Here I am responding to Wojowu's request in a comment; I don't really consider this usefully addressing the question raised.

I thought of a simpler construction. Take a square with mirrored sides, and
poke a point-hole in the interior of the top edge. When a ray from outside
with an irrational slope (w.r.t. the box sides) enters through the hole, it is
trapped. For if it escaped through the hole after $k$ reflections, then it
would contradict the irrationality of the slope. It is known that the path of
this "billiard" is aperiodic (a result due to G.A.Galperin, I believe).
<hr />
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
[![RaysInSquare][1]][1]
<hr />
And of course almost all rays have irrational slope.

But as soon as the point-hole is enlarged to a positive-width hole, the ray
will eventually escape (Poincaré recurrence).


  [1]: https://i.sstatic.net/Wy5sx.jpg