Доклады Академии наукДоклады Академии наук0869-5652The Russian Academy of Sciences1354210.31857/S0869-56524854415-421Research ArticleOuter billiards outside a regular dodecagonRukhovichF. D.dprpavlin@gmail.comMoscow Institute of Physics and Technology2205201948544154211906201919062019Copyright © 2019, Russian academy of sciences2019<p class="a"><span lang="EN-US">The existence of an aperiodic orbit for an outer billiard outside a regular dodecagon is proved. It is shown that almost all orbits of such an outer billiard are periodic, and all possible periods are explicitly listed. The proofs of the theorems make use of computer calculations. </span></p>outer billiardsrenormalization schemepiecewise isometrydynamical systemsвнешние биллиардыренормализационная схемакусочная изометриядинамические системы[Rukhovich F. Outer Billiards Outside Regular Dodecagon: Computer Proof of Periodicity of Almost All Orbits and Existence of an Aperiodic Point. arXiv:1809. 03791.][Rukhovich F. Outer Billiards. arXiv:1505.06332.][Табачников С. Внешние биллиарды // УМН. 1993. Т. 48. В. 6 (294). С. 75-102.][Bedaride N., Cassaigne J. Outer Billiards Outside Regular Polygons // J. London Math. Soc. 2011. V. 84. Iss. 2. P. 303-324. https://doi.org/10.1112/jlms/jdr010, arXiv:0912.0563.][Tabachnikov S. Geometry and Billiards. Student Mathematical Library. Providence (RI): Amer. Math. Soc., 2005. V. 30.][Tabachnikov S. On the Dual Billiard Problem // Adv. Math. 1995. V. 115 (2). P. 221-249.][Schwartz R. E. Outer Billiards, Arithmetic Graph and the Octagon. arXiv:1006.2782, 2010.][Schwartz R. E. The Octagonal PETs // Math. Surv. and Monogr. 2014. V. 197.]