 09115 YiChiuan CHEN
 On Topological Entropy of Billiard Tables with Small Inner Scatterers
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Jul 13, 09

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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 antiintegrable limit with the theory of LyusternikShnirel'man, we show that a billiard system in this class generically admits a set of nondegenerate antiintegrable orbits which corresponds bijectively to a topological Markov chain of arbitrarily large topological entropy. The antiintegrable 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 welldefined at the antiintegrable 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.
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