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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]
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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

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