American Institute of Mathematical Sciences

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  2018, 12: 193-222. doi: 10.3934/jmd.2018008

Seifert manifolds admitting partially hyperbolic diffeomorphisms

Andy Hammerlindl 1, , Rafael Potrie 2, and  Mario Shannon 2,3,

1. 

School of Mathematical Sciences, Monash University, Victoria 3800, Australia
URL: http://users.monash.edu.au/~ahammerl/
2. 

CMAT, Facultad de Ciencias, Universidad de la República, Igua 4225, Montevideo 11400, Uruguay
URL: www.cmat.edu.uy/~rpotrie
3. 

Institute Mathèmatique de Burgogne, Dijon, France

AH: Partially supported by the Australian Research Council.
RP: Partially supported by CSIC group 618, MathAmSud-Physeco, and the Australian Research Council.
MS: Partially supported by CSIC group 618

Received  May 30, 2017 Revised  March 15, 2018 Published  June 2018

We characterize which 3-dimensional Seifert manifolds admit transitive partially hyperbolic diffeomorphisms. In particular, a circle bundle over a higher-genus surface admits a transitive partially hyperbolic diffeomorphism if and only if it admits an Anosov flow.

Keywords: Partially hyperbolic diffeomorphisms, Seifert spaces.
Mathematics Subject Classification: Primary: 37D30, 37C15; Secondary: 57R30, 55R05.
Citation: Andy Hammerlindl, Rafael Potrie, Mario Shannon. Seifert manifolds admitting partially hyperbolic diffeomorphisms. Journal of Modern Dynamics, 2018, 12: 193-222. doi: 10.3934/jmd.2018008
References:
[1]

T. Barbot, Flots d'Anosov sur les variétés graphées au sens de Waldhausen, Ann. Inst. Fourier (Grenoble), 46 (1996), 1451-1517. doi: 10.5802/aif.1556. Google Scholar

[2]

T. Barbot, Actions de groupes sur les 1-variétés non séparées et feuilletages de codimension un, Ann. Fac. Sci. Toulouse Math.(6), 7 (1998), 559-597. doi: 10.5802/afst.911. Google Scholar

[3]

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

[4]

C. Bonatti, K. Parwani and R. Potrie, Anomalous partially hyperbolic diffeomorphisms Ⅰ: Dynamically coherent examples, Ann. Sci. Éc. Norm. Supér.(4), 49 (2016), 1387-1402. doi: 10.24033/asens.2311. Google Scholar

[5]

C. Bonatti, A. Gogolev and R. Potrie, Anomalous partially hyperbolic diffeomorphisms Ⅱ: Stably ergodic examples, Invent. Math., 206 (2016), 801-836. doi: 10.1007/s00222-016-0663-7. Google Scholar

[6]

C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie, Anomalous partially hyperbolic diffeomorphisms Ⅲ: Abundance and incoherence, arXiv: 1706.04962.Google Scholar

[7]

J. Bowden, Contact structures, deformations and taut foliations, Geom. Topol., 20 (2016), 697-746. doi: 10.2140/gt.2016.20.697. Google Scholar

[8]

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

[9]

M. Brittenham, Essential laminations in seifert fibered spaces, Topology, 32 (1993), 61-85. doi: 10.1016/0040-9383(93)90038-W. Google Scholar

[10]

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

[11]

K. Burns and A. Wilkinson, Dynamical coherence and center bunching, Discrete Contin. Dyn. Syst., 22 (2008), 89-100. doi: 10.3934/dcds.2008.22.89. Google Scholar

[12]

D. Calegari, Foliations and the Geometry of 3-Manifolds, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2007. Google Scholar

[13]

A. Candel and L. Conlon, Foliations I, Graduate Studies in Mathematics, 23, American Mathematical Society, Providence, RI, 2000; Foliations II, Graduate Studies in Mathematics, 60, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/060. Google Scholar

[14]

P. Carrasco, F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, Partially hyperbolic dynamics in dimension 3, arXiv: 1501.00932.Google Scholar

[15]

S. Choi, Geometric Structures on 2-Orbifolds: Exploration of Discrete Symmetry, MSJ Memoirs, 27, Mathematical Society of Japan, Tokyo, 2012. doi: 10.1142/e035. Google Scholar

[16]

D. Eisenbud, U. Hirsch and W. Neumann, Transverse foliations of Seifert bundles and self-homeomorphism of the circle, Comment. Math. Helv., 56 (1981), 638-660. doi: 10.1007/BF02566232. Google Scholar

[17]

É. Ghys, Flots d'Anosov sur les 3-variétés fibrées en cercles, Ergodic Theory Dynam. Systems, 4 (1984), 67-80. doi: 10.1017/S0143385700002273. Google Scholar

[18]

N. Gourmelon, Adapted metrics for dominated splittings, Ergodic Theory Dynam. Systems, 27 (2007), 1839-1849. doi: 10.1017/S0143385707000272. Google Scholar

[19]

A. Hammerlindl, Horizontal vector fields and Seifert fiberings, arXiv: 1803.09922.Google Scholar

[20]

A. Hammerlindl and R. Potrie, Pointwise partial hyperbolicity in three-dimensional nilmanifolds, J. Lond. Math. Soc.(2), 89 (2014), 853-875. doi: 10.1112/jlms/jdu013. Google Scholar

[21]

A. Hammerlindl and R. Potrie, Classification of partially hyperbolic diffeomorphisms in three dimensional manifolds with solvable fundamental group, J. Topol., 8 (2015), 842-870. doi: 10.1112/jtopol/jtv009. Google Scholar

[22]

A. Hammerlindl and R. Potrie, Partial hyperbolicity and classification: A survey, Ergodic Theory Dynam. Systems, 38 (2018), 401-443. doi: 10.1017/etds.2016.50. Google Scholar

[23]

A. Hatcher, Notes on basic 3-manifold topology, Available from: http://www.math.cornell.edu/~hatcher.Google Scholar

[24]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. Google Scholar

[25]

M. Jankins and W. Neumann, Lectures on Seifert manifolds, Brandeis Lecture Notes, 2, Brandeis University, Waltham, MA, 1983. Google Scholar

[26]

G. Levitt, Feuilletages des variétés de dimension 3 qui sont fibrés en circles, Comment. Math. Helv., 53 (1978), 572-594. doi: 10.1007/BF02566099. Google Scholar

[27]

G. Levitt, Foliations and laminations on hyperbolic surfaces, Topology, 22 (1983), 119-135. doi: 10.1016/0040-9383(83)90023-X. Google Scholar

[28]

K. Mann, Spaces of surface group representations, Invent. Math., 201 (2015), 669-710. doi: 10.1007/s00222-014-0558-4. Google Scholar

[29]

R. Naimi, Foliations transverse to fibers of Seifert manifolds, Comment. Math. Helv., 69 (1994), 155-162. doi: 10.1007/BF02564479. Google Scholar

[30]

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

[31]

F. Rodriguez Hertz, M. A. 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

[32]

F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, A non-dynamically coherent example on $\mathbb{T}^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1023-1032. doi: 10.1016/j.anihpc.2015.03.003. Google Scholar

[33]

P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc., 15 (1983), 401-487. doi: 10.1112/blms/15.5.401. Google Scholar

[34]

V. V. Solodov, Components of topological foliations, (Russian) Mat. Sb. (N.S.), 119 (1982), 340–354, 447. Google Scholar

show all references

References:
[1]

T. Barbot, Flots d'Anosov sur les variétés graphées au sens de Waldhausen, Ann. Inst. Fourier (Grenoble), 46 (1996), 1451-1517. doi: 10.5802/aif.1556. Google Scholar

[2]

T. Barbot, Actions de groupes sur les 1-variétés non séparées et feuilletages de codimension un, Ann. Fac. Sci. Toulouse Math.(6), 7 (1998), 559-597. doi: 10.5802/afst.911. Google Scholar

[3]

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

[4]

C. Bonatti, K. Parwani and R. Potrie, Anomalous partially hyperbolic diffeomorphisms Ⅰ: Dynamically coherent examples, Ann. Sci. Éc. Norm. Supér.(4), 49 (2016), 1387-1402. doi: 10.24033/asens.2311. Google Scholar

[5]

C. Bonatti, A. Gogolev and R. Potrie, Anomalous partially hyperbolic diffeomorphisms Ⅱ: Stably ergodic examples, Invent. Math., 206 (2016), 801-836. doi: 10.1007/s00222-016-0663-7. Google Scholar

[6]

C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie, Anomalous partially hyperbolic diffeomorphisms Ⅲ: Abundance and incoherence, arXiv: 1706.04962.Google Scholar

[7]

J. Bowden, Contact structures, deformations and taut foliations, Geom. Topol., 20 (2016), 697-746. doi: 10.2140/gt.2016.20.697. Google Scholar

[8]

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

[9]

M. Brittenham, Essential laminations in seifert fibered spaces, Topology, 32 (1993), 61-85. doi: 10.1016/0040-9383(93)90038-W. Google Scholar

[10]

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

[11]

K. Burns and A. Wilkinson, Dynamical coherence and center bunching, Discrete Contin. Dyn. Syst., 22 (2008), 89-100. doi: 10.3934/dcds.2008.22.89. Google Scholar

[12]

D. Calegari, Foliations and the Geometry of 3-Manifolds, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2007. Google Scholar

[13]

A. Candel and L. Conlon, Foliations I, Graduate Studies in Mathematics, 23, American Mathematical Society, Providence, RI, 2000; Foliations II, Graduate Studies in Mathematics, 60, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/060. Google Scholar

[14]

P. Carrasco, F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, Partially hyperbolic dynamics in dimension 3, arXiv: 1501.00932.Google Scholar

[15]

S. Choi, Geometric Structures on 2-Orbifolds: Exploration of Discrete Symmetry, MSJ Memoirs, 27, Mathematical Society of Japan, Tokyo, 2012. doi: 10.1142/e035. Google Scholar

[16]

D. Eisenbud, U. Hirsch and W. Neumann, Transverse foliations of Seifert bundles and self-homeomorphism of the circle, Comment. Math. Helv., 56 (1981), 638-660. doi: 10.1007/BF02566232. Google Scholar

[17]

É. Ghys, Flots d'Anosov sur les 3-variétés fibrées en cercles, Ergodic Theory Dynam. Systems, 4 (1984), 67-80. doi: 10.1017/S0143385700002273. Google Scholar

[18]

N. Gourmelon, Adapted metrics for dominated splittings, Ergodic Theory Dynam. Systems, 27 (2007), 1839-1849. doi: 10.1017/S0143385707000272. Google Scholar

[19]

A. Hammerlindl, Horizontal vector fields and Seifert fiberings, arXiv: 1803.09922.Google Scholar

[20]

A. Hammerlindl and R. Potrie, Pointwise partial hyperbolicity in three-dimensional nilmanifolds, J. Lond. Math. Soc.(2), 89 (2014), 853-875. doi: 10.1112/jlms/jdu013. Google Scholar

[21]

A. Hammerlindl and R. Potrie, Classification of partially hyperbolic diffeomorphisms in three dimensional manifolds with solvable fundamental group, J. Topol., 8 (2015), 842-870. doi: 10.1112/jtopol/jtv009. Google Scholar

[22]

A. Hammerlindl and R. Potrie, Partial hyperbolicity and classification: A survey, Ergodic Theory Dynam. Systems, 38 (2018), 401-443. doi: 10.1017/etds.2016.50. Google Scholar

[23]

A. Hatcher, Notes on basic 3-manifold topology, Available from: http://www.math.cornell.edu/~hatcher.Google Scholar

[24]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. Google Scholar

[25]

M. Jankins and W. Neumann, Lectures on Seifert manifolds, Brandeis Lecture Notes, 2, Brandeis University, Waltham, MA, 1983. Google Scholar

[26]

G. Levitt, Feuilletages des variétés de dimension 3 qui sont fibrés en circles, Comment. Math. Helv., 53 (1978), 572-594. doi: 10.1007/BF02566099. Google Scholar

[27]

G. Levitt, Foliations and laminations on hyperbolic surfaces, Topology, 22 (1983), 119-135. doi: 10.1016/0040-9383(83)90023-X. Google Scholar

[28]

K. Mann, Spaces of surface group representations, Invent. Math., 201 (2015), 669-710. doi: 10.1007/s00222-014-0558-4. Google Scholar

[29]

R. Naimi, Foliations transverse to fibers of Seifert manifolds, Comment. Math. Helv., 69 (1994), 155-162. doi: 10.1007/BF02564479. Google Scholar

[30]

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

[31]

F. Rodriguez Hertz, M. A. 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

[32]

F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, A non-dynamically coherent example on $\mathbb{T}^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1023-1032. doi: 10.1016/j.anihpc.2015.03.003. Google Scholar

[33]

P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc., 15 (1983), 401-487. doi: 10.1112/blms/15.5.401. Google Scholar

[34]

V. V. Solodov, Components of topological foliations, (Russian) Mat. Sb. (N.S.), 119 (1982), 340–354, 447. Google Scholar

Figure 1.  The concatenation on the left is coherently oriented and the one on the right is not
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Figure 2.  Cutting a curve in the concatenation
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Figure 3.  Constructing a vector field in a section of the bundle
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Figure 4.  A map from the bundle to the unit tangent bundle
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