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High-order persistence of resonant caustics in perturbed circular billiards

Part of: Low-dimensional dynamical systems Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems

Published online by Cambridge University Press:  23 October 2025

COMLAN EDMOND KOUDJINAN  and
RAFAEL RAMÍREZ-ROS Open the ORCID record for RAFAEL RAMÍREZ-ROS [Opens in a new window]
COMLAN EDMOND KOUDJINAN
Affiliation:
Department of Mathematics, University of Toronto , Ontario, Canada (e-mail: koudjinanedmond@gmail.com)
RAFAEL RAMÍREZ-ROS*
Affiliation:
Departament de Matemàtiques, Universitat Politècnica de Catalunya , Barcelona, Spain
*
e-mail: rafael.ramirez@upc.edu
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Abstract

We find necessary and sufficient conditions for high-order persistence of resonant caustics in perturbed circular billiards. The main tool is a perturbation theory based on the Bialy–Mironov generating function for convex billiards. All resonant caustics with period q persist up to order $\lceil q/n \rceil -1$ under any polynomial deformation of the circle of degree n.

Keywords

convex billiardstwist mapsperiodic orbitsinvariant curvesperturbation theory

MSC classification

Primary: 37E40: Twist maps
Secondary: 37J40: Perturbations, normal forms, small divisors, KAM theory, Arnol'd diffusion

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 29
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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References

Avila, A., De Simoi, J. and Kaloshin, V.. An integrable deformation of an ellipse of small eccentricity is an ellipse. Ann. of Math. (2) 184 (2016), 527–558.10.4007/annals.2016.184.2.5CrossRefGoogle Scholar
Baryshnikov, Y. and Zharnitsky, V.. Sub-Riemannian geometry and periodic orbits in classical billiards. Math. Res. Lett. 13 (2006), 587–598.10.4310/MRL.2006.v13.n4.a8CrossRefGoogle Scholar
Bialy, M.. Convex billiards and a theorem by E. Hopf. Math. Z. 124 (1993), 147–154.10.1007/BF02572397CrossRefGoogle Scholar
Bialy, M.. Gutkin billiard tables in higher dimensions and rigidity. Nonlinearity 31 (2018), 2281–2293.10.1088/1361-6544/aaaf4dCrossRefGoogle Scholar
Bialy, M.. Mather $\beta$ -function for ellipses and rigidity. Entropy 24 (2022), 1600.10.3390/e24111600CrossRefGoogle ScholarPubMed
Bialy, M.. Effective rigidity away from the boundary for centrally symmetric billiards. Ergod. Th. & Dynam. Sys. 44 (2024), 1741–1756.10.1017/etds.2023.70CrossRefGoogle Scholar
Bialy, M. and Mironov, A. E.. Angular billiard and algebraic Birkhoff conjecture. Adv. Math. 313 (2017), 102–126.10.1016/j.aim.2017.04.001CrossRefGoogle Scholar
Bialy, M. and Mironov, A. E.. The Birkhoff–Poritsky conjecture for centrally-symmetric billiard tables. Ann. of Math. (2) 196 (2022), 389–413.10.4007/annals.2022.196.1.2CrossRefGoogle Scholar
Bialy, M., Mironov, A. E. and Shalom, L.. Outer billiards with the dynamics of a standard shift on a finite number of invariant curves. Exp. Math. 30 (2019), 469–474.10.1080/10586458.2018.1563514CrossRefGoogle Scholar
Bialy, M. and Tabachnikov, S.. Dan Reznik’s identities and more. Eur. J. Math. 8 (2022), 1341–1354.10.1007/s40879-020-00428-7CrossRefGoogle Scholar
Bialy, M. and Tsodikovich, D.. Locally maximising orbits for the non-standard generating function of convex billiards and applications. Nonlinearity 36 (2023), 2001.10.1088/1361-6544/acbb50CrossRefGoogle Scholar
Bialy, M. and Tsodikovich, D.. Billiard tables with rotational symmetry. Int. Math. Res. Not. IMRN 2023 (2023), 3970–4003.10.1093/imrn/rnab366CrossRefGoogle Scholar
Bolsinov, A., Matveev, V. S., Miranda, E. and Tabachnikov, S.. Open problems, questions and challenges in finite-dimensional integrable systems. Philos. Trans. Roy. Soc. A 376 (2018), 20170430.10.1098/rsta.2017.0430CrossRefGoogle ScholarPubMed
Chen, F. and Wang, Q.. High-order Melnikov method for time-periodic equations. Adv. Nonlinear Stud. 17 (2017), 793–818.10.1515/ans-2017-6017CrossRefGoogle Scholar
Chen, F. and Wang, Q.. High order Melnikov method: theory and application. J. Differential Equations 267 (2019), 1095–1128.10.1016/j.jde.2019.02.003CrossRefGoogle Scholar
Cyr, V.. A number theoretic question arising in the geometry of plane curves and in billiard dynamics. Proc. Amer. Math. Soc. 140 (2012), 3035–3040.10.1090/S0002-9939-2012-11258-4CrossRefGoogle Scholar
Damasceno, J., Dias Carneiro, M. and Ramírez-Ros, R.. The billiard inside an ellipse deformed by the curvature flow. Proc. Amer. Math. Soc. 145 (2017), 705–719.10.1090/proc/13351CrossRefGoogle Scholar
Delshams, A. and Ramírez-Ros, R.. Exponentially small splitting of separatrices for perturbed integrable standard-like maps. J. Nonlinear Sci. 8 (1998), 317–352.10.1007/s003329900054CrossRefGoogle Scholar
Fierobe, C., Kaloshin, V. and Sorrentino, A.. Lecture notes on Birkhoff billiards: dynamics, integrability and spectral rigidity. Modern Aspects of Dynamical Systems. Ed. C. Bonanno, A. Sorrentino and C. Ulcigrai. Springer Nature, Berlin, 2024, pp. 1–57.10.1007/978-3-031-62014-0_1CrossRefGoogle Scholar
Fierobe, C. and Sorrentino, A.. On the existence of periodic invariant curves for analytic families of twist maps and billiards. Preprint, 2024, arXiv:2407.17090.Google Scholar
Gelfreich, V. G.. A proof of the exponentially small transversality of the separatrices for the standard map. Comm. Math. Phys. 201 (1999), 155–216.10.1007/s002200050553CrossRefGoogle Scholar
Gutkin, E.. Capillary floating and the billiard ball problem. J. Math. Fluid Mech. 14 (2012), 363–382.10.1007/s00021-011-0071-0CrossRefGoogle Scholar
Hezari, H. and Zelditch, S.. One can hear the shape of ellipses of small eccentricity. Ann. of Math. (2) 196 (2022), 1083–1134.10.4007/annals.2022.196.3.4CrossRefGoogle Scholar
Huang, G. and Kaloshin, V.. On the finite dimensionality of integrable deformations of strictly convex integrable billiard tables. Mosc. Math. J. 19 (2019), 307–327.10.17323/1609-4514-2019-19-2-307-327CrossRefGoogle Scholar
Huang, G., Kaloshin, V. and Sorrentino, A.. Nearly circular domains which are integrable close to the boundary are ellipses. Geom. Funct. Anal. 28 (2018), 334–392.10.1007/s00039-018-0440-4CrossRefGoogle Scholar
Innami, N.. Convex curves whose points are vertices of billiard triangles. Kodai Math. J. 11 (1988), 17–24.10.2996/kmj/1138038814CrossRefGoogle Scholar
Kaloshin, V., Koudjinan, C. E. and Zhang, K.. Birkhoff conjecture for nearly centrally symmetric domains. Geom. Func. Anal. 34 (2024), 1973–2007.10.1007/s00039-024-00695-6CrossRefGoogle Scholar
Kaloshin, V. and Sorrentino, A.. On the integrability of Birkhoff billiards. Philos. Trans. Roy. Soc. A 376 (2018), 20170419.10.1098/rsta.2017.0419CrossRefGoogle ScholarPubMed
Kaloshin, V. and Sorrentino, A.. On the local Birkhoff conjecture for convex billiards. Ann. of Math. (2) 188 (2018), 315–380.10.4007/annals.2018.188.1.6CrossRefGoogle Scholar
Kaloshin, V. and Sorrentino, A.. Inverse problems and rigidity questions in billiard dynamics. Ergod. Th. & Dynam. Sys. 42 (2021), 1–34.Google Scholar
Kaloshin, V. and Zhang, K.. Density of convex billiards with rational caustics. Nonlinearity 31 (2018), 5214.10.1088/1361-6544/aadc12CrossRefGoogle Scholar
Knill, O.. On nonconvex caustics of convex billiards. Elem. Math. 53 (1998), 89–106.10.1007/s000170050038CrossRefGoogle Scholar
Lazutkin, V. F.. The existence of caustics for a billiard problem in a convex domain. Izd Akad. Nauk SSSR Ser. Mat. 37 (1973), 185–214.Google Scholar
Levi, M. and Moser, J. K.. A Lagrangian proof of the invariant curve theorem for twist mappings. Proc. Symp. Pure Math. 69 (2001), 733–746.10.1090/pspum/069/1858552CrossRefGoogle Scholar
Martín, P., Sauzin, D. and Seara, T. M.. Exponentially small splitting of separatrices in the perturbed McMillan map. Discrete Contin. Dyn. Syst. 31 (2011), 301–372.10.3934/dcds.2011.31.301CrossRefGoogle Scholar
Martín, P., Tamarit-Sariol, A. and Ramírez-Ros, R.. On the length and area spectrum of analytic convex domains. Nonlinearity 29 (2016), 198–231.10.1088/0951-7715/29/1/198CrossRefGoogle Scholar
Martín, P., Tamarit-Sariol, A. and Ramírez-Ros, R.. Exponentially small asymptotic formulas for the length spectrum in some billiard tables. Exp. Math. 25 (2016), 416–440.10.1080/10586458.2015.1076361CrossRefGoogle Scholar
Meiss, J. D.. Symplectic maps, variational principles, and transport. Rev. Mod. Phys. 64 (1992), 795–848.10.1103/RevModPhys.64.795CrossRefGoogle Scholar
Pinto-de-Carvalho, S. and Ramírez-Ros, R.. Nonpersistence of resonant caustics in perturbed elliptic billiards. Ergod. Th. & Dynam. Sys. 33 (2013), 1876–1890.10.1017/S0143385712000417CrossRefGoogle Scholar
Popov, G.. Invariants of the length spectrum and spectral invariants of planar convex domains. Comm. Math. Phys. 161 (1994), 335–364.10.1007/BF02099782CrossRefGoogle Scholar
Ramírez-Ros, R.. Break-up of resonant invariant curves in billiards and dual billiards associated to perturbed circular tables. Physica D 214 (2006), 278–287.10.1016/j.physd.2005.12.007CrossRefGoogle Scholar
Tabachnikov, S.. Billiards (Panorama Synthétique, 1). SMF, Paris, 1995.Google Scholar
Wang, Q.. Exponentially small splitting: a direct approach. J. Differential Equations 269 (2020), 954–1036.10.1016/j.jde.2019.12.028CrossRefGoogle Scholar
Zhang, J.. Coexistence of period 2 and 3 caustics for deformative nearly circular billiard maps. Discrete Contin. Dyn. Syst. 39 (2019), 6419–6440.10.3934/dcds.2019278CrossRefGoogle Scholar