Journal of Modern Dynamics (JMD)

Mean action and the Calabi invariant

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

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Michael Hutchings - Department of Mathematics, 970 Evans Hall, University of California, Berkeley, CA 94720, United States (email)

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.

Received: December 2015;      Revised: August 2016;      Available Online: November 2016.