Absolutely Periodic Billiard Orbits of Arbitrarily High Order in Smooth Strictly Convex Domains

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Callis, Keagan Graham
Kaloshin, Vadim Y
Billiard orbits in smooth (C∞) strictly convex domains in R2 are a special class ofsmooth area preserving twist diffeomorphisms of the cylinder. These maps are determined by the domain Ω on which the billiard orbit resides, and properties of the billiard map can thus lead to conclusions on various mathematical objects which involve the same domain. For instance, properties of the periodic orbits of the billiard map such as (1) the degeneracy of a periodic orbit or (2) the measure of the set of all periodic orbits can lead to conclusions on the asymptotic expansion of the Laplace spectrum of the domain. In this work we show that by an arbitrarily small perturbation in the C∞ norm of the domain can create a domain containing a periodic orbit which is highly degenerate. This result can be viewed as extending Newhouse phenomena which was previously obtained within the class of smooth area preserving diffeomorphisms to the more restricted class of billiard maps. The methods used to carry these perturbations over to the class of billiard maps is by perturbations of the domain Ω. Thus in this work we also explore how high order perturbations of the boundary of the domain, specifically the curvature κ of the boundary, effect the higher order jets of the corresponding billiard map. The billiard map near the boundary is almost integrable for smooth strictly convex domains. We use this fact to perform a small preliminary perturbation which yields a domains with a periodic orbit containing a (quadratic) Homoclinic Tangency. The main technique in obtaining Newhouse phenomena is by unfolding generically these Homoclinic Tangencies. We thus show how one is able to unfold these Homoclinic Tangencies by perturbations of the curvature. At the same time, we show how one is able to perform these perturbations without destroying other billiard orbits in consideration.