## 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

Zbl 0744.58041
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### References:

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