N. Chernov

Stadia are popular models of chaotic billiards introduced by Bunimovich in 1974. They are analogous to dispersing billiards due to Sinai, but their fundamental technical characteristics are quite different. Recently many new results were obtained for various chaotic billiards, including sharp bounds on correlations and probabilistic limit theorems, and these results require new, more powerful technical apparatus. We present that apparatus here, in the context of stadia, and prove "regularity" properties.

We investigate whether a search light, $S$, illuminating a tiny angle ("cone") with vertex $A$ inside a bounded region $Q \in \mathbb{R}^2$ with the mirror boundary $\partial Q$, will eventually illuminate the entire region $Q$. It is assumed that light rays hitting the corners of $Q$ terminate. We prove that: $(\mathbf{I})$ 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 $\partial Q$) or the region between two confocal hyperbolas will be illuminated; $(\mathbf{II})$ if $Q =$ a square, or $(\mathbf{III})$ if $Q =$ a dispersing (Sinai) or semidespirsing billiards, then the entire region $Q$ is will be illuminated.