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15 April 2021 Dynamical convexity and closed orbits on symmetric spheres
Viktor L. Ginzburg, Leonardo Macarini
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Duke Math. J. 170(6): 1201-1250 (15 April 2021). DOI: 10.1215/00127094-2020-0097

Abstract

Our main theme here is the dynamics of Reeb flows with symmetries on the standard contact sphere. We introduce the notion of strong dynamical convexity for contact forms invariant under a group action, supporting the standard contact structure, and prove that in dimension 2n+1 any such contact form satisfying a condition slightly weaker than strong dynamical convexity has at least n+1 simple closed Reeb orbits. For contact forms with antipodal symmetry, we prove that strong dynamical convexity is a consequence of ordinary convexity. In dimension 5 or greater, we construct examples of antipodally symmetric, dynamically convex contact forms which are not strongly dynamically convex, and thus not contactomorphic to convex ones via a contactomorphism commuting with the antipodal map. Finally, we relax this condition on the contactomorphism furnishing a condition that has nonempty C1-interior.

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Viktor L. Ginzburg. Leonardo Macarini. "Dynamical convexity and closed orbits on symmetric spheres." Duke Math. J. 170 (6) 1201 - 1250, 15 April 2021. https://doi.org/10.1215/00127094-2020-0097

Information

Received: 15 December 2019; Revised: 18 August 2020; Published: 15 April 2021
First available in Project Euclid: 23 March 2021

Digital Object Identifier: 10.1215/00127094-2020-0097

Subjects:
Primary: 37J11
Secondary: 37J55 , 53D40

Keywords: Closed orbits , Conley–Zehnder index , dynamical convexity , equivariant symplectic homology , Reeb flows

Rights: Copyright © 2021 Duke University Press

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Vol.170 • No. 6 • 15 April 2021
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