Search Light in Billiard Tables

We investigate whether a search light, S, illuminating a tiny angle (“cone”) with vertex A inside a bounded region Q ∈ IR with the mirror boundary ∂Q, will eventually illuminate the entire region Q. It is assumed that light rays hitting the corners of Q terminate. We prove that: (I) if Q = a circle or an ellipse, then either the entire Q or an annulus between two concentric circles/confocal ellipses (one of which is ∂Q) or the region between two confocal hyperbolas will be illuminated; (II) if Q = a square, or (III) if Q = a dispersing (Sinai) or semidespirsing billiards, then the entire region Q is will be illuminated.