Nihonkai Mathematical Journal

Differentiability of invariant circles for strongly integrable convex billiards

Nobuhiro Innami

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Let $C$ be a closed convex curve of class $C^2$ in the plane. We consider the domain bounded by $C$ a billiard table. Assume that the convex billiard of $C$ is integrable and satisfies a certain property. The property is that the limiting leaves are either closed curves or discrete points in the phase space. Then the set of points with irrational slopes make invariant circles of class $C^1$. If the sets of points with rational slopes do not make invariant circles, then they contains two invariant circles such that they are of class $C^1$ except at finitely many points in $C$.

Article information

Nihonkai Math. J., Volume 24, Number 1 (2013), 1-17.

First available in Project Euclid: 5 September 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C22: Geodesics [See also 58E10]
Secondary: 37E40: Twist maps

Geometry of geodesics Convex billiards Integrable convex billiards Invarient circles


Innami, Nobuhiro. Differentiability of invariant circles for strongly integrable convex billiards. Nihonkai Math. J. 24 (2013), no. 1, 1--17.

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