Deviations for generalized tiling billiards in cyclic polygons

Abstract

This work continues the study of tiling billiards, a class of dynamical system introduced by Davis et al. in 2018. We develop the study of generalized tiling billiards in a cyclic polygon. This work shows that the behavior of generalized tiling billiards in cyclic N-gons with N > 4 is considerably different from that of triangular and quadrilateral tiling billiards studied before. Indeed, we exhibit an open set of generalized tiling billiard trajectories deviating sublinearly from their asymptotic direction, whereas for N = 3 or 4 almost every trajectory stays at a bounded distance from a line. Moreover, we establish the rate of deviations both in the generic case and in some non generic cases.

Figures (13)

• Figure 1: Trajectories of regular tiling billiards
• Figure 2: Example of two trajectories of a polygonal tiling billiard
• Figure 3: Examples of trajectories in triangle tiling billiards
• Figure 4: The way we put polygons together to play tiling billiards will not tile the plane with $N$-gons, $N\geqslant 5$.
• Figure 5: Examples of trajectories in a generalized pentagonal tiling billiard

Theorems & Definitions (35)

• Theorem 1.1: BSDFI18, PR19, HPR22
• Lemma 1.1
• Theorem A
• Theorem B
• Remark 1.2
• Theorem C
• Definition 2.1: Generalized tiling billiard
• Remark 2.2
• Definition 2.3
• Theorem 2.4: Theorem 3.3 of BSDFI18