Exponential instability for a class of dispersing billiards

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    Abstract

    The billiard in the exterior of a finite disjoint union K of strictly convex bodies in R-d with smooth boundaries is considered. The existence of global constants 0 <delta <1 and C > 0 is established such that if two billiard trajectories have n successive reflections from the same convex components of K, then the distance between their jth reflection points is less than C(delta(j) + delta(n-j)) for a sequence of integers j with uniform density in (1,2,..., n). Consequently, the billiard ball map (although not continuous in general) is expansive. As applications, an asymptotic of the number of prime closed billiard trajectories is proved which generalizes a result of Morita [Mor], and it is shown that the topological entropy of the billiard flow does not exceed log(s - I)/a, where s is the number of convex components of K and a is the minimal distance between different convex components of K.
    Original languageEnglish
    Pages (from-to)201-226
    JournalErgodic Theory and Dynamical Systems
    Volume19
    DOIs
    Publication statusPublished - 1999

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