article published yes Vadim Kaloshin author FE553552-CDE8-11E9-B324-C0EBE56974250000-0002-6051-2628 Edmond Koudjinan author 52DF3E68-AEFA-11EA-95A4-124A3DDC885E0000-0003-2640-4049 Ke Zhang author VaKa department Spectral rigidity and integrability for billiards and geodesic flows project 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 C0-integrability condition used in their paper. https://research-explorer.ista.ac.at/download/18483/18833/2024_GeometricFunctionalAnalysis_Kaloshin.pdf application/pdfno Springer Nature2024 eng 1016-443X 1420-8970 2306.12301 00132980420000110.1007/s00039-024-00695-6 341973-2007 Kaloshin V, Koudjinan E, Zhang K. 2024. Birkhoff conjecture for nearly centrally symmetric domains. Geometric and Functional Analysis. 34, 1973–2007. Kaloshin V, Koudjinan E, Zhang K. Birkhoff conjecture for nearly centrally symmetric domains. <i>Geometric and Functional Analysis</i>. 2024;34:1973-2007. doi:<a href="https://doi.org/10.1007/s00039-024-00695-6">10.1007/s00039-024-00695-6</a> Kaloshin, V., Koudjinan, E., &#38; Zhang, K. (2024). Birkhoff conjecture for nearly centrally symmetric domains. <i>Geometric and Functional Analysis</i>. Springer Nature. <a href="https://doi.org/10.1007/s00039-024-00695-6">https://doi.org/10.1007/s00039-024-00695-6</a> V. Kaloshin, E. Koudjinan, and K. Zhang, “Birkhoff conjecture for nearly centrally symmetric domains,” <i>Geometric and Functional Analysis</i>, vol. 34. Springer Nature, pp. 1973–2007, 2024. V. Kaloshin, E. Koudjinan, K. Zhang, Geometric and Functional Analysis 34 (2024) 1973–2007. Kaloshin, Vadim, Edmond Koudjinan, and Ke Zhang. “Birkhoff Conjecture for Nearly Centrally Symmetric Domains.” <i>Geometric and Functional Analysis</i>. Springer Nature, 2024. <a href="https://doi.org/10.1007/s00039-024-00695-6">https://doi.org/10.1007/s00039-024-00695-6</a>. Kaloshin, Vadim, et al. “Birkhoff Conjecture for Nearly Centrally Symmetric Domains.” <i>Geometric and Functional Analysis</i>, vol. 34, Springer Nature, 2024, pp. 1973–2007, doi:<a href="https://doi.org/10.1007/s00039-024-00695-6">10.1007/s00039-024-00695-6</a>. 184832024-10-27T23:01:45Z2025-09-08T14:27:45Z