# 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.

##### References

More filters

••

TL;DR: In this paper, the authors consider some examples of domains in the d-dimensional Euclidean space, d≥3, that generate billiards with chaotic behavior and show that time correlations tend to zero in the limit t → ∞.

34 citations

••

TL;DR: In this article, it was shown that billiards in a class of regions in R n, n>2 with focusing and flat boundary components have nonvanishing Lyapunov exponents.

Abstract: We give the affirmative answer to the long-standing question whether or not the mechanism of defocusing can produce a chaotic behavior in high-dimensional Hamiltonian systems. To do this we prove that billiards in a class of regions in R n , n>2, with focusing and flat boundary components have nonvanishing Lyapunov exponents.

33 citations

••

TL;DR: In this paper, the authors constructed linearly stable periodic orbits in a class of billiard systems in 3D domains with boundaries containing semispheres arbitrarily far apart, and showed that the results about planar billiard system in domains with convex boundaries which have nonvanishing Lyapunov exponents cannot be easily extended to 3D.

Abstract: We construct linearly stable periodic orbits in a class of billiard systems in 3 dimensional domains with boundaries containing semispheres arbitrarily far apart. It shows that the results about planar billiard systems in domains with convex boundaries which have nonvanishing Lyapunov exponents cannot be easily extended to 3 dimensions.

29 citations

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

...This publication inspired objections that it can be not the case.(11) Finally, it has been proven for some classes of high-dimensional regions that the mechanism of defocusing does work in high dimensions as well12−16 and can generate chaoticity and ergodicity of corresponding billiards and therefore the strong illumination property....

[...]

••

TL;DR: In this article, a class of 3D regions with focusing components was described to generate a billiard system with non-vanishing Lyapunov exponents, and the long standing question whether chaotic motion caused by defocusing can be produced in more than two dimensions was answered.

Abstract: We describe a class of 3-dimensional regions with focusing components that generate a billiard system with non-vanishing Lyapunov exponents. To do this we answer affirmatively the long standing question whether or not the chaotic motion caused by defocusing can be produced in more than two dimensions.

26 citations

••

22 citations