Topological entropy for Reeb vector fields in dimension three via open book decompositions (Q1729927)

A cooriented contact structure $\xi$ on a closed $3$-manifold $M$ is a tangential $2$-plane field, globally given as the kernel of a contact form $\alpha$, i.e. $\xi=\ker \alpha$ with $\alpha\wedge\,d \alpha$ nowhere vanishing. A choice of a contact form $\alpha$ for $\xi$ gives rise to the Reeb vector field $R_\alpha$ by $i_{R_\alpha}d\alpha=0$ and $\alpha(R_\alpha)=1$. \par A main challenge in contact geometry is the investigation of the dynamical system given by $R_\alpha$. A celebrated result by \textit{C. H. Taubes} [Geom. Topol. 11, 2117--2202 (2007; Zbl 1135.57015)] solves the $3$-dimensional Weinstein conjecture in the affirmative, stating that any Reeb vector field on any closed $3$-manifold admits at least one periodic orbit. \par From the work in the present paper it follows that on a large class of contact $3$-manifolds any Reeb vector field generates a chaotic dynamical system. \par More precisely, the authors consider Reeb flows on the class of contact manifolds supported by an open book with connected binding and pseudo-Anosov monodromy with fractional Dehn twist coefficient $k/n$, for $k\geq5$. One can argue that the above class of contact manifolds contains ``almost all'' contact manifold, since any contact manifold admits such an open book decomposition with $k\geq1$ [\textit{V. Colin} and \textit{K. Honda}, Geom. Dedicata 132, 95--103 (2008; Zbl 1151.57017)]. (However, all contact structures as above with $k\geq2$ are tight [\textit{V. Colin} and \textit{K. Honda}, J. Eur. Math. Soc. (JEMS) 15, No. 2, 443--507 (2013; Zbl 1266.57013)].) \par In the main result of this article the authors compute the topological entropy of the dynamical system associated to any Reeb flow of a contact $3$-manifold supported by an open book with connected binding and pseudo-Anosov monodromy with fractional Dehn twist coefficient $k/n$, for $k\geq5$ to be non-vanishing. \par The topological entropy is a non-negative real number associated to a dynamical system measuring its complexity. In the present paper the authors review the definition of topological entropy by \textit{R. Bowen} [Trans. Am. Math. Soc. 153, 401--414 (1971; Zbl 0212.29201)] and \textit{E. I. Dinaburg} [Sov. Math., Dokl. 11, 13--16 (1970; Zbl 0196.26401); translation from Dokl. Akad. Nauk SSSR 190, 19--22 (1970)]. An equivalent definition was given earlier in [\textit{R. L. Adler} et al., Trans. Am. Math. Soc. 114, 309--319 (1965; Zbl 0127.13102)]. \par By [\textit{A. Katok}, Publ. Math., Inst. Hautes Étud. Sci. 51, 137--173 (1980; Zbl 0445.58015)] and [\textit{O. M. Sarig}, J. Am. Math. Soc. 26, No. 2, 341--426 (2013; Zbl 1280.37031)] any dynamical system with positive topological entropy, generated by a non-vanish\-ing vector field on a $3$-manifold contains a Smale horseshoe [\textit{S. Smale}, Bull. Am. Math. Soc., New Ser. 55, No. 3, 331--336 (original pagination) (2018; Zbl 1398.37018)], the prototype of a chaotic system. In particular, this implies that the number of hyperbolic periodic orbits grows exponentially with respect to the period of the orbit. \par In the proof the authors construct a Legendrian knot $\Lambda$ close to the stable foliation of the pseudo-Anosov monodromy on the page together with a Legendrian push-off $\hat\Lambda$ of $\Lambda$. Under the condition $k\geq5$ the authors can deduce a growth rate of the strip Legendrian contact homologies from $\Lambda$ to $\hat\Lambda$ (and prove its well-definedness). From this they deduce any Reeb flow to have positive topological entropy.