# American Institute of Mathematical Sciences

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). Google Scholar [2] M. R. R. Alves, Cylindrical contact homology and topological entropy,, to appear in Geometry & Topology, (). Google Scholar [3] D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature,, Trudy Mat. Inst. Steklov., 90 (1967). Google Scholar [4] T. Barthelmé and S. R. Fenley, Counting periodic orbits of Anosov flows in free homotopy classes,, , (2015). Google Scholar [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. Google Scholar [6] F. Bourgeois, A survey of contact homology,, in New Perspectives and Challenges in Symplectic Field Theory, (2009). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [11] S. R. Fenley, Anosov flows in 3-manifolds,, Ann. of Math. (2), 139 (1994), 79. doi: 10.2307/2946628. Google Scholar [12] S. R. Fenley, Homotopic indivisibility of closed orbits of $3$-dimensional Anosov flows,, Math. Z., 225 (1997), 289. doi: 10.1007/PL00004313. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137. Google Scholar [21] A. Katok, Entropy and closed geodesies,, Ergodic Theory Dynam. Systems, 2 (1982), 339. doi: 10.1017/S0143385700001656. Google Scholar [22] W. Klingenberg, Riemannian manifolds with geodesic flow of Anosov type,, Ann. of Math. (2), 99 (1974), 1. doi: 10.2307/1971011. Google Scholar [23] Y. Lima and O. Sarig, Symbolic dynamics for three dimensional flows with positive topological entropy,, , (2014). Google Scholar [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. Google Scholar [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. Google Scholar [26] A. Manning, Topological entropy for geodesic flows,, Ann. of Math. (2), 110 (1979), 567. doi: 10.2307/1971239. Google Scholar [27] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos,, CRC Press, (1995). Google Scholar [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. Google Scholar [29] A. Vaugon, Contact homology of contact Anosov flows,, preprint. Available from: , (). Google Scholar

show all references

##### References:
 [1] M. R. R. Alves, Legendrian contact homology and topological entropy,, , (2014). Google Scholar [2] M. R. R. Alves, Cylindrical contact homology and topological entropy,, to appear in Geometry & Topology, (). Google Scholar [3] D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature,, Trudy Mat. Inst. Steklov., 90 (1967). Google Scholar [4] T. Barthelmé and S. R. Fenley, Counting periodic orbits of Anosov flows in free homotopy classes,, , (2015). Google Scholar [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. Google Scholar [6] F. Bourgeois, A survey of contact homology,, in New Perspectives and Challenges in Symplectic Field Theory, (2009). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [11] S. R. Fenley, Anosov flows in 3-manifolds,, Ann. of Math. (2), 139 (1994), 79. doi: 10.2307/2946628. Google Scholar [12] S. R. Fenley, Homotopic indivisibility of closed orbits of $3$-dimensional Anosov flows,, Math. Z., 225 (1997), 289. doi: 10.1007/PL00004313. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137. Google Scholar [21] A. Katok, Entropy and closed geodesies,, Ergodic Theory Dynam. Systems, 2 (1982), 339. doi: 10.1017/S0143385700001656. Google Scholar [22] W. Klingenberg, Riemannian manifolds with geodesic flow of Anosov type,, Ann. of Math. (2), 99 (1974), 1. doi: 10.2307/1971011. Google Scholar [23] Y. Lima and O. Sarig, Symbolic dynamics for three dimensional flows with positive topological entropy,, , (2014). Google Scholar [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. Google Scholar [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. Google Scholar [26] A. Manning, Topological entropy for geodesic flows,, Ann. of Math. (2), 110 (1979), 567. doi: 10.2307/1971239. Google Scholar [27] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos,, CRC Press, (1995). Google Scholar [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. Google Scholar [29] A. Vaugon, Contact homology of contact Anosov flows,, preprint. Available from: , (). Google Scholar
 [1] Enoch Humberto Apaza Calla, Bulmer Mejia Garcia, Carlos Arnoldo Morales Rojas. Topological properties of sectional-Anosov flows. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4735-4741. doi: 10.3934/dcds.2015.35.4735 [2] Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147 [3] Dou Dou, Meng Fan, Hua Qiu. Topological entropy on subsets for fixed-point free flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6319-6331. doi: 10.3934/dcds.2017273 [4] Yong Fang. Thermodynamic invariants of Anosov flows and rigidity. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1185-1204. doi: 10.3934/dcds.2009.24.1185 [5] Mark Pollicott. Closed orbits and homology for $C^2$-flows. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 529-534. doi: 10.3934/dcds.1999.5.529 [6] Alfonso Artigue. Discrete and continuous topological dynamics: Fields of cross sections and expansive flows. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5911-5927. doi: 10.3934/dcds.2016059 [7] Yong Fang. Rigidity of Hamenstädt metrics of Anosov flows. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1271-1278. doi: 10.3934/dcds.2016.36.1271 [8] Mário Bessa, Jorge Rocha. Three-dimensional conservative star flows are Anosov. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 839-846. doi: 10.3934/dcds.2010.26.839 [9] Oliver Butterley, Carlangelo Liverani. Smooth Anosov flows: Correlation spectra and stability. Journal of Modern Dynamics, 2007, 1 (2) : 301-322. doi: 10.3934/jmd.2007.1.301 [10] Mark Pollicott. Ergodicity of stable manifolds for nilpotent extensions of Anosov flows. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 599-604. doi: 10.3934/dcds.2002.8.599 [11] Christian Bonatti, Nancy Guelman. Axiom A diffeomorphisms derived from Anosov flows. Journal of Modern Dynamics, 2010, 4 (1) : 1-63. doi: 10.3934/jmd.2010.4.1 [12] Michael Usher. Floer homology in disk bundles and symplectically twisted geodesic flows. Journal of Modern Dynamics, 2009, 3 (1) : 61-101. doi: 10.3934/jmd.2009.3.61 [13] Yong Fang. Quasiconformal Anosov flows and quasisymmetric rigidity of Hamenst$\ddot{a}$dt distances. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3471-3483. doi: 10.3934/dcds.2014.34.3471 [14] Artur O. Lopes, Vladimir A. Rosas, Rafael O. Ruggiero. Cohomology and subcohomology problems for expansive, non Anosov geodesic flows. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 403-422. doi: 10.3934/dcds.2007.17.403 [15] Yong Ji, Ercai Chen, Yunping Wang, Cao Zhao. Bowen entropy for fixed-point free flows. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6231-6239. doi: 10.3934/dcds.2019271 [16] Jaeyoo Choy, Hahng-Yun Chu. On the dynamics of flows on compact metric spaces. Communications on Pure & Applied Analysis, 2010, 9 (1) : 103-108. doi: 10.3934/cpaa.2010.9.103 [17] Jeffrey Boland. On rigidity properties of contact time changes of locally symmetric geodesic flows. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 645-650. doi: 10.3934/dcds.2000.6.645 [18] Peter Albers, Urs Frauenfelder. Floer homology for negative line bundles and Reeb chords in prequantization spaces. Journal of Modern Dynamics, 2009, 3 (3) : 407-456. doi: 10.3934/jmd.2009.3.407 [19] Al Momin. Contact homology of orbit complements and implied existence. Journal of Modern Dynamics, 2011, 5 (3) : 409-472. doi: 10.3934/jmd.2011.5.409 [20] Kurt Vinhage. On the rigidity of Weyl chamber flows and Schur multipliers as topological groups. Journal of Modern Dynamics, 2015, 9: 25-49. doi: 10.3934/jmd.2015.9.25

2018 Impact Factor: 0.295

## Metrics

• PDF downloads (13)
• HTML views (0)
• Cited by (2)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]