Amie Wilkinson asked me the following question some time ago:
Given a smooth convex Jordan curve , consider the billiard map , let be the projection.
a) If , does this imply is a circle?
(Yes, in fact we only need to fix the second component of points in for a chosen pair of points . Classical geometry)
b) If , does this imply is a circle?
(No, my example was a cute construction that attaches six circular arcs together)
c) What’s the smallest set s.t. if fixes the second component on then has to be a circle?
I am still thinking about c)…My guess is that any sub interval would work, and of course any dense subset inside a given set works equally well as the whole set…
But is it possible to have only finitely many angles? Maybe even two angles?