TY - JOUR

T1 - Nearly Circular Domains Which Are Integrable Close to the Boundary Are Ellipses

AU - Huang, Guan

AU - Kaloshin, Vadim

AU - Sorrentino, Alfonso

N1 - Publisher Copyright:
© 2018, Springer International Publishing AG, part of Springer Nature.

PY - 2018/4/1

Y1 - 2018/4/1

N2 - The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of an ellipse of small eccentricity which preserves integrability near the boundary, is itself an ellipse. This extends the result in Avila et al. (Ann Math 184:527–558, ADK16), where integrability was assumed on a larger set. In particular, it shows that (local) integrability near the boundary implies global integrability. One of the crucial ideas in the proof consists in analyzing Taylor expansion of the corresponding action-angle coordinates with respect to the eccentricity parameter, deriving and studying higher order conditions for the preservation of integrable rational caustics.

AB - The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of an ellipse of small eccentricity which preserves integrability near the boundary, is itself an ellipse. This extends the result in Avila et al. (Ann Math 184:527–558, ADK16), where integrability was assumed on a larger set. In particular, it shows that (local) integrability near the boundary implies global integrability. One of the crucial ideas in the proof consists in analyzing Taylor expansion of the corresponding action-angle coordinates with respect to the eccentricity parameter, deriving and studying higher order conditions for the preservation of integrable rational caustics.

UR - http://www.scopus.com/inward/record.url?scp=85044193861&partnerID=8YFLogxK

U2 - 10.1007/s00039-018-0440-4

DO - 10.1007/s00039-018-0440-4

M3 - Article

AN - SCOPUS:85044193861

SN - 1016-443X

VL - 28

SP - 334

EP - 392

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

IS - 2

ER -