2024/07/24 by Fierobe, Corentin, Sorrentino, Alfonso
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2407.17090
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.