systems, 23 (2003), 395. doi: 10.1017/S0143385702001499. Google Scholar [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. Google Scholar [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. Google Scholar [9] M. Brin and A. Manning, Anosov diffeomorphisms with pinched spectrum,, in Dynamical Systems and Turbulence, 898 (1980), 48. Google Scholar [10] M. Brin and Ja. B. Pesin, Partially hyperbolic dynamical systems,, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] A. Candel and L. Conlon, Foliations I and II,, Graduate Studies in Mathematics, (2003). doi: 10.1090/gsm/060. Google Scholar [16] S. Crovisier, Perturbation de la dynamique de difféomorphismes en topologie $C^1$,, Astérisque, 354 (2013). Google Scholar [17] L. J. Díaz, E. R. Pujals and R. Ures, Partial hyperbolicity and robust transitivity,, Acta Math., 183 (1999), 1. doi: 10.1007/BF02392945. Google Scholar [18] D. Dolgopyat and A. Wilkinson, Stable accesibility is $C^1$-dense. Geometric methods in dynamics. II,, Astérisque, 287 (2003), 33. Google Scholar [19] J. Franks, Anosov diffeomorphisms,, in 1970 Global Analysis (Proc. Sympos. Pure Math., (1968), 61. Google Scholar [20] A. Hammerlindl, Leaf conjugacies in the torus,, Ergodic Th. and Dynam. Sys., 33 (2013), 896. doi: 10.1017/etds.2012.171. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [24] M. A. Rodriguez Hertz, F. Rodriguez Hertz and R. Ures, A non dynamically coherent example in $\mathbbT^3$,, in preparation., (). Google Scholar [25] M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds,, Lecture Notes in Math., (1977). Google Scholar [26] R. Mañe, An ergodic closing lemma,, Ann. of Math. (2), 116 (1982), 503. doi: 10.2307/2007021. Google Scholar [27] K. Parwani, On $3$-manifolds that support partially hyperbolic diffeomorphisms,, Nonlinearity, 23 (2010), 589. doi: 10.1088/0951-7715/23/3/009. Google Scholar [28] J. F. Plante, Foliations of $3$-manifolds with solvable fundamental group,, Invent. Math., 51 (1979), 219. doi: 10.1007/BF01389915. Google Scholar [29] R. Potrie, Partial Hyperbolicity and Attracting Regions in 3-Dimensional Manifolds,, Ph.D. Thesis, (2012). Google Scholar [30] I. Richards, On the classification of noncompact surfaces,, Trans. Amer. Math. Soc., 106 (1963), 259. doi: 10.1090/S0002-9947-1963-0143186-0. Google Scholar [31] V. V. Solodov, Components of topological foliations,, Math. USSR-Sbornik, 47 (1984), 329. doi: 10.1070/SM1984v047n02ABEH002645. Google Scholar [32] P. Walters, Anosov diffeomorphisms are topologically stable,, Topology, 9 (1970), 71. doi: 10.1016/0040-9383(70)90051-0. Google Scholar
 [1] Wenjun Zhang, Bernd Krauskopf, Vivien Kirk. How to find a codimension-one heteroclinic cycle between two periodic orbits. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2825-2851. doi: 10.3934/dcds.2012.32.2825 [2] Boris Hasselblatt, Yakov Pesin, Jörg Schmeling. Pointwise hyperbolicity implies uniform hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2819-2827. doi: 10.3934/dcds.2014.34.2819 [3] Keith Burns, Amie Wilkinson. Dynamical coherence and center bunching. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 89-100. doi: 10.3934/dcds.2008.22.89 [4] Sergey Kryzhevich, Sergey Tikhomirov. Partial hyperbolicity and central shadowing. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2901-2909. doi: 10.3934/dcds.2013.33.2901 [5] Michael L. Frankel, Victor Roytburd. Dynamical structure of one-phase model of solid combustion. Conference Publications, 2005, 2005 (Special) : 287-296. doi: 10.3934/proc.2005.2005.287 [6] Ji Li, Kening Lu, Peter W. Bates. Invariant foliations for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3639-3666. doi: 10.3934/dcds.2014.34.3639 [7] Yakov Pesin. On the work of Dolgopyat on partial and nonuniform hyperbolicity. Journal of Modern Dynamics, 2010, 4 (2) : 227-241. doi: 10.3934/jmd.2010.4.227 [8] Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Raúl Ures. Partial hyperbolicity and ergodicity in dimension three. Journal of Modern Dynamics, 2008, 2 (2) : 187-208. doi: 10.3934/jmd.2008.2.187 [9] Jérôme Buzzi, Todd Fisher. Entropic stability beyond partial hyperbolicity. Journal of Modern Dynamics, 2013, 7 (4) : 527-552. doi: 10.3934/jmd.2013.7.527 [10] Anna-Lena Horlemann-Trautmann, Alessandro Neri. A complete classification of partial MDS (maximally recoverable) codes with one global parity. Advances in Mathematics of Communications, 2020, 14 (1) : 69-88. doi: 10.3934/amc.2020006 [11] Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809 [12] Boris Hasselblatt and Jorg Schmeling. Dimension product structure of hyperbolic sets. Electronic Research Announcements, 2004, 10: 88-96. [13] Thierry Barbot, Carlos Maquera. On integrable codimension one Anosov actions of $\RR^k$. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 803-822. doi: 10.3934/dcds.2011.29.803 [14] Michael Brin, Dmitri Burago, Sergey Ivanov. Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus. Journal of Modern Dynamics, 2009, 3 (1) : 1-11. doi: 10.3934/jmd.2009.3.1 [15] Andy Hammerlindl. Partial hyperbolicity on 3-dimensional nilmanifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3641-3669. doi: 10.3934/dcds.2013.33.3641 [16] Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006 [17] Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5203-5219. doi: 10.3934/dcds.2015.35.5203 [18] Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281 [19] Tatsuya Arai. The structure of dendrites constructed by pointwise P-expansive maps on the unit interval. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 43-61. doi: 10.3934/dcds.2016.36.43 [20] Patrick Bonckaert, Timoteo Carletti, Ernest Fontich. On dynamical systems close to a product of $m$ rotations. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 349-366. doi: 10.3934/dcds.2009.24.349

2018 Impact Factor: 0.295