Journal of Modern Dynamics (JMD)

Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds
Pages: 497 - 509, Volume 10, 2016

doi:10.3934/jmd.2016.10.497      Abstract        References        Full text (196.1K)           Related Articles

Marcelo R. R. Alves - Institut de Mathématiques, Université de Neuchâtel, Rue Émile Argand 11, CP 158, 2000 Neuchâtel, Switzerland (email)

1 M. R. R. Alves, Legendrian contact homology and topological entropy, arXiv:1410.3381, (2014).
2 M. R. R. Alves, Cylindrical contact homology and topological entropy, to appear in Geometry & Topology, arXiv:1410.3380.
3 D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209pp.       
4 T. Barthelmé and S. R. Fenley, Counting periodic orbits of Anosov flows in free homotopy classes, arXiv:1505.07999, (2015).
5 F. Bourgeois, Contact homology and homotopy groups of the space of contact structures, Math. Res. Lett., 13 (2006), 71-85.       
6 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.       
7 F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki and E. Zehnder, Compactness results in symplectic field theory, Geom. Topol., 7 (2003), 799-888.       
8 V. Colin and K. Honda, Reeb vector fields and open book decompositions, J. Eur. Math. Soc. (JEMS), 15 (2013), 443-507.       
9 D. Dragnev, Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations, Comm. Pure Appl. Math., 57 (2004), 726-763.       
10 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.       
11 S. R. Fenley, Anosov flows in 3-manifolds, Ann. of Math. (2), 139 (1994), 79-115.       
12 S. R. Fenley, Homotopic indivisibility of closed orbits of $3$-dimensional Anosov flows, Math. Z., 225 (1997), 289-294.       
13 P. Foulon and B. Hasselblatt, Contact Anosov flows on hyperbolic 3-manifolds, Geom. Topol., 17 (2013), 1225-1252.       
14 U. Frauenfelder and F. Schlenk, Volume growth in the component of the Dehn-Seidel twist, Geom. Funct. Anal. (GAFA), 15 (2005), 809-838.       
15 U. Frauenfelder and F. Schlenk, Fiberwise volume growth via Lagrangian intersections, J. Symplectic Geom., 4 (2006), 117-148.       
16 H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math., 114 (1993), 515-563.       
17 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.       
18 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.       
19 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.       
20 A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.       
21 A. Katok, Entropy and closed geodesies, Ergodic Theory Dynam. Systems, 2 (1982), 339-365.       
22 W. Klingenberg, Riemannian manifolds with geodesic flow of Anosov type, Ann. of Math. (2), 99 (1974), 1-13.       
23 Y. Lima and O. Sarig, Symbolic dynamics for three dimensional flows with positive topological entropy, arXiv:1408.3427, (2014).
24 L. Macarini and G. P. Paternain, Equivariant symplectic homology of Anosov contact structures, Bull. Braz. Math. Soc. (N.S.), 43 (2012), 513-527.       
25 L. Macarini and F. Schlenk, Positive topological entropy of Reeb flows on spherizations, Math. Proc. Cambridge Philos. Soc., 151 (2011), 103-128.       
26 A. Manning, Topological entropy for geodesic flows, Ann. of Math. (2), 110 (1979), 567-573.       
27 C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, Boca Raton, FL, 1995.       
28 O. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy, J. Amer. Math. Soc., 26 (2013), no. 2, 341-426.       
29 A. Vaugon, Contact homology of contact Anosov flows, preprint. Available from: http://anne.vaugon.vwx.fr/Anosov.pdf.

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