An integrable deformation of an ellipse of small eccentricity is an ellipse

Artur Avila; Jacopo De Simoi; Vadim Kaloshin

doi:https://doi.org/10.4007/annals.2016.184.2.5

Pages 527-558 from Volume 184 (2016), Issue 2 by Artur Avila, Jacopo De Simoi, Vadim Kaloshin

Abstract

The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we show that a version of this conjecture is true for tables bounded by small perturbations of ellipses of small eccentricity.

Mathematical Subject Classification

Primary 2000: 37D50, 37E40, 37J35, 70H06

Milestones

Received: 21 January 2015
Revised: 20 January 2016
Accepted: 5 February 2016
Published online: 29 July 2016

Authors

Artur Avila

CNRS, IMJ-PRG, UMR 7586, Université Paris Diderot, Sorbonne Paris Cité, Sorbonnes Universités, UPMC Univ Paris 06, Paris, France and IMPA, Rio de Janeiro, Brasil

Jacopo De Simoi

University of Toronto, Toronto, ON, Canada

Vadim Kaloshin

University of Maryland, College Park, MD