The billiard inside an ellipse deformed by the curvature flow

Damasceno, Josué; Dias Carneiro, Mario; Ramírez-Ros, Rafael

doi:10.1090/proc/13351

The billiard inside an ellipse deformed by the curvature flow
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by Josué Damasceno, Mario J. Dias Carneiro and Rafael Ramírez-Ros PDF

Proc. Amer. Math. Soc. 145 (2017), 705-719 Request permission

Abstract:

The billiard dynamics inside an ellipse is integrable. It has zero topological entropy, four separatrices in the phase space, and a continuous family of convex caustics: the confocal ellipses. We prove that the curvature flow destroys the integrability, increases the topological entropy, splits the separatrices in a transverse way, and breaks all resonant convex caustics.

References

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972.
George D. Birkhoff, Dynamical systems, American Mathematical Society Colloquium Publications, Vol. IX, American Mathematical Society, Providence, R.I., 1966. With an addendum by Jurgen Moser. MR 0209095
Keith Burns and Howard Weiss, A geometric criterion for positive topological entropy, Comm. Math. Phys. 172 (1995), no. 1, 95–118. MR 1346373
Pablo S. Casas and Rafael Ramírez-Ros, The frequency map for billiards inside ellipsoids, SIAM J. Appl. Dyn. Syst. 10 (2011), no. 1, 278–324. MR 2788926, DOI 10.1137/10079389X
Shau-Jin Chang and Richard Friedberg, Elliptical billiards and Poncelet’s theorem, J. Math. Phys. 29 (1988), no. 7, 1537–1550. MR 946326, DOI 10.1063/1.527900
Kai-Seng Chou and Xi-Ping Zhu, The curve shortening problem, Chapman & Hall/CRC, Boca Raton, FL, 2001. MR 1888641, DOI 10.1201/9781420035704
R. Cushman, Examples of nonintegrable analytic Hamiltonian vector fields with no small divisors, Trans. Amer. Math. Soc. 238 (1978), 45–55. MR 478223, DOI 10.1090/S0002-9947-1978-0478223-8
Amadeu Delshams and Rafael Ramírez-Ros, Poincaré-Mel′nikov-Arnol′d method for analytic planar maps, Nonlinearity 9 (1996), no. 1, 1–26. MR 1373998, DOI 10.1088/0951-7715/9/1/001
Amadeu Delshams and Rafael Ramírez-Ros, Melnikov potential for exact symplectic maps, Comm. Math. Phys. 190 (1997), no. 1, 213–245. MR 1484553, DOI 10.1007/s002200050239
M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), no. 1, 69–96. MR 840401
Matthew A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987), no. 2, 285–314. MR 906392
John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. MR 709768, DOI 10.1007/978-1-4612-1140-2
Valeriĭ V. Kozlov and Dmitriĭ V. Treshchëv, Billiards, Translations of Mathematical Monographs, vol. 89, American Mathematical Society, Providence, RI, 1991. A genetic introduction to the dynamics of systems with impacts; Translated from the Russian by J. R. Schulenberger. MR 1118378, DOI 10.1016/0021-8928(91)90103-2
Dan Jane, An example of how the Ricci flow can increase topological entropy, Ergodic Theory Dynam. Systems 27 (2007), no. 6, 1919–1932. MR 2371602, DOI 10.1017/S0143385707000211
Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
V. F. Lazutkin, Existence of caustics for the billiard problem in a convex domain, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 186–216 (Russian). MR 0328219
Sônia Pinto-de-Carvalho and Rafael Ramírez-Ros, Non-persistence of resonant caustics in perturbed elliptic billiards, Ergodic Theory Dynam. Systems 33 (2013), no. 6, 1876–1890. MR 3122155, DOI 10.1017/S0143385712000417
Rafael Ramírez-Ros, Break-up of resonant invariant curves in billiards and dual billiards associated to perturbed circular tables, Phys. D 214 (2006), no. 1, 78–87. MR 2200796, DOI 10.1016/j.physd.2005.12.007
Serge Tabachnikov, Billiards, Panor. Synth. 1 (1995), vi+142 (English, with English and French summaries). MR 1328336
E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469, DOI 10.1017/CBO9780511608759