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.