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arXiv · 2407.17090

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

Corentin Fierobe·Alfonso Sorrentino
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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.

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Corentin Fierobe, Alfonso Sorrentino. 2024-07-24. On the existence of periodic invariant curves for analytic families of twist maps and billiards. https://arxiv.org/abs/2407.17090

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