Geodesic
Outer billiards outside regular decagon: periodicity of almost all orbits and existence of aperiodic orbit
Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 406-441.

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Outer billiards were introduced by B. Neumann in 1950s and became popular in 1970s due to J. Moser; Moser considered outer, or dual, billiard as toy model of celestial mechanics. The problem of stability of the Solar system has such a property that ‘`it’s easy to write $n$ equations of particles motion down but hard to understand this motion intuitively"; according to this, Moser suggested to consider Neumann's outer billiard problem which has the same property. One of classical examples of dynamical systems is an outer billiard outside regular $n$-gon; in particular, this billiard is connected with problems of existence of aperiodic trajectory and of fullness of periodic points. These problems resolved only for a few number of a special cases. In case $n=3,4,6$ table is a lattice polygon; as a consequence, there are no aperiodic points, and periodic points form a set of full measure. In 1993, S. Tabachnikov was managed to find an aperiodic points in case of regual pentagon; it was done using renomalization scheme — method which has a fundamental importance in research of self-similar dynamical systems. According to R. Schwartz, cases which are next by complexity are $n = 10, 8, 12$; in these cases, and also in case $n = 5$, it's possible to build a renomalization scheme which, as R. Scwartz writes, “allows one to give (at least in principle) a complete description of what is going on.” Later, author was managed to discover self-similar sturctures and build renormalization scheme for cases of regular octagon and dodecagon. This article is devoted to outer billiard outside regular decagon. The existence of an aperiodic orbit for an outer billiard outside a regular octagon is proved. Additionally, almost all orbits of such an outer billiard are proved to be periodic. All possible periods are explicitly listed. The work is based on classical technology of search and research of renormalization scheme. Periodic structures which occur in case $n = 10$ are similar to periodic structures in case $n = 5$, but has their own features.
Keywords: outer billiards, self-similarity, aperiodic point. 
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F. D. Rukhovich. Outer billiards outside regular decagon: periodicity of almost all orbits and existence of aperiodic orbit. Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 406-441. http://geodesic.mathdoc.fr/item/CHEB_2019_20_2_a32/

[1] Rukhovich F., “Outer billiards outside a regular octagon: periodicity of almost all orbits and existence of an aperiodic orbit”, Doklady Mathematics, 98:1 (2018), 334–337 | Zbl

[2] Rukhovich F., Outer billiards outside regular dodecagon: computer proof of periodicity of almost all orbits and existence of an aperiodic point, 2018, arXiv: 1809.03791

[3] Tabachnikov S., “Outer billiards”, Russian Math. Surveys, 48:6 (294) (1993), 75–102 | MR | Zbl

[4] J. Moser, Is the solar system stable?, Math. Intell., 1 (1978), 65–71 | DOI | MR

[5] Schwartz R. E., Outer billiards on kites, Annals of Mathematics Studies, 171, Princeton University Press, Princeton, NJ, 2009 | MR | Zbl

[6] Dolgopyat D., Fayad B., “Unbounded orbits for semicircular outer billiard”, Ann. Henri Poincare, 10:2 (2009), 357–375 | MR | Zbl

[7] Bedaride N., Cassaigne J., “Outer billiards outside regular regular polygons”, Journal of the London Mathematical Society, 2011 | MR | Zbl

[8] Tabachnikov S., Geometry and Billiards, Student Mathematical Library, 30, American Mathematical Society, Providence, RI, 2005 | MR | Zbl

[9] Tabachnikov S., “On the dual billiard problem”, Adv. Math., 115:2 (1995), 221–249 | MR | Zbl

[10] Bedaride N., Cassaigne J., Outer billiards outside regular regular polygons, 2011, arXiv: 0912.0563 | MR | Zbl

[11] R. E. Schwartz, Outer Billiards, Arithmetic Graph and the Octagon, 2010, arXiv: 1006.2782

[12] Schwartz R. E., The octagonal PETs, Mathematical Surveys and Monographs, 197, American Mathematical Society, Providence, RI, 2014 | MR

[13] N. Pytheas Fogg, Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Mathematics, 1794, eds. V. Berthé, S. Ferenczi, C. Mauduit, A. Siegel, Springer-Verlag, Berlin, 2002 | MR | Zbl

[14] Shaidenko. A, F. Vivaldi, “Global stability of a class of discontinuous dual billiards”, Comm. Math. Phys., 110 (1987), 625–640 | MR | Zbl

[15] Kolodziej R., “The antibilliard outside a polygon”, Bull. Pol. Acad. Sci., 37 (1989), 163–168 | MR | Zbl

[16] Gutkin E., Simanyi N., “Dual polygonal billiards and necklace dynamics”, Comm. Math. Phys., 143 (1991), 431–450 | MR

[17] Jeong I. J., “Outer billiards with contraction: regular polygons”, Dynamical Systems, 33:4 (2015), 565–580 | DOI | MR