Outer billiards outside regular polygons: tame case

Ph. D. Rukhovich

Izvestiya. Mathematics , Tome 86 (2022) no. 3, pp. 508-559

Résumé

We consider the periodicity problem, that is, the existence of an aperiodic point and fullness of measure of the set of periodic points for outer billiards outside regular $n$-gons. The lattice cases $n=3,4,6$ are trivial: no aperiodic points exist and the set of periodic points is of full measure. The cases $n=5,10,8,12$ (and only these cases) are regarded as tame. The periodicity problems were solved for $n=5$ in a breakthrough paper by Tabachnikov, who pioneered a renormalization-scheme method for studying the arising self-similar structures. The case $n=10$ is similar to $n=5$ and was studied earlier by the present author. The present paper is devoted to the remaining cases $n=8,12$. We establish the existence of an aperiodic orbit in outer billiards outside regular octagons and dodecagons and prove that almost all trajectories of these outer billiards are periodic. In the regular dodecagon case we give a rigorous computer-assisted proof. We establish equivalence between the outer billiards outside a regular $n$-gon and a regular $n/2$-gon, where $n$ is even and $n/2$ is odd. Our investigation is based on Tabachnikov's renormalization scheme.

Détail

Keywords: aperiodic point, piecewise isometry, first return map, renormalization scheme.
Mots-clés : outer billiard