Topological entropy for Reeb vector fields in dimension three via open book decompositions
[Entropie topologique des champs de Reeb en dimension 3 via les livres ouverts]
Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 119-148.

On associe à toute décomposition en livre ouvert d’une variété de contact close (M,ξ) de dimension 3, de monodromie pseudo-Anosov, de reliure connexe et de coefficient de Dehn fractionnaire c=k/n, un nœud legendrien Λ proche du feuilletage stable d’une page accompagné d’un petit translaté legendrien Λ ^. Lorsque k5, on applique les techniques de [CH13] pour montrer que l’homologie de contact legendrienne cylindrique de ΛΛ ^ est bien définie et a une propriété de croissance exponentielle. Le travail [Alv19] implique alors que tout champ de Reeb pour ξ a une entropie topologique non nulle.

Given an open book decomposition of a closed contact three manifold (M,ξ) with pseudo-Anosov monodromy, connected binding, and fractional Dehn twist coefficient c=k/n, we construct a Legendrian knot Λ close to the stable foliation of a page, together with a small Legendrian pushoff Λ ^. When k5, we apply the techniques of [CH13] to show that the strip Legendrian contact homology of ΛΛ ^ is well-defined and has an exponential growth property. The work [Alv19] then implies that all Reeb vector fields for ξ have positive topological entropy.

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DOI : 10.5802/jep.89
Classification : 57M50, 37B40, 53C15
Keywords: Topological entropy, contact structure, open book decomposition, mapping class group, Reeb dynamics, pseudo-Anosov, contact homology
Mot clés : Entropie topologique, structure de contact, livre ouvert, groupe de difféotopie, dynamique de Reeb, pseudo-Anosov, homologie de contact
Alves, Marcelo R.R. 1 ; Colin, Vincent 2 ; Honda, Ko 3

1 Département de Mathématique, Université Libre de Bruxelles, CP 218, Boulevard du Triomphe, B-1050 Bruxelles, Belgique
2 Université de Nantes, UMR 6629 du CNRS 44322 Nantes, France
3 University of California, Los Angeles Los Angeles, CA 90095, USA
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     title = {Topological entropy for {Reeb} vector fields in dimension three via open book decompositions},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Alves, Marcelo R.R.; Colin, Vincent; Honda, Ko. Topological entropy for Reeb vector fields in dimension three via open book decompositions. Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 119-148. doi : 10.5802/jep.89. http://www.numdam.org/articles/10.5802/jep.89/

[Abb99] Abbas, C. Finite energy surfaces and the chord problem, Duke Math. J., Volume 96 (1999) no. 2, pp. 241-316 | DOI | MR | Zbl

[Alv16a] Alves, M. R. R. Cylindrical contact homology and topological entropy, Geom. Topol., Volume 20 (2016) no. 6, pp. 3519-3569 | DOI | MR | Zbl

[Alv16b] Alves, M. R. R. Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds, J. Modern Dyn., Volume 10 (2016), pp. 497-509 | DOI | MR | Zbl

[Alv19] Alves, M. R. R. Legendrian contact homology and topological entropy, J. Topol. Anal. (2019) (to appear, doi:10.1142/S1793525319500031, arXiv:1410.3381) | DOI | MR | Zbl

[BEE12] Bourgeois, F.; Ekholm, T.; Eliashberg, Y. Effect of Legendrian surgery, Geom. Topol., Volume 16 (2012) no. 1, pp. 301-389 (With an appendix by S. Ganatra and M. Maydanskiy) | DOI | MR | Zbl

[BEH + 03] Bourgeois, F.; Eliashberg, Y.; Hofer, H.; Wysocki, K.; Zehnder, E. Compactness results in symplectic field theory, Geom. Topol., Volume 7 (2003), pp. 799-888 | DOI | MR | Zbl

[BH15] Bao, E.; Honda, K. Semi-global Kuranishi charts and the definition of contact homology (2015) (arXiv:1512.00580)

[Bou09] Bourgeois, F. A survey of contact homology, New perspectives and challenges in symplectic field theory (CRM Proc. Lecture Notes), Volume 49, American Mathematical Society, Providence, RI, 2009, pp. 45-71 | DOI | MR | Zbl

[Bow70] Bowen, R. Topological entropy and axiom A, Global Analysis (Berkeley, Calif., 1968) (Proc. Sympos. Pure Math.), Volume XIV, American Mathematical Society, Providence, R.I., 1970, pp. 23-41 | Zbl

[CH08] Colin, V.; Honda, K. Stabilizing the monodromy of an open book decomposition, Geom. Dedicata, Volume 132 (2008), pp. 95-103 | DOI | MR | Zbl

[CH13] Colin, V.; Honda, K. Reeb vector fields and open book decompositions, J. Eur. Math. Soc. (JEMS), Volume 15 (2013) no. 2, pp. 443-507 | DOI | MR | Zbl

[Che02] Chekanov, Y. Differential algebra of Legendrian links, Invent. Math., Volume 150 (2002) no. 3, pp. 441-483 | DOI | MR | Zbl

[EGH00] Eliashberg, Y.; Givental, A.; Hofer, H. Introduction to symplectic field theory, Geom. Funct. Anal. (2000), pp. 560-673 Special volume GAFA 2000 (Tel Aviv, 1999), Part II | MR | Zbl

[Ekh08] Ekholm, T. Rational symplectic field theory over 2 for exact Lagrangian cobordisms, J. Eur. Math. Soc. (JEMS), Volume 10 (2008) no. 3, pp. 641-704 | DOI | Zbl

[FLP12] Fathi, A.; Laudenbach, F.; Poénaru, Valentin Thurston’s work on surfaces, Mathematical Notes, 48, Princeton University Press, Princeton, NJ, 2012 (Translated from the 1979 French original) | MR | Zbl

[FM12] Farb, B.; Margalit, D. A primer on mapping class groups, Princeton Mathematical Series, 49, Princeton University Press, Princeton, NJ, 2012 | MR

[FOOO09] Fukaya, K.; Oh, Y.-G.; Ohta, H.; Ono, K. Lagrangian intersection Floer theory: anomaly and obstruction. Part II, AMS/IP Studies in Advanced Mathematics, 46, American Mathematical Society, Providence, RI, 2009, pp. 397-805 | MR | Zbl

[FS05] Frauenfelder, U.; Schlenk, F. Volume growth in the component of the Dehn-Seidel twist, Geom. Funct. Anal., Volume 15 (2005) no. 4, pp. 809-838 | DOI | MR | Zbl

[FS06] Frauenfelder, U.; Schlenk, F. Fiberwise volume growth via Lagrangian intersections, J. Symplectic Geom., Volume 4 (2006) no. 2, pp. 117-148 | DOI | MR | Zbl

[Gir02] Giroux, E. Géométrie de contact: de la dimension trois vers les dimensions supérieures, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 405-414 | MR | Zbl

[HL88] Halperin, S.; Lemaire, J.-M. Notions of category in differential algebra, Algebraic topology—rational homotopy (Louvain-la-Neuve, 1986) (Lect. Notes in Math.), Volume 1318, Springer, Berlin, 1988, pp. 138-154 | DOI | MR | Zbl

[Hof93] Hofer, H. Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math., Volume 114 (1993) no. 3, pp. 515-563 | DOI | MR | Zbl

[HWZ96] Hofer, H.; Wysocki, K.; Zehnder, E. Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 13 (1996) no. 3, pp. 337-379 Correction: Ibid. 15 (1998) no. 4, p. 535–538 | DOI | MR | Zbl

[HWZ07] Hofer, H.; Wysocki, K.; Zehnder, E. A general Fredholm theory. I. A splicing-based differential geometry, J. Eur. Math. Soc. (JEMS), Volume 9 (2007) no. 4, pp. 841-876 | DOI | MR | Zbl

[Kat80] Katok, A. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. Inst. Hautes Études Sci. (1980) no. 51, pp. 137-173 | DOI | MR | Zbl

[Kat82] Katok, A. Entropy and closed geodesics, Ergodic Theory Dynam. Systems, Volume 2 (1982) no. 3-4, pp. 339-365 | DOI | MR | Zbl

[KH95] Katok, A.; Hasselblatt, B. Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995 | MR | Zbl

[LS19] Lima, Y.; Sarig, O. M. Symbolic dynamics for three-dimensional flows with positive topological entropy, J. Eur. Math. Soc. (JEMS), Volume 21 (2019) no. 1, pp. 199-256 | DOI | MR | Zbl

[MS11] Macarini, L.; Schlenk, F. Positive topological entropy of Reeb flows on spherizations, Math. Proc. Cambridge Philos. Soc., Volume 151 (2011) no. 1, pp. 103-128 | DOI | MR | Zbl

[New89] Newhouse, S. E. Continuity properties of entropy, Ann. of Math. (2), Volume 129 (1989) no. 2, pp. 215-235 Correction: Ibid. 131 (1990) no. 2, p. 409–410 | DOI | MR | Zbl

[Par15] Pardon, J. Contact homology and virtual fundamental cycles (2015) (arXiv:1508.03873)

[Par16] Pardon, J. An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves, Geom. Topol., Volume 20 (2016) no. 2, pp. 779-1034 | DOI | MR | Zbl

[Sar13] Sarig, O. M. Symbolic dynamics for surface diffeomorphisms with positive entropy, J. Amer. Math. Soc., Volume 26 (2013) no. 2, pp. 341-426 | DOI | MR | Zbl

[Sta] Differential Graded Algebra, Stacks Project (http://stacks.math.columbia.edu/download/dga.pdf)

[Yom87] Yomdin, Y. Volume growth and entropy, Israel J. Math., Volume 57 (1987) no. 3, pp. 285-300 | DOI | MR | Zbl

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