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

Seifert manifolds admitting partially hyperbolic diffeomorphisms

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.

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.

[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.

[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.

[4]

C. BonattiK. 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.

[5]

C. BonattiA. 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.

[6]

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

[7]

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

[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.

[9]

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

[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.

[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.

[12]

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

[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.

[14]

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

[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.

[16]

D. EisenbudU. 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.

[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.

[18]

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

[19]

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

[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.

[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.

[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.

[23]

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

[24]

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

[25]

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

[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.

[27]

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

[28]

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

[29]

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

[30]

K. Parwani, On 3-manifolds that support partially hyperbolic diffeomorphisms, Nonlinearity, 23 (2010), 589-606. doi: 10.1088/0951-7715/23/3/009.

[31]

F. Rodriguez HertzM. 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.

[32]

F. Rodriguez HertzM. 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.

[33]

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

[34]

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

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.

[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.

[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.

[4]

C. BonattiK. 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.

[5]

C. BonattiA. 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.

[6]

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

[7]

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

[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.

[9]

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

[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.

[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.

[12]

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

[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.

[14]

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

[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.

[16]

D. EisenbudU. 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.

[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.

[18]

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

[19]

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

[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.

[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.

[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.

[23]

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

[24]

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

[25]

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

[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.

[27]

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

[28]

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

[29]

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

[30]

K. Parwani, On 3-manifolds that support partially hyperbolic diffeomorphisms, Nonlinearity, 23 (2010), 589-606. doi: 10.1088/0951-7715/23/3/009.

[31]

F. Rodriguez HertzM. 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.

[32]

F. Rodriguez HertzM. 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.

[33]

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

[34]

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

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