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Billiards on rational-angled triangles. (English) Zbl 0967.37019
Billiards in polygons are notoriously hard to investigate – they are on the “border” between chaotic and regular dynamical behavior. Polygons whose angles are rational multiple of $$\pi$$ (“rational polygons”) are very regular, their phase space is foliated by invariant surfaces. Still, the motion on an invariant surface is not exactly integrable, unless the polygon tiles the plane by reflections. Veech introduced the so called “lattice polygons” that allow a precise description of the motion on invariant surfaces. But the class of lattice polygons itself is not well understood. This paper is devoted to an explicit description of lattice triangle, mostly acute and right ones. The authors give some criteria for a triangle to be a lattice one and find new examples of lattice triangles.

##### MSC:
 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
##### Keywords:
billiards; rational polygons; lattice polygons
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