Mean action and the Calabi invariant

Michael  Hutchings

doi:10.3934/jmd.2016.10.511

Article Contents

2016, Volume 10: 511-539. Doi: 10.3934/jmd.2016.10.511

This issue Previous Article Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds Next Article New infinite families of pseudo-Anosov maps with vanishing Sah-Arnoux-Fathi invariant

Mean action and the Calabi invariant

Michael Hutchings^1,

1.
Department of Mathematics, 970 Evans Hall, University of California, Berkeley, CA 94720

Received: November 30, 2015

Revised: July 31, 2016

Published: October 2016

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Abstract

Given an area-preserving diffeomorphism of the closed unit disk which is a rotation near the boundary, one can naturally define an ``action'' function on the disk which agrees with the rotation number on the boundary. The Calabi invariant of the diffeomorphism is the average of the action function over the disk. Given a periodic orbit of the diffeomorphism, its ``mean action'' is defined to be the average of the action function over the orbit. We show that if the Calabi invariant is less than the boundary rotation number, then the infimum over periodic orbits of the mean action is less than or equal to the Calabi invariant. The proof uses a new filtration on embedded contact homology determined by a transverse knot, which may be of independent interest. (An analogue of this filtration can be defined for any other version of contact homology in three dimensions that counts holomorphic curves.)

Keywords:

Calabi invariant,
embedded contact homology.

Mathematics Subject Classification: Primary: 37J10, 53D42; Secondary: 57R17, 57R58.

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