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Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds

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  • Let $(M, \xi)$ be a compact contact 3-manifold and assume that there exists a contact form $\alpha_0$ on $(M, \xi)$ whose Reeb flow is Anosov. We show this implies that every Reeb flow on $(M, \xi)$ has positive topological entropy, answering a question raised in [2]. Our argument builds on previous work of the author [2] and recent work of Barthelmé and Fenley [4]. This result combined with the work of Foulon and Hasselblatt [13] is then used to obtain the first examples of hyperbolic contact 3-manifolds on which every Reeb flow has positive topological entropy.
    Mathematics Subject Classification: Primary: 37J05, 37D20, 53D42; Secondary: 37B40, 53D35.


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  • [1]

    M. R. R. Alves, Legendrian contact homology and topological entropy, arXiv:1410.3381, (2014).


    M. R. R. Alves, Cylindrical contact homology and topological entropy, to appear in Geometry & Topology, arXiv:1410.3380.


    D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209pp.


    T. Barthelmé and S. R. Fenley, Counting periodic orbits of Anosov flows in free homotopy classes, arXiv:1505.07999, (2015).


    F. Bourgeois, Contact homology and homotopy groups of the space of contact structures, Math. Res. Lett., 13 (2006), 71-85.doi: 10.4310/MRL.2006.v13.n1.a6.


    F. Bourgeois, A survey of contact homology, in New Perspectives and Challenges in Symplectic Field Theory, CRM Proc. Lecture Notes, 49, Amer. Math. Soc., Providence, RI, 2009.


    F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki and E. Zehnder, Compactness results in symplectic field theory, Geom. Topol., 7 (2003), 799-888.doi: 10.2140/gt.2003.7.799.


    V. Colin and K. Honda, Reeb vector fields and open book decompositions, J. Eur. Math. Soc. (JEMS), 15 (2013), 443-507.doi: 10.4171/JEMS/365.


    D. Dragnev, Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations, Comm. Pure Appl. Math., 57 (2004), 726-763.doi: 10.1002/cpa.20018.


    Y. Eliashberg, A. Givental and H. Hofer, Introduction to symplectic field theory, in Visions in Mathematics. GAFA 2000 Special Volume, Part II, Birkhäuser Basel, 2010, 560-673.doi: 10.1007/978-3-0346-0425-3_4.


    S. R. Fenley, Anosov flows in 3-manifolds, Ann. of Math. (2), 139 (1994), 79-115.doi: 10.2307/2946628.


    S. R. Fenley, Homotopic indivisibility of closed orbits of $3$-dimensional Anosov flows, Math. Z., 225 (1997), 289-294.doi: 10.1007/PL00004313.


    P. Foulon and B. Hasselblatt, Contact Anosov flows on hyperbolic 3-manifolds, Geom. Topol., 17 (2013), 1225-1252.doi: 10.2140/gt.2013.17.1225.


    U. Frauenfelder and F. Schlenk, Volume growth in the component of the Dehn-Seidel twist, Geom. Funct. Anal. (GAFA), 15 (2005), 809-838.doi: 10.1007/s00039-005-0526-7.


    U. Frauenfelder and F. Schlenk, Fiberwise volume growth via Lagrangian intersections, J. Symplectic Geom., 4 (2006), 117-148.doi: 10.4310/JSG.2006.v4.n2.a1.


    H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math., 114 (1993), 515-563.doi: 10.1007/BF01232679.


    H. Hofer, K. Wysocki and E. Zehnder, Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 337-379.


    H. Hofer, K. Wysocki and E. Zehnder, Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math. (2), 157 (2003), 125-255.doi: 10.4007/annals.2003.157.125.


    U. Hryniewicz, A. Momin and P. A. S. Salomão, A Poincaré-Birkhoff theorem for tight Reeb flows on $S^3$, Invent. Math., 199 (2015), no. 2, 333-422.doi: 10.1007/s00222-014-0515-2.


    A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.


    A. Katok, Entropy and closed geodesies, Ergodic Theory Dynam. Systems, 2 (1982), 339-365.doi: 10.1017/S0143385700001656.


    W. Klingenberg, Riemannian manifolds with geodesic flow of Anosov type, Ann. of Math. (2), 99 (1974), 1-13.doi: 10.2307/1971011.


    Y. Lima and O. Sarig, Symbolic dynamics for three dimensional flows with positive topological entropy, arXiv:1408.3427, (2014).


    L. Macarini and G. P. Paternain, Equivariant symplectic homology of Anosov contact structures, Bull. Braz. Math. Soc. (N.S.), 43 (2012), 513-527.doi: 10.1007/s00574-012-0024-0.


    L. Macarini and F. Schlenk, Positive topological entropy of Reeb flows on spherizations, Math. Proc. Cambridge Philos. Soc., 151 (2011), 103-128.doi: 10.1017/S0305004111000119.


    A. Manning, Topological entropy for geodesic flows, Ann. of Math. (2), 110 (1979), 567-573.doi: 10.2307/1971239.


    C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, Boca Raton, FL, 1995.


    O. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy, J. Amer. Math. Soc., 26 (2013), no. 2, 341-426.doi: 10.1090/S0894-0347-2012-00758-9.


    A. Vaugon, Contact homology of contact Anosov flows, preprint. Available from: http://anne.vaugon.vwx.fr/Anosov.pdf.

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