Elliptic islands in strictly convex billiards

Abstract

This paper addresses the question of genericity of existence of elliptic islands for the billiard map associated to strictly convex closed curves. More precisely, we study 2-periodic orbits of billiards associated to C 5 closed and strictly convex curves and show that the existence of elliptic islands is a dense property on the subset of those billiards having an elliptic 2-periodic point. Our main tools are normal perturbations, the Birkhoff Normal Form for elliptic fixed points and Moser's Twist Theorem.

The First Birkhoff Coefficient and the Stability of 2-Periodic Orbits on Billiards

“…Unfortunately, one cannot assure that at least one of the others is elliptic. In fact, there are many examples where all 2-periodic orbits are isolated and hyperbolic (see, for instance [5] or [9]). …”

“…On the other hand, ellipticity is an open property, in the sense that if a billiard associated to a C 2 strictly convex curve α has an elliptic 2-periodic orbit, then any strictly convex curve sufficiently C 2 -close to α generates a billiard with an elliptic 2-periodic orbit [5]. So, a large class of strictly convex billiards has elliptic 2-periodic orbits.…”

Ergodicity of the generalized lemon billiards

“…Later the existence of such islands has been proved rigorously 11,22 for almost all billiard tables, see also Proposition II.2. As we will see in Subsections IV A-IV C, this island undergoes significant developments as the parameter B moves away from 1.…”

“…Recently, the nonlinear stability of the elliptic case has been treated11,22 by Dias Carneiro, Oliffson Kamphorst, and Pinto-de-Carvalho. In fact, the following restates a result in Ref.…”

Ergodicity in Umbrella Billiards

“…On a bounded region Q ⊂ R 2 (the billiard table), an infinitesimal particle moves along segments at unit speed, changing direction according to the law of specular reflection upon collisions at boundaries. The essential link in billiards between the geometry of the table and the dynamics of the system facilitates a robust model which has proved useful in approaching problems ranging from the foundations of the Boltzmann's ergodic hypothesis [9], to the description of shell effects in semiclassical physics [5], to the design of microwave resonators in quantum chaos [33], and many other other applications [1,17,23,25,26]. In particular, ergodic properties are determined by the shape of the table, producing a spectrum of behaviors from completely integrable to strongly chaotic.…”