Let S * Q be the spherization of a closed connected manifold of dimension at least two. Consider a contactomorphism ϕ that can be reached by a contact isotopy that is everywhere positively transverse to the contact structure. In other words, ϕ is the time-1-map of a time-dependent Reeb flow. We show that the volume growth of ϕ is bounded from below by the topological complexity of the loop space of Q. Denote by ΩQ 0 (q) the component of the based loop space that contains the constant loop.Theorem. If the fundamental group of Q or the homology of ΩQ 0 (q) grows exponentially, then the volume growth of ϕ is exponential, and thus its topological entropy is positive.A similar statement holds for polynomial growths. This result generalizes work of Dinaburg, Gromov, Paternain and Petean on geodesic flows and of Macarini, Frauenfelder, Labrousse and Schlenk on Reeb flows.Our main tool is a version of Rabinowitz-Floer homology developed by Albers and Frauenfelder.
Topological entropy is not lower semi-continuous: small perturbation of the dynamical system can lead to a collapse of entropy. In this note we show that for some special classes of dynamical systems (geodesic flows, Reeb flows, positive contactomorphisms) topological entropy at least is stable in the sense that there exists a nontrivial continuous lower bound, given that a certain homological invariant grows exponentially.
Abstract. The classical Bott-Samelson theorem states that if on a Riemannian manifold all geodesics issuing from a certain point return to this point, then the universal cover of the manifold has the cohomology ring of a compact rank one symmetric space. This result on geodesic flows has been generalized to Reeb flows and partially to positive Legendrian isotopies by FrauenfelderLabrousse-Schlenk. We prove the full theorem for positive Legendrian isotopies.
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