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

Guan Huang YMSC,Tsinghua University Vadim Kaloshin University of Maryland, College Park Alfonso Sorrentino Università degli Studi di Roma “Tor Vergata”

Dynamical Systems mathscidoc:1804.11002

GAFA, 28, (2), 334–392, 2018.4
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.
Integrable billiards, Birkhoff conjecture
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@inproceedings{guan2018nearly,
  title={Nearly Circular Domains Which Are Integrable Close to the Boundary Are Ellipses},
  author={Guan Huang, Vadim Kaloshin, and Alfonso Sorrentino},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180429112839126699076},
  booktitle={GAFA},
  volume={28},
  number={2},
  pages={334–392},
  year={2018},
}
Guan Huang, Vadim Kaloshin, and Alfonso Sorrentino. Nearly Circular Domains Which Are Integrable Close to the Boundary Are Ellipses. 2018. Vol. 28. In GAFA. pp.334–392. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180429112839126699076.
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