DCDS
Fractal bodies invisible in 2 and 3 directions
Alexander Plakhov Vera Roshchina
Discrete & Continuous Dynamical Systems - A 2013, 33(4): 1615-1631 doi: 10.3934/dcds.2013.33.1615
We study the problem of invisibility for bodies with a mirror surface in the framework of geometrical optics. We show that for any two given directions it is possible to construct a two-dimensional fractal body invisible in these directions. Moreover, there exists a three-dimensional fractal body invisible in three orthogonal directions. The work continues the previous study in [1,12], where two-dimensional bodies invisible in one direction and three-dimensional bodies invisible in one and two orthogonal directions were constructed.
keywords: billiards. problems of minimal resistance Invisibility in geometrical optics
DCDS
Mathematical retroreflectors
Alexander Plakhov
Discrete & Continuous Dynamical Systems - A 2011, 30(4): 1211-1235 doi: 10.3934/dcds.2011.30.1211
Retroreflectors are optical devices that reverse the direction of incident beams of light. Here we present a collection of billiard type retroreflectors consisting of four objects; three of them are asymptotically perfect retroreflectors, and the fourth one is a retroreflector which is very close to perfect. Three objects of the collection have recently been discovered and published or submitted for publication. The fourth object --- notched angle --- is a new one; a proof of its retroreflectivity is given.
keywords: Billiards problems of maximum resistance. retroreflectors shape optimization
JMD
Perfect retroreflectors and billiard dynamics
Pavel Bachurin Konstantin Khanin Jens Marklof Alexander Plakhov
Journal of Modern Dynamics 2011, 5(1): 33-48 doi: 10.3934/jmd.2011.5.33
We construct semi-infinite billiard domains which reverse the direction of most incoming particles. We prove that almost all particles will leave the open billiard domain after a finite number of reflections. Moreover, with high probability the exit velocity is exactly opposite to the entrance velocity, and the particle's exit point is arbitrarily close to its initial position. The method is based on asymptotic analysis of statistics of entrance times to a small interval for irrational circle rotations. The rescaled entrance times have a limiting distribution in the limit when the length of the interval vanishes. The proof of the main results follows from the study of related limiting distributions and their regularity properties.
keywords: billiards dynamical renormalization retroreflectors. homogeneous flow Recurrence circle rotation

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