Thanks for the clarification. Isn't it necessary for the evolute (locus of centers of curvature) to cut the ellipse for there to be 4-point intersections of the ellipse with circles centered on it? That limits you to eccentricity < sqrt(1/2). Even in that case I would be worried about starting points such that the orbit, initially determined without branching by 2-point intersections, would be quasiperiodic and avoid the regions near the ends of the minor axis where the 4-point intersections occur. Veit On Nov 20, 2010, at 7:04 PM, Bill Thurston wrote:
There is a continuum if you vary the starting point, but I was trying to say something about how many there are with a fixed starting point. The number varies according to the starting point, but I think my argument is a good sketch of a proof that for any fixed ellipse that is not a circle and any fixed starting point p on the ellipse, the number of n-gons through p grows exponentially with n. Bill Thurston