Construction of \(K\)-stable foliations for two-dimensional dispersing billiards without eclipse. (English) Zbl 1055.37037

Morita, Takehiko

J. Math. Soc. Japan 56, No. 3, 803-831 (2004).

Summary: Let \(T\) be the billiard map for a two-dimensional dispersing billiard without eclipse. We show that the nonwandering set \(\Omega^+\) for \(T\) has a hyperbolic structure quite similar to that of the horseshoe. We construct a sort of stable foliation for \((\Omega^+,T)\) each leaf of which is a \(K\)-decreasing curve. We call the foliation a \(K\)-stable foliation for \((\Omega^+, T)\). Moreover, we prove that the foliation is Lipschitz continuous with respect to the Euclidean distance in the so-called \((r,\varphi)\)-coordinates. It is well known that we can not always expect the existence of such a Lipschitz continuous invariant foliation for a dynamical system even if the dynamical system itself is smooth. Therefore, we keep our construction as elementary and selfcontained as possible so that one can see the concrete structure of the set \(\Omega^+\) and why the \(K\)-stable foliation turns out to be Lipschitz continuous.

MSC:

37D50	Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
37D05	Dynamical systems with hyperbolic orbits and sets
37D10	Invariant manifold theory for dynamical systems

Keywords:

billiard map; Lipschitz continuous invariant foliation