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Elliptic islands on the elliptical stadium

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  • The elliptical stadium is a plane region bounded by a curve constructed by joining two half-ellipses with axes $a > 1$ and $b = 1$, by two parallel segments of equal length $2h$. The corresponding billiard problem defines a two-parameter family of discrete dynamical systems through the maps $T_{a,h}$.
    We investigate the existence of elliptic islands for a special family of periodic orbits of $T_{a,h}$. The hyperbolic character of those orbits were studied in [2] for $1 < a < \sqrt 2$ and here we look for the elliptical character for every $a > 1$.
    We prove that, for $a < \sqrt 2$, the lower bound for chaos $h = H(a)$ found in [2] is the upper bound of ellipticity for this special family. For $a > \sqrt 2$ we prove that there is no upper bound on h for the existence of elliptic islands.
    Mathematics Subject Classification: 37E40, 70K42.


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