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?

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