Birkhoff Conjecture for Nearly Centrally Symmetric Domains

• Published in: Geometric and functional analysis Vol. 34; no. 6; pp. 1973 - 2007
• Main Authors: Kaloshin, V., Koudjinan, C. E., Zhang, Ke
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2024; Springer Nature B.V
• Subjects: Analysis; Mathematics; Mathematics and Statistics; Birkhoff conjecture; Integrability; Birkhoff billiard
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-024-00695-6

In this paper we prove a perturbative version of a remarkable Bialy–Mironov (Ann. Math. 196(1):389–413, 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. 196(1):389–413, 2022 ) with a local result by Kaloshin–Sorrentino (Ann. Math. 188(1):315–380, 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. 196(1):389–413, 2022 ) by proving that a notion of rational integrability is equivalent to the C 0 -integrability condition used in their paper.