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.

Huang, G., Kaloshin, V., Sorrentino, A. (2018). Nearly Circular Domains Which Are Integrable Close to the Boundary Are Ellipses. GEOMETRIC AND FUNCTIONAL ANALYSIS, 28(2), 334-392 [10.1007/s00039-018-0440-4].

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

Sorrentino, A
2018-04-01

Abstract

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.
apr-2018
Online ahead of print
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - ANALISI MATEMATICA
Settore MAT/03 - GEOMETRIA
Settore MAT/07 - FISICA MATEMATICA
English
Con Impact Factor ISI
https://link.springer.com/article/10.1007/s00039-018-0440-4?wt_mc=Internal.Event.1.SEM.ArticleAuthorOnlineFirst
Huang, G., Kaloshin, V., Sorrentino, A. (2018). Nearly Circular Domains Which Are Integrable Close to the Boundary Are Ellipses. GEOMETRIC AND FUNCTIONAL ANALYSIS, 28(2), 334-392 [10.1007/s00039-018-0440-4].
Huang, G; Kaloshin, V; Sorrentino, A
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/195472
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