Symplectic geometry of Anosov flows in dimension 3 and bi-contact topology.(English)Zbl 1553.37054
Summary: We give a purely contact and symplectic geometric characterization of Anosov flows in dimension 3 and discuss a framework to use tools from contact and symplectic geometry and topology in the study of Anosov dynamics. We also discuss some uniqueness results regarding the underlying (bi)-contact structures for an Anosov flow and give a characterization of Anosovity based on Reeb flows.
MSC:
| 37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |
| 37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |
| 53D05 | Symplectic manifolds (general theory) |
| 53D10 | Contact manifolds (general theory) |
| 53E50 | Flows related to symplectic and contact structures |
| 57R17 | Symplectic and contact topology in high or arbitrary dimension |
Cite
References:
| [1] | Anosov, D. V., Ergodic properties of geodesic flows on closed Riemannian manifolds of negative curvature, Dokl. Akad. Nauk SSSR, 151, 6, 1250-1252, 1963 ·Zbl 0135.40402 |
| [2] | Anosov, D. Victorovich, Geodesic flows on closed Riemannian manifolds of negative curvature, Tr. Mat. Inst. Steklova, 90, 5, 1967 ·Zbl 0163.43604 |
| [3] | Arroyo, A.; Rodriguez Hertz, F., Homoclinic bifurcations and uniform hyperbolicity for three-dimensional flows, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 20, 5, 2003 ·Zbl 1045.37006 |
| [4] | Asaoka, M., Regular projectively Anosov flows on three-dimensional manifolds, Ann. Inst. Fourier, 60, 5, 2010 ·Zbl 1202.37030 |
| [5] | Asaoka, M.; Dufraine, E.; Noda, T., Homotopy classes of total foliations, Comment. Math. Helv., 87, 2, 271-302, 2012 ·Zbl 1244.57049 |
| [6] | Barbot, T., Caractérisation des flots d’Anosov en dimension 3 par leurs feuilletages faibles, Ergod. Theory Dyn. Syst., 15, 2, 247-270, 1995 ·Zbl 0826.58025 |
| [7] | Barthelmé, T., Anosov Flows in Dimension 3, Preliminary Version, 2017 |
| [8] | Barthelmé, T.; Fenley, S. R., Knot theory of \(\mathbb{R} \)-covered Anosov flows: homotopy versus isotopy of closed orbits, J. Topol., 7, 3, 677-696, 2014 ·Zbl 1311.37016 |
| [9] | Béguin, F.; Bonatti, C.; Yu, B., Building Anosov flows on 3-manifolds, Geom. Topol., 21, 3, 1837-1930, 2017 ·Zbl 1375.37083 |
| [10] | Blair, D. E.; Peronne, D., Conformally Anosov flows in contact metric geometry, Balk. J. Geom. Appl., 3, 2, 33-46, 1998 ·Zbl 0955.53044 |
| [11] | Bonatti, C.; Bowden, J.; Potrie, R., Some Remarks on Projective Anosov Flows in Hyperbolic 3-Manifolds, 2018 MATRIX Annals, 359-369, 2020, Springer: Springer Cham ·Zbl 1447.57020 |
| [12] | Bonatti, C.; Díaz, L. J.; Viana, M., Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective, vol. 102, 2006, Springer Science & Business Media |
| [13] | Bowden, J., Approximating \(C^0\)-foliations by contact structures, Geom. Funct. Anal., 26, 1255-1296, 2016 ·Zbl 1362.57037 |
| [14] | Brunella, M., Separating the basic sets of a nontransitive Anosov flow, Bull. Lond. Math. Soc., 25, 5, 487-490, 1993 ·Zbl 0790.58028 |
| [15] | A. Cioba, C. Wendl, Unknotted Reeb orbits and nicely embedded holomorphic curves, Preprint arXiv, 2016. ·Zbl 1436.53062 |
| [16] | Colin, V.; Firmo, S., Paires de structures de contact sur les variétés de dimension trois, Algebraic Geom. Topol., 11, 5, 2627-2653, 2011 ·Zbl 1234.57019 |
| [17] | De La Llave, R., Rigidity of higher-dimensional conformal Anosov systems, Ergod. Theory Dyn. Syst., 22, 6, 1845-1870, 2002 ·Zbl 1028.37021 |
| [18] | Doering, C., Persistently transitive vector fields on three-dimensional manifolds, Dynamical Systems and Bifurcation Theory. Dynamical Systems and Bifurcation Theory, Pitman Res. Notes Math. Ser., 160, 59-89, 1987 ·Zbl 0631.58016 |
| [19] | Eliashberg, Y., A few remarks about symplectic filling, Geom. Topol., 8, 1, 277-293, 2004 ·Zbl 1067.53070 |
| [20] | Eliashberg, Y., Classification of overtwisted contact structures on 3-manifolds, Invent. Math., 98, 3, 623-637, 1989 ·Zbl 0684.57012 |
| [21] | Eliashberg, Y., Contact 3-manifolds twenty years since J. Martinet’s work, Ann. Inst. Fourier, 42, 1-2, 1992 ·Zbl 0756.53017 |
| [22] | Eliashberg, Y., Filling by holomorphic discs and its applications, (Geometry of Low-Dimensional Manifolds: Volume 2: Symplectic Manifolds and Jones-Witten Theory, 1990), 45 ·Zbl 0731.53036 |
| [23] | Eliashberg, Y., Unique holomorphically fillable contact structure on the 3-torus, Int. Math. Res. Not., 1996, 2, 77-82, 1996 ·Zbl 0852.58034 |
| [24] | Eliashberg, Y.; Thurston, W. P., Confoliations, vol. 13, 1998, American Mathematical Soc. ·Zbl 0893.53001 |
| [25] | Etnyre, J. B., On symplectic fillings, Algebraic Geom. Topol., 4, 1, 73-80, 2004 ·Zbl 1078.53074 |
| [26] | Etnyre, J.; Ghrist, R., Tight contact structures and Anosov flows, Topol. Appl., 124, 2, 211-219, 2002 ·Zbl 1028.53080 |
| [27] | Fenley, S. R., Anosov flows in 3-manifolds, Ann. Math., 139, 1, 79-115, 1994 ·Zbl 0796.58039 |
| [28] | Fenley, S. R., Quasigeodesic Anosov flows and homotopic properties of flow lines, J. Differ. Geom., 41, 2, 479-514, 1995 ·Zbl 0832.57009 |
| [29] | Foulon, P.; Hasselblatt, B., Contact Anosov flows on hyperbolic 3-manifolds, Geom. Topol., 17, 2, 1225-1252, 2013 ·Zbl 1277.37057 |
| [30] | Gay, D. T., Four-dimensional symplectic cobordisms containing three-handles, Geom. Topol., 10, 3, 1749-1759, 2006 ·Zbl 1129.53061 |
| [31] | Geiges, H., An Introduction to Contact Topology, vol. 109, 2008, Cambridge University Press ·Zbl 1153.53002 |
| [32] | Giroux, E., Structures de contact en dimension trois et bifurcations des feuilletages de surfaces, Invent. Math., 141, 3, 615-689, 2000 ·Zbl 1186.53097 |
| [33] | Gromov, M., Pseudo holomorphic curves in symplectic manifolds, Invent. Math., 82, 2, 307-347, 1985 ·Zbl 0592.53025 |
| [34] | Hasselblatt, B., Regularity of the Anosov splitting and of horospheric foliations, Ergod. Theory Dyn. Syst., 14, 4, 645-666, 1994 ·Zbl 0821.58032 |
| [35] | Haefliger, A., Variétés feuilletées, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (3), 16, 367-397, 1962, (in French) ·Zbl 0122.40702 |
| [36] | Hirsch, M. W.; Pugh, C. C.; Shub, M., Invariant Manifolds, vol. 583, 2006, Springer |
| [37] | Hofer, H., Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math., 114, 1, 515-563, 1993 ·Zbl 0797.58023 |
| [38] | Hofer, H.; Wysocki, K.; Zehnder, E., Unknotted periodic orbits for Reeb flows on the three-sphere, Topol. Methods Nonlinear Anal., 7, 2, 219-244, 1996 ·Zbl 0898.58018 |
| [39] | Honda, Ko., On the classification of tight contact structures I, Geom. Topol., 4, 1, 309-368, 2000 ·Zbl 0980.57010 |
| [40] | Honda, Ko., On the classification of tight contact structures II, J. Differ. Geom., 55, 1, 83-143, 2000 ·Zbl 1038.57007 |
| [41] | Hozoori, S., Dynamics and topology of conformally Anosov contact 3-manifolds, Differ. Geom. Appl., 73, Article 101679 pp., 2020 ·Zbl 1460.53066 |
| [42] | Hozoori, S., On Anosovity, divergence and bi-contact surgery, 2021, preprint ·Zbl 1537.37039 |
| [43] | Hozoori, S., Ricci curvature, Reeb flows and contact 3-manifolds, J. Geom. Anal., 1-26, 2021 ·Zbl 1514.53131 |
| [44] | Hurder, S.; Katok, A., Differentiability, rigidity and Godbillon-Vey classes for Anosov flows, Publ. Math. Inst. Hautes Études Sci., 72, 1, 5-61, 1990 ·Zbl 0725.58034 |
| [45] | Kanai, M., Differential-geometric studies on dynamics of geodesic and frame flows, Jpn. J. Math. New Ser., 19, 1, 1-30, 1993 ·Zbl 0798.58055 |
| [46] | Kazez, W.; Roberts, R., \( C^0\) approximations of foliations, Geom. Topol., 21, 6, 3601-3657, 2017 ·Zbl 1381.57014 |
| [47] | Margulis, G. A., On Some Aspects of the Theory of Anosov Systems, 1-71, 2004, Springer: Springer Berlin, Heidelberg ·Zbl 1140.37010 |
| [48] | Massot, P.; Niederkrüger, K.; Wendl, C., Weak and strong fillability of higher dimensional contact manifolds, Invent. Math., 192, 2, 287-373, 2013 ·Zbl 1277.57026 |
| [49] | McDuff, D., Symplectic manifolds with contact type boundaries, Invent. Math., 103, 1, 651-671, 1991 ·Zbl 0719.53015 |
| [50] | Mosher, L., Dynamical systems and the homology norm of a 3-manifold, I: efficient intersection of surfaces and flows, Duke Math. J., 65, 3, 449-500, 1992 ·Zbl 0754.58030 |
| [51] | Mitsumatsu, Y., Anosov flows and non-Stein symplectic manifolds, Ann. Inst. Fourier, 45, 5, 1995 ·Zbl 0988.57521 |
| [52] | Noda, T., Projectively Anosov flows with differentiable (un) stable foliations, Ann. Inst. Fourier, 50, 5, 2000 ·Zbl 1023.37014 |
| [53] | Plante, J. F.; Thurston, W. P., Anosov flows and the fundamental group, Topology, 11, 2, 147-150, 1972 ·Zbl 0246.58014 |
| [54] | Perrone, D., Torsion and conformally Anosov flows in contact Riemannian geometry, J. Geom., 83, 1-2, 164-174, 2005 ·Zbl 1085.53073 |
| [55] | Pujals, E., From hyperbolicity to dominated splitting, Inst. Mat. Pura Appl., 2006 |
| [56] | Pujals, E. R.; Sambarino, M., On the dynamics of dominated splitting, Ann. Math., 675-739, 2009 ·Zbl 1178.37032 |
| [57] | Sambarino, M., A (Short) Survey on Dominated Splittings, Mathematical Congress of the Americas, vol. 656, 2016, American Mathematical Soc. ·Zbl 1378.37066 |
| [58] | Simić, S., Codimension one Anosov flows and a conjecture of Verjovsky, Ergod. Theory Dyn. Syst., 17, 5, 1211-1231, 1997 ·Zbl 0903.58026 |
| [59] | Sullivan, D., On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, (Riemann Surfaces and Related Topics, Proceedings of the 1978 Stony Brook Conference, State Univ. New York, Stony Brook, 1981, Princeton Univ. Press) ·Zbl 0567.58015 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
