Absolutely Periodic Billiard Orbits of Arbitrarily High Order in Smooth Strictly Convex Domains
Absolutely Periodic Billiard Orbits of Arbitrarily High Order in Smooth Strictly Convex Domains
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Date
2023
Authors
Callis, Keagan Graham
Advisor
Kaloshin, Vadim Y
Citation
Abstract
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.