There's a longstanding open problem concerning whether or not it's possible to trap all the light from a point source using a finite collection of circles/lines whose sides are mirrors. This seems highly related to the theory of billiards, in particular the ergodic properties of billiards. It's been known since the 60s that billiards in domains with convex (inwards) walls are ergodic. This would seemingly be enough to prove that light escapes in the circle case, since we could place the circles on a torus and wait for the light from the light source to diffuse to a point which would correspond to escaping in the plane. The only problem is that the rays emitted from a single point are a set of measure zero. So my question is: does anybody familiar with ergodic theory, in general or billiards specifically, see a way to patch this argument?
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