Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-02-05T03:35:18.972Z Has data issue: false hasContentIssue false
  • English
  • Français

Reeb orbits that force topological entropy

Part of: Symplectic geometry, contact geometry Topological dynamics

Published online by Cambridge University Press:  02 September 2021

MARCELO R. R. ALVES  and
ABROR PIRNAPASOV Open the ORCID record for ABROR PIRNAPASOV [Opens in a new window]
MARCELO R. R. ALVES*
Affiliation:
Department of Mathematics, University of Antwerp, Campus Middelheim, Middelheimlaan 1, BE-2020Antwerpen, Belgium
ABROR PIRNAPASOV
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum Lehrstuhl X (Analysis), Fach 55 Gebäude IB, Etage 3, Raum 59 D-44780Bochum, Germany (e-mail: [email protected])
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop a forcing theory of topological entropy for Reeb flows in dimension three. A transverse link L in a closed contact $3$ -manifold $(Y,\xi )$ is said to force topological entropy if $(Y,\xi )$ admits a Reeb flow with vanishing topological entropy, and every Reeb flow on $(Y,\xi )$ realizing L as a set of periodic Reeb orbits has positive topological entropy. Our main results establish topological conditions on a transverse link L, which imply that L forces topological entropy. These conditions are formulated in terms of two Floer theoretical invariants: the cylindrical contact homology on the complement of transverse links introduced by Momin [A. Momin. J. Mod. Dyn.5 (2011), 409–472], and the strip Legendrian contact homology on the complement of transverse links, introduced by Alves [M. R. R. Alves. PhD Thesis, Université Libre de Bruxelles, 2014] and further developed here. We then use these results to show that on every closed contact $3$ -manifold that admits a Reeb flow with vanishing topological entropy, there exist transverse knots that force topological entropy.

Keywords

topological entropyReeb dynamicscontact homologytransverse knots

MSC classification

Primary: 37B40: Topological entropy 53D42: Symplectic field theory; contact homology
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 10 , October 2022 , pp. 3025 - 3068
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abbas, C.. Finite energy surfaces and the chord problem. Duke Math. J. 96(2) (1999), 241–316.CrossRefGoogle Scholar
Abbas, C.. Pseudoholomorphic strips in symplectizations. II. Fredholm theory and transversality. Comm. Pure Appl. Math. 57(1) (2004), 1–58.CrossRefGoogle Scholar
Abbondandolo, A., Alves, M. R. R., Saglam, M. and Schlenk, F.. Entropy collapse versus entropy rigidity for Reeb and Finsler flows. Preprint, 2021, arXiv:2103.01144.Google Scholar
Alves, M. R. R.. Growth rate of Legendrian contact homology and dynamics of Reeb flows. PhD Thesis, Université Libre de Bruxelles, 2014.Google Scholar
Alves, M. R. R.. Cylindrical contact homology and topological entropy. Geom. Topol. 20(6) (2016), 3519–3569.CrossRefGoogle Scholar
Alves, M. R. R.. Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds. J. Mod. Dyn. 10 (2016), 497–509.CrossRefGoogle Scholar
Alves, M. R. R.. Legendrian contact homology and topological entropy. J. Topol. Anal. 11(1) (2019), 53–108.CrossRefGoogle Scholar
Alves, M. R. R., Colin, V. and Honda, K.. Topological entropy for Reeb vector fields in dimension three via open book decompositions. J. Éc. polytech. Math. 6 (2019), 119–148.CrossRefGoogle Scholar
Alves, M. R. R., Hryniewicz, U. L., Pirnapasov, A. and Salomão, P. A. S.. A Denvir–Mackay theorem for Reeb flows. Work in preparation.Google Scholar
Alves, M. R. R. and Meiwes, M.. Dynamically exotic contact spheres of dimensions ≥7. Comment. Math. Helv. 94(3) (2019), 569–622.CrossRefGoogle Scholar
Audin, M. and Damian, M.. Morse Theory and Floer Homology (Universitext) . Springer, London, 2014.CrossRefGoogle Scholar
Berger, M.. A Panoramic View of Riemannian Geometry. Springer, Berlin, 2003.CrossRefGoogle Scholar
Bourgeois, F., Eliashberg, Y., Hofer, H., Wysocki, K. and Zehnder, E.. Compactness results in symplectic field theory. Geom. Topol. 7 (2003), 799–888.CrossRefGoogle Scholar
Bowen, R.. Entropy and the fundamental group. The Structure of Attractors in Dynamical Systems (Lecture Notes in Mathematics, 668). Eds. Markley, Nelson G., Martin, John C. and Perrizo, William. Springer, Berlin, 1978, pp. 21–29.CrossRefGoogle Scholar
Boyland, P.. Topological methods in surface dynamics. Topology Appl. 58(3) (1994), 223–298.CrossRefGoogle Scholar
Colin, V., Dehornoy, P. and Rechtman, A.. On the existence of supporting broken book decompositions for contact forms in dimension 3. Preprint, 2020, arXiv:2001.01448.Google Scholar
Colin, V. and Honda, K.. Stabilizing the monodromy of an open book decomposition. Geom. Dedicata 132(1) (2008), 95–103.CrossRefGoogle Scholar
Dahinden, L.. Lower complexity bounds for positive contactomorphisms. Israel J. Math. 224(1) (2018), 367–383.CrossRefGoogle Scholar
Dahinden, L.. Positive topological entropy of positive contactomorphisms. J. Symplectic Geom. 18(3) (2020), 691–732.CrossRefGoogle Scholar
Denvir, J. and Mackay, R. S.. Consequences of contractible geodesics on surfaces. Trans. Amer. Math. Soc. 350(11) (1998), 4553–4568.CrossRefGoogle Scholar
Dimitroglou Rizell, G.. Legendrian ambient surgery and legendrian contact homology. J. Symplectic Geom. 14(3) (2016), 811–901.CrossRefGoogle Scholar
Dragnev, D. L.. Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations. Comm. Pure Appl. Math. 57(6) (2004), 726–763.CrossRefGoogle Scholar
Ekholm, T.. Rational symplectic field theory over ${\mathbb{Z}}_2$ for exact Lagrangian cobordisms. J. Eur. Math. Soc. (JEMS) 10(3) (2008), 641–704.CrossRefGoogle Scholar
Eliashberg, Y., Givental, A. and Hofer, H.. Introduction to symplectic field theory. Geom. Funct. Anal. Number Special Volume (Part II) (2000), 560–673.Google Scholar
Foulon, P., Hasselblatt, B. and Vaugon, A.. Orbit growth of contact structures after surgery. Preprint, 2019, arXiv:1910.14357.Google Scholar
Geiges, H.. An Introduction to Contact Topology (Cambridge Studies in Advanced Mathematics, 109). Cambridge University Press, Cambridge, 2008.CrossRefGoogle Scholar
Gromov, M.. Pseudo holomorphic curves in symplectic manifolds. Invent. Math. 82(2) (1985), 307–347.CrossRefGoogle Scholar
Hofer, H.. Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three. Invent. Math. 114(3) (1993), 515–563.CrossRefGoogle Scholar
Hofer, H., Wysocki, K. and Zehnder, E.. Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics. Ann. Inst. H. Poincaré Anal. Non Linéaire 13(3) (1996), 337–379.CrossRefGoogle Scholar
Hryniewicz, U. L.. A note on Schwartzman–Fried–Sullivan theory, with an application. J. Fixed Point Theory Appl. 22 (2020), 25.CrossRefGoogle Scholar
Hryniewicz, U. L., Momin, A. and Salomão, P. A. S.. A Poincaré–Birkhoff theorem for tight Reeb flows on S 3 . Invent. Math. 199 (2014), 333–422.CrossRefGoogle Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and Its Applications, 54). Cambridge University Press, Cambridge, 1997.Google Scholar
Lazzarini, L.. Existence of a somewhere injective pseudo-holomorphic disc. Geom. Funct. Anal. 10(4) (2000), 829–862.CrossRefGoogle Scholar
Li, T. Y. and Yorke, J. A.. Period three implies chaos. The Theory of Chaotic Attractors. Eds. Hunt, B. R., Li, T. Y., Kennedy, J. A. and Nusse, H. E.. Springer, New York, 2004, pp. 77–84.CrossRefGoogle Scholar
Macarini, L. and Schlenk, F.. Positive topological entropy of Reeb flows on spherizations. Math. Proc. Cambridge Philos. Soc. 151(1) (2011), 103–128.CrossRefGoogle Scholar
Meiwes, M.. Rabinowitz Floer homology, leafwise intersections, and topological entropy. PhD Thesis, Ruprecht-Karls-Universität Heidelberg, 2018.Google Scholar
Momin, A.. Contact homology of orbit complements and implied existence. J. Mod. Dyn. 5 (2011), 409–472.CrossRefGoogle Scholar
Wendl, C.. Automatic transversality and orbifolds of punctured holomorphic curves in dimension four. Comment. Math. Helv. 85(2) (2010), 347–407.CrossRefGoogle Scholar
Young, L. S.. Entropy of continuous flows on compact 2-manifolds. Topology 16(4) (1977), 469—471.CrossRefGoogle Scholar