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

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

1. 

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

Received  January 2016 Revised  August 2016 Published  November 2016

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.
Citation: Marcelo R. R. Alves. Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds. Journal of Modern Dynamics, 2016, 10: 497-509. doi: 10.3934/jmd.2016.10.497
References:
[1]

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

[2]

M. R. R. Alves, Cylindrical contact homology and topological entropy,, to appear in Geometry & Topology, ().

[3]

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

[4]

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

[5]

F. Bourgeois, Contact homology and homotopy groups of the space of contact structures,, Math. Res. Lett., 13 (2006), 71. 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, (2009).

[7]

F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki and E. Zehnder, Compactness results in symplectic field theory,, Geom. Topol., 7 (2003), 799. 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. 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. 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, (2000), 560. 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. doi: 10.2307/2946628.

[12]

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

[13]

P. Foulon and B. Hasselblatt, Contact Anosov flows on hyperbolic 3-manifolds,, Geom. Topol., 17 (2013), 1225. 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. 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. 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. 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.

[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. 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), 333. 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.

[21]

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

[22]

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

[23]

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

[24]

L. Macarini and G. P. Paternain, Equivariant symplectic homology of Anosov contact structures,, Bull. Braz. Math. Soc. (N.S.), 43 (2012), 513. 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. doi: 10.1017/S0305004111000119.

[26]

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

[27]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos,, CRC Press, (1995).

[28]

O. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy,, J. Amer. Math. Soc., 26 (2013), 341. doi: 10.1090/S0894-0347-2012-00758-9.

[29]

A. Vaugon, Contact homology of contact Anosov flows,, preprint. Available from: , ().

show all references

References:
[1]

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

[2]

M. R. R. Alves, Cylindrical contact homology and topological entropy,, to appear in Geometry & Topology, ().

[3]

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

[4]

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

[5]

F. Bourgeois, Contact homology and homotopy groups of the space of contact structures,, Math. Res. Lett., 13 (2006), 71. 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, (2009).

[7]

F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki and E. Zehnder, Compactness results in symplectic field theory,, Geom. Topol., 7 (2003), 799. 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. 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. 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, (2000), 560. 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. doi: 10.2307/2946628.

[12]

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

[13]

P. Foulon and B. Hasselblatt, Contact Anosov flows on hyperbolic 3-manifolds,, Geom. Topol., 17 (2013), 1225. 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. 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. 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. 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.

[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. 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), 333. 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.

[21]

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

[22]

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

[23]

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

[24]

L. Macarini and G. P. Paternain, Equivariant symplectic homology of Anosov contact structures,, Bull. Braz. Math. Soc. (N.S.), 43 (2012), 513. 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. doi: 10.1017/S0305004111000119.

[26]

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

[27]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos,, CRC Press, (1995).

[28]

O. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy,, J. Amer. Math. Soc., 26 (2013), 341. doi: 10.1090/S0894-0347-2012-00758-9.

[29]

A. Vaugon, Contact homology of contact Anosov flows,, preprint. Available from: , ().

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