On topological entropy of billiard tables with small inner scatterers

Abstract

We present in this paper an approach to studying the topological entropy of a class of billiard systems. In this class, any billiard table consists of strictly convex domain in the plane and strictly convex inner scatterers. Combining the concept of anti-integrable limit with the theory of Lyusternik-Shnirel'man, we show that a billiard system in this class generically admits a set of non-degenerate anti-integrable orbits which corresponds bijectively to a topological Markov chain of arbitrarily large topological entropy. The anti-integrable limit is the singular limit when scatterers shrink to points. In order to get around the singular limit and so as to apply the implicit function theorem, on auxiliary circles encircling these scatterers we define a length functional whose critical points are well-defined at the anti-integrable limit and give rise to billiard orbits when the scatterers are not points. Consequently, we prove the topological entropy of the first return map to the scatterers can be made arbitrarily large provided the inner scatterers are sufficiently small.

Degenerate billiards in celestial mechanics

“…In [7] it is proved that for small ε > 0 nondegenerate collision chains of this degenerate billiard are shadowed by trajectories of the billiard system in Ω ε . For a discrete set N , this was shown earlier in [14], see also [12].…”

Linear and nonlinear stability of periodic orbits in annular billiards

“…The computations are identical to above, so we suppress them and proceed to give the result tr DB 2n+2 (z 0 ) = 2 − 16nδ 2 nR 2 − R sin kπ n − nδ 2 R 2 sin 2 kπ n (15) Setting δ = 0 we again see that the corresponding periodic orbit is parabolic. For δ = 0, the same analysis as in the paragraph after (14) shows that the condition nR 2 − R sin kπ n − nδ 2 > 0 is necessary and sufficient for existence of linearly stable orbits. This yields the inequality…”

“…The same arguments as the ones following (14) imply that for given k and n with 0 < δ ≤ sin kπ n 2n , the allowed radius range is sin kπ n + sin 2 kπ n + 4n 2 δ 2 2n…”

“…Then it is clear that R ± < 0. Since in the billiard context we have R > 0, the physically allowed solutions of (14) correspond to R satisfying (13). Consider now f < 0, corresponding to δ > sin π n 2n .…”

“…Denote by f = sin 2 π n −4n 2 δ 2 4nδ 2 the coefficient of R 2 in (14). The formal solutions of (14) are…”