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arXiv · 2306.12301

Birkhoff Conjecture for nearly centrally symmetric domains

Vadim Kaloshin·Comlan Edmond Koudjinan·Ke Zhang
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Abstract

In this paper we prove a perturbative version of a remarkable Bialy-Mironov (Ann.Math:389-413(196), 2022) result. They prove non perturbative Birkhoff conjecture for centrally-symmetric convex domains, namely, a centrally-symmetric convex domain with integrable billiard is ellipse. We combine techniques from Bialy-Mironov (Ann.Math:389-413(196), 2022) with a local result by Kaloshin-Sorrentino (Ann.Math:315-380(188), 2018) and show that a domain close enough to a centrally-symmetric one with integrable billiard is ellipse. To combine these results we derive a slight extension of Bialy-Mironov (Ann.Math:389-413(196), 2022) by proving that a notion of rational integrability is equivalent to the $C^0$-integrability condition used in their paper.

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Vadim Kaloshin, Comlan Edmond Koudjinan, Ke Zhang. 2023-06-21. Birkhoff Conjecture for nearly centrally symmetric domains. https://arxiv.org/abs/2306.12301

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