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Chaotic properties of the elliptical stadium. (English) Zbl 0843.58084

Summary: The elliptical stadium is a curve constructed by joining two half-ellipses, with half axes \(a>1\) and \(b=1\), by two straight segments of equal length \(2h\).
V. J. Donnay [ibid. 141, No. 2, 225-257 (1991; Zbl 0744.58041)] has shown that if \(1< a< \sqrt {2}\) and if \(h\) is big enough, then the corresponding billiard map has a positive Lyapunov exponent almost everywhere; moreover, \(h\to \infty\) as \(a\to \sqrt {2}\).
In this work we prove that if \(1<a< \sqrt {4-2 \sqrt {2}}\), then \(h> 2a^2 \sqrt {a^2-1}\) assures the positiveness of a Lyapunov exponent. And we conclude that, for these values of \(a\) and \(h\), the elliptical stadium billiard mapping is ergodic and has the \(K\)-property.

MSC:

37A99 Ergodic theory
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry

Citations:

Zbl 0744.58041
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References:

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