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

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

Rafael Potrie 1,

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.
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), 259-319. 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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