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

Marcelo R. R.  Alves

doi:10.3934/jmd.2016.10.497

Article Contents

2016, Volume 10: 497-509. Doi: 10.3934/jmd.2016.10.497

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

Marcelo R. R. Alves^1,

1.
Institut de Mathématiques, Université de Neuchâtel, Rue Émile Argand 11, CP 158, 2000 Neuchâtel

Received: January 2016

Revised: August 2016

Published: October 2016

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Abstract

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.

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Mathematics Subject Classification: Primary: 37J05, 37D20, 53D42; Secondary: 37B40, 53D35.

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References

[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.doi: 10.4310/MRL.2006.v13.n1.a6.
[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.doi: 10.2140/gt.2003.7.799.
[8]	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.
[9]	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.
[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.doi: 10.1007/978-3-0346-0425-3_4.
[11]	S. R. Fenley, Anosov flows in 3-manifolds, Ann. of Math. (2), 139 (1994), 79-115.doi: 10.2307/2946628.
[12]	S. R. Fenley, Homotopic indivisibility of closed orbits of $3$-dimensional Anosov flows, Math. Z., 225 (1997), 289-294.doi: 10.1007/PL00004313.
[13]	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.
[14]	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.
[15]	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.
[16]	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.
[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.doi: 10.4007/annals.2003.157.125.
[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.doi: 10.1007/s00222-014-0515-2.
[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.doi: 10.1017/S0143385700001656.
[22]	W. Klingenberg, Riemannian manifolds with geodesic flow of Anosov type, Ann. of Math. (2), 99 (1974), 1-13.doi: 10.2307/1971011.
[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.doi: 10.1007/s00574-012-0024-0.
[25]	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.
[26]	A. Manning, Topological entropy for geodesic flows, Ann. of Math. (2), 110 (1979), 567-573.doi: 10.2307/1971239.
[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.doi: 10.1090/S0894-0347-2012-00758-9.
[29]	A. Vaugon, Contact homology of contact Anosov flows, preprint. Available from: http://anne.vaugon.vwx.fr/Anosov.pdf.