Reeb orbits that force topological entropy

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.

Hofer's geometry and topological entropy

“…Some of these results generalize to positive contactomorphisms [Dah18], and results on the dependence of some lower bounds on the topological entropy with respect to their positive contact Hamiltonians have been obtained in [Dah21]. Forcing results for Reeb flows are obtained in [AP22]. A related discussion and results on questions of -stability of the topological entropy of geodesic flows can be found in [ADMM22].…”

Generic properties of $3$-dimensional Reeb flows: Birkhoff sections and entropy

“…In [1] a condition on the fractional Dehn twist coefficients of a supporting open book decomposition is given to guarantee that every Reeb vector field, possibly degenerate, for the given supported contact structure has positive topological entropy. Alves and Pirnapasov [3] proved that any contact 3-manifold admits transverse links that force positive topological entropy when realized as periodic Reeb orbits. Alves and Meiwes obtain in [2] contact structures in higher dimensional spheres such that the Reeb flows of all defining contact forms have positive topological entropy.…”

“…With the above formula, the relation between the dynamics of X L on ∂D L and the linearized dynamics along the various orbits γ ⊂ L becomes precise, since b(t, θ) is exactly the same as the function appearing in (3). A solution…”

Hofer's geometry and topological entropy

“…Some of these results generalize to positive contactomorphisms [19], and results on the dependence of some lower bounds on topological entropy with respect to their positive contact Hamiltonians has been obtained in [20]. Forcing results for Reeb flows are obtained in [10]. A related discussion and results on questions of C 0 -stability of the topological entropy of geodesic flows can be found in [7].…”