Cusps of caustics by reflection in ellipses

This paper is concerned with the billiard version of Jacobi's last geometric statement and its generalizations. Given a non-focal point (Formula presented.) inside an elliptic billiard table, one considers the family of rays emanating from (Formula presented.) and the caustic (Formula presented.) of the reflected family after (Formula presented.) reflections off the ellipse, for each positive integer (Formula presented.). It is known that (Formula presented.) has at least four cusps and it has been conjectured that it has exactly four (ordinary) cusps. The present paper presents a proof of this conjecture in the special case when the ellipse is a circle. In the case of an arbitrary ellipse, we give an explicit description of the location of four of the cusps of (Formula presented.), though we do not prove that these are the only cusps.