Q. Is it possible to trap all the light from one point source by a finite collection of two-sided disjoint segment mirrors?
I posed this question in several forums before (e.g., here and in an earlier MO question), and it has remained unsolved. But I've recently become re-interested in it. Let me first clarify the question. It seems best to treat the mirrors as open segments (i.e., not including their endpoints), but insist that they are disjoint as closed segments. And the point source of light should be disjoint from the closed segments.
$6$ mirrors. Lightray starts at center, exits (green) after $46$ reflections.
Of course a finite number of rays can be trapped periodically, and less obviously a finite number of rays can be trapped nonperiodically. But it seems quite impossible to trap all rays from a single fixed point. Because of segment disjointness, there are paths to $\infty$, and it seems likely that some ray will hew closely enough to some path to escape to $\infty$. So I believe the answer to my question is No.
Perhaps application of Poincaré recurrence could lead to a proof, but I cannot see it.
Related: Can we trap light in a polygonal room?.