# Illumination problem and absolutely focusing mirrors

19 Nov 2001-Vol. 4446, pp 185-192

TL;DR: In this paper, the authors consider the illumination and the strong illumination properties for closed bounded regions of Euclidean spaces, and they show how the regions with different illumination properties should be designed.

Abstract: We consider the illumination and the strong illumination properties for closed bounded regions of Euclidean spaces. These properties are intimately connected with a problem of chaoticity of the corresponding billiards. It is shown that there are only two mechanisms of chaoticity in billiard systems, which are called the mechanism of dispersing and the mechanism of defocusing. Our results show how the regions with different illumination properties should be designed. Especially each focusing mirror in the boundary of a region must be an absolutely focusing one. The notion of absolutely focusing mirrors is a new one in the geometric optic and it plays a key role for the illumination problem.

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TL;DR: In this paper, the B-property for two-dimensional domains with focusing and neutral regular components is proved and some examples of three and more dimensional domains with billiards obeying this property are also considered.

Abstract: For billiards in two dimensional domains with boundaries containing only focusing and neutral regular components and satisfacting some geometrical conditionsB-property is proved. Some examples of three and more dimensional domains with billiards obeying this property are also considered.

574 citations

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TL;DR: On considere un systeme billard de 2 objets de masses m 1 and m 2, on montre que pour un ensemble dense de paires (m 1,m 2 ) ce systeme est ergodique.

Abstract: On considere un systeme billard de 2 objets de masses m 1 et m 2 . On montre que pour un ensemble dense de paires (m 1 ,m 2 ) ce systeme est ergodique

314 citations

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TL;DR: In this paper, a large class of billiards with convex pieces of the boundary which have nonvanishing Lyapunov exponents was introduced, where the exponents have non-vanishing exponents.

Abstract: We introduce a large class of billiards with convex pieces of the boundary which have nonvanishing Lyapunov exponents.

237 citations

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214 citations

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TL;DR: In this paper, a system of caustics is found for a plane convex domain with a sufficiently smooth boundary; the Caustics are close to the boundary and occupy a set of positive measure.

Abstract: A system of caustics is found for a plane convex domain with a sufficiently smooth boundary; the caustics are close to the boundary and occupy a set of positive measure. A caustic is a convex smooth curve lying in the domain and possessing the property that a tangent to it becomes another tangent to the same curve after reflection from the boundary according to the law of geometrical optics.

184 citations