On the existence of periodic invariant curves for analytic families of twist maps and billiards

TL;DR

This paper proves that in analytic families of twist maps, invariant curves with specific rotation numbers either exist for all parameters or only at discrete points, and applies this to billiard models.

Contribution

It extends previous results on invariant curves in twist maps to dimension 2 and applies the findings to various billiard systems.

Findings

Invariant curves exist for all parameters or a discrete set.

The set of maps with a given invariant curve is a strict analytic subset.

Application to rational caustics in billiard models.

Abstract

In this paper we prove that in any analytic one-parameter family of twist maps of the annulus, homotopically invariant curves filled with periodic points corresponding to a given rotation number, either exist for all values of the parameters or at most for a discrete subset. Moreover, we show that the set of analytic twist maps having such an invariant curve of a given rotation number is a strict analytic subset of the set of analytic twist maps. The first result extends, in dimension 2, a previous result by Arnaud, Massetti and Sorrentino. We then apply our result to rational caustics of billiards, considering several models such as Birkhoff billiards, outer billiards and symplectic billiards.