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), 259-319. 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), 355-418. 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, 102, Mathematical Physics, III, Springer-Verlag, Berlin, 2005.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] M. Brin, On dynamical coherence, Ergodic Theory Dynam. Systems, 23 (2003), 395-401. 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, Cambridge Univ. Press, Cambridge, 2004, 307-312.  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), 1-11. 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, Warwick 1980 (Coventry, 1979/1980), Lecture Notes in Mathematics, 898, Springer, Berlin-New York, 1981, 48-53.  Google Scholar [10] M. Brin and Ja. B. Pesin, Partially hyperbolic dynamical systems, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212.  Google Scholar [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.  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), 75-88. 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), 89-100. 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), 451-489. doi: 10.4007/annals.2010.171.451.  Google Scholar [15] A. Candel and L. 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(2), 116 (1982), 503-540. doi: 10.2307/2007021.  Google Scholar [27] K. Parwani, On $3$-manifolds that support partially hyperbolic diffeomorphisms, Nonlinearity, 23 (2010), 589-606. 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), 219-230. doi: 10.1007/BF01389915.  Google Scholar [29] R. Potrie, Partial Hyperbolicity and Attracting Regions in 3-Dimensional Manifolds, Ph.D. Thesis, arXiv:1207.1822, 2012. Google Scholar [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.  Google Scholar [31] V. V. Solodov, Components of topological foliations, Math. USSR-Sbornik, 47 (1984), 329-343. doi: 10.1070/SM1984v047n02ABEH002645.  Google Scholar [32] P. Walters, Anosov diffeomorphisms are topologically stable, Topology, 9 (1970), 71-78. doi: 10.1016/0040-9383(70)90051-0.  Google Scholar

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References:
 [1] P. Berger, Persistence of laminations, Bull. Braz. Math. Soc. (N.S.), 41 (2010), 259-319. 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), 355-418. 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, 102, Mathematical Physics, III, Springer-Verlag, Berlin, 2005.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] M. Brin, On dynamical coherence, Ergodic Theory Dynam. Systems, 23 (2003), 395-401. 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, Cambridge Univ. Press, Cambridge, 2004, 307-312.  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), 1-11. 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, Warwick 1980 (Coventry, 1979/1980), Lecture Notes in Mathematics, 898, Springer, Berlin-New York, 1981, 48-53.  Google Scholar [10] M. Brin and Ja. B. Pesin, Partially hyperbolic dynamical systems, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212.  Google Scholar [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.  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), 75-88. 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), 89-100. 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), 451-489. doi: 10.4007/annals.2010.171.451.  Google Scholar [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.  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), 1-43. 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), 33-60.  Google Scholar [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.  Google Scholar [20] A. Hammerlindl, Leaf conjugacies in the torus, Ergodic Th. and Dynam. Sys., 33 (2013), 896-933. 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), 853-875. 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, Aspects of Mathematics, E3, Friedr. Vieweg & Sohn, Braunschweig, 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), 185-202. 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., 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar [26] R. Mañe, An ergodic closing lemma, Ann. of Math. (2), 116 (1982), 503-540. doi: 10.2307/2007021.  Google Scholar [27] K. Parwani, On $3$-manifolds that support partially hyperbolic diffeomorphisms, Nonlinearity, 23 (2010), 589-606. 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), 219-230. doi: 10.1007/BF01389915.  Google Scholar [29] R. Potrie, Partial Hyperbolicity and Attracting Regions in 3-Dimensional Manifolds, Ph.D. Thesis, arXiv:1207.1822, 2012. Google Scholar [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.  Google Scholar [31] V. V. Solodov, Components of topological foliations, Math. USSR-Sbornik, 47 (1984), 329-343. doi: 10.1070/SM1984v047n02ABEH002645.  Google Scholar [32] P. Walters, Anosov diffeomorphisms are topologically stable, Topology, 9 (1970), 71-78. doi: 10.1016/0040-9383(70)90051-0.  Google Scholar
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