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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 |
| 2. | CMAT, Facultad de Ciencias, Universidad de la República, Igua 4225, Montevideo 11400, Uruguay |
| 3. | Institute Mathèmatique de Burgogne, Dijon, France |
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
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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.(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. 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. |
| [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.(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 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. |
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. 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. |
| [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.(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 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. |
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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