# 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 article, a description of a large class of plane billiards with the Pesin region of measure one is given, and open conditions including properly those founded by Wojtkowski [W1] for C4 focusing boundaries are obtained.

Abstract: We give a description of a large class of plane billiards with Pesin region of measure one. Open conditions including properly those founded by Wojtkowski [W1] forC4 focusing boundaries are obtained. Lyapunov's forms, introduced by Lewowicz, are used.

85 citations

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TL;DR: In this paper, the authors introduce the notion of a focusing arc and show that such arcs can be used to construct convex 2D planar billiards with positive Lyapunov exponent almost everywhere.

Abstract: A new open class of convex 2 dimensional planar billiards with positive Lyapunov exponent almost everywhere is constructed. We introduce the notion of a focusing arc and show that such arcs can be used to build billiard systems with positive Lyapunov exponents. We prove that under smallC6 perturbations, focusing arcs remain focusing and thereby show that perturbations of the Bunimovich stadium billiard have positive Lyapunov exponents.

83 citations

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TL;DR: In this article, it was shown that such dynamical systems are systems of A N Kolmogorov (K-systems) if the "perturbation" satisfies certain conditions which have an intuitive geometric interpretation.

Abstract: One considers dynamical systems generated by billiards which are "perturbations" of dispersing billiards It is shown that such dynamical systems are systems of A N Kolmogorov (K-systems), if the "perturbation" satisfies certain conditions which have an intuitive geometric interpretation Bibliography: 12 items

60 citations

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01 Jan 1992TL;DR: In this paper, the authors consider focusing curves Γf (of class C∝, α≥3) such that each incoming infinitesimal beam of parallel rays focuses after hitting the last time in the series of consecutive reflections from it.

Abstract: We consider focusing curves Γf (of class C∝, α≥3) such that each incoming infinitesimal beam of parallel rays focuses after hitting Γf for the last time in the series of consecutive reflections from it. We call such curves absolutely focusing. We prove some characteristic properties of absolutely focusing curves and show that these remain absolutely focusing under small C3-(C4)-perturbations if this component has constant (nonconstant) curvature. We also present examples of absolutely focusing curves and consider the applications of these curves to some classes of continuous fractions.

49 citations

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TL;DR: Ergodicity of two-dimensional billiards which satisfy some general conditions is proved in this paper, and this theorem is applied to one concrete class of billiard that contains, in particular, billiard players in the "stadium".

Abstract: Ergodicity of two-dimensional billiards which satisfy some general conditions is proved. This theorem is applied to one concrete class of billiards that contains, in particular, billiards in the “stadium”.

43 citations

### "Illumination problem and absolutely..." refers background in this paper

...If we drop the condition of convexity then there are big classes of regions which contain only focusing or two or all three kinds of boundary components and for which chaoticity, ergodicity (and therefore the strong illumination) is proven.(8)...

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...General conditions for chaoticity, ergodicity and therefore for the strong illumination property for billiards in two-dimensional regions were formulated.(8,21) They could be briefly described as the following three conditions....

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