most recent 30 from http://mathoverflow.net 2013-05-24T15:51:09Z http://mathoverflow.net/feeds/question/53852 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/53852/billiards-with-incompatible-regions mjqxxxx 2011-01-31T02:55:43Z 2011-01-31T02:55:43Z

<p>An <a href="http://mathoverflow.net/questions/53641/dense-orbits-in-billiards" rel="nofollow">existing question</a> asks whether "almost every" two-dimensional billiard possesses at least one orbit that is dense in its interior. My question is about the following set of strong counter-examples: two-dimensional billiards that contain a pair of open regions that are <em>incompatible</em>, in that no orbit intersects both regions. <a href="http://mathworld.wolfram.com/IlluminationProblem.html" rel="nofollow">Penrose's unilluminable room</a> is one example: an orbit that passes through the upper half of the room cannot pass through either of the square culs-de-sac in the bottom half. That example seems to rely on non-generic properties of the boundary shape (e.g., the inclusion of elliptical arcs with precisely placed foci), but perhaps this intuition is wrong.</p> <p>How generic is the property of having incompatible regions? Are there known examples of billiards where this property is robust under arbitrary small perturbations of the boundary?</p>