Seifert manifolds admitting partially hyperbolic diffeomorphisms

Andy Hammerlindl; Rafael Potrie; Mario Shannon

doi:10.3934/jmd.2018008

Article Contents

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

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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 29, 2017

Revised: March 14, 2018

Published: June 2018

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Abstract

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:

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