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: |
[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.
![]() ![]() |