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Geometric and Functional Analysis

, Volume 28, Issue 2, pp 334–392 | Cite as

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

  • Guan Huang
  • Vadim KaloshinEmail author
  • Alfonso Sorrentino
Article

Abstract

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.

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References

  1. ADK16.
    Avila, A., De Simoi, J., Kaloshin, V.: An integrable deformation of an ellipse of small eccentricity is an ellipse. Ann. Math. 184, 527–558 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. BAN88.
    V. Bangert Mather sets for twist maps and geodesics on tori. Dynamics reported, Vol. 1, volume 1 of Dyn. Rep. Ser. Dyn. Syst. Appl., pp. 1–56. Wiley, Chichester, (1988)Google Scholar
  3. BIA93.
    M. Bialy Convex billiards and a theorem by E. Hopf. Math. Z, (1)124 (1993) 147–154Google Scholar
  4. BM16.
    M. Bialy and A. Mironov Angular Billiard and Algebraic Birkhoff conjecture. Preprint (2016)Google Scholar
  5. BIR27.
    G. D. Birkhoff On the periodic motions of dynamical systems. Acta Math., (1)50 (1927), 359–379Google Scholar
  6. BFM98.
    Bolsinov, A.V., Fomenko, A.T., Matveev, V.S.: Two-dimensional Riemannian metrics with an integrable geodesic flow. Local and global geometries. Mat. Sb, (10)189: 5–32, (1998); translation. Sb. Math. 189(9–10), 1441–1466 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. BI94.
    D. Burago and S. Ivanov. Riemannian tori without conjugate points are flat. Geom. Funct. Anal., (3)4 (1994), 259–269 53C20 (53C22)Google Scholar
  8. CF88.
    Chang, S.-J., Friedberg, R.: Elliptical billiards and Poncelet's theorem. J. Math. Phys. 29, 1537–1550 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  9. CK17.
    L. Corsi and V. Kaloshin. Locally integrable non-Liouville analytic geodesic flows. Preprint (2017)Google Scholar
  10. DCR17.
    Damasceno, J., Dias, M.J.: Carneiro and R. Ramírez-Ros. The billiard inside an ellipse deformed by the curvature flow. Proc. Am. Math. Soc. 145, 705–719 (2017)CrossRefzbMATHGoogle Scholar
  11. DR96.
    A. Delshams and R. Ramírez-Ros. Poincaré-Melnikov-Arnold method for analytic planar maps. Nonlinearity, (1)9 (1996), 1–26Google Scholar
  12. DKW17.
    J. De Simoi, V. Kaloshin and Q. Wei. Deformational spectral rigidity among \(\mathbb{Z}_2\)-symmetric domains close to the circle (Appendix B coauthored with H. Hezari), Ann. Math., (1)186 (2017), 277–314Google Scholar
  13. GER31.
    S. A. Gershgorin. Über die Abgrenzung der Eigenwerte einer Matrix. Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk, 6 (1931), 749–754Google Scholar
  14. GT01.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, New York (2001)zbMATHGoogle Scholar
  15. GK95.
    Gutkin, E., Katok, A.: Caustics for inner and outer billiards. Commun. Math. Phys. 173, 101–133 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  16. GUT03.
    E. Gutkin Billiard dynamics: a survey with the emphasis on open problems. Regul. Chaotic Dyn., (1)8, (2003) 1–13Google Scholar
  17. HAL77.
    Halpern, B.: Strange billiard tables. Trans. Am. Math. Soc. 232, 297–305 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  18. HKS16.
    G. Huang, V. Kaloshin and A. Sorrentino On Marked Length Spectrum of Generic Strictly Convex Billiard Tables. Preprint (2016)Google Scholar
  19. INN02.
    Innami, N.: Geometry of geodesics for convex billiards and circular billiards. Nihonkai Math. J. 13, 73–120 (2002)MathSciNetzbMATHGoogle Scholar
  20. KS.
    V. Kaloshin and A. Sorrentino On Local Birkhoff Conjecture for Convex Billiards arXiv:1612.09194
  21. LAZ73.
    V. F. Lazutkin. Existence of caustics for the billiard problem in a convex domain. (Russian) Izv. Akad. Nauk SSSR Ser. Mat, 37 (1973), 186–216Google Scholar
  22. MAT82.
    J. N. Mather. Glancing billiards. Ergodic Theory Dyn. Syst. (3–4)2 (1982), 397–403Google Scholar
  23. MAT90.
    J. N. Mather. Differentiability of the minimal average action as a function of the rotation number. Bol. Soc. Brasil. Mat. (N.S.), 21 (1990), 59–70Google Scholar
  24. MF94.
    J. N. Mather and G. Forni. Action minimizing orbits in Hamiltonian systems. Transition to chaos in classical and quantum mechanics (Montecatini Terme, 1991), Lecture Notes in Math., Vol. 1589 (1994), 92–186Google Scholar
  25. MOS03.
    Moser, J.: Selected Chapters of the Calculus of Variations. Lectures in Mathematics, ETH, Zurich (2003)CrossRefzbMATHGoogle Scholar
  26. DR13.
    S. P. de Carvalho and R. Ramírez-Ros. Non-persistence of resonant caustics in perturbed elliptic billiards. Ergodic Theory Dyn. Syst., (6)33 (2013), 1876–1890Google Scholar
  27. POR50.
    Poritsky, H.: The billiard ball problem on a table with a convex boundary–an illustrative dynamical problem. Ann. Math. 51, 446–470 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  28. RAM06.
    Ramírez-Ros, R.: Break-up of resonant invariant curves in billiards and dual billiards associated to perturbed circular tables. Phys. D 214, 78–87 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  29. SIB04.
    K. F. Siburg. The principle of least action in geometry and dynamics. Lecture Notes in Mathematics Vol.1844, xiii+ 128 pp, Springer, (2004)Google Scholar
  30. SOR15.
    A. Sorrentino. Computing Mather's beta-function for Birkhoff billiards. Discrete Contin. Dyn. Syst. Ser. A (10)35 (2015), 5055– 5082Google Scholar
  31. SORR15.
    A. Sorrentino. Action-Minimizing Methods in Hamiltonian Dynamics. An Introduction to Aubry-Mather Theory. Mathematical Notes Series Vol. 50, Princeton University Press, (2015)Google Scholar
  32. TAB95.
    S. Tabachnikov. Billiards. Panor. Synth, No. 1 (1995) vi+ 142 ppGoogle Scholar
  33. TAB05.
    S. Tabachnikov. Geometry and billiards. Student Mathematical Library Vol.30 (2005) xii+ 176 pp, American Mathematical SocietyGoogle Scholar
  34. TAB96.
    M. B. Tabanov. New ellipsoidal confocal coordinates and geodesics on an ellipsoid. J. Math. Sci., (6)82 (1996), 3851–3858Google Scholar
  35. TRE13.
    Treschev, D.: Billiard map and rigid rotation. Phys. D 255, 31–34 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  36. WOJ94.
    M. P. Wojtkowski. Two applications of Jacobi fields to the billiard ball problem. J. Differ. Geom. , (1)40 (1994), 155–164Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Guan Huang
    • 1
  • Vadim Kaloshin
    • 2
    Email author
  • Alfonso Sorrentino
    • 3
  1. 1.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA
  3. 3.Dipartimento di MatematicaUniversità degli Studi di Roma “Tor Vergata”RomeItaly

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