Four cusps of caustics by reflection

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This talk is concerned with a billiard version of Jacobis Last Geometric Statement and its generalizations. Given a point O inside an oval billiard table (or mirror), one considers the family of rays emanating from O and the caustic (or envelope) of the reflected family of rays after n reflections off the walls of the table. I will describe two related statements:

(1) Theorem: for a generic O this caustic has at least 4 cusps for each positive integer n.

(2) Conjecture: for an elliptic table there are exactly four (ordinary) cusps.

I will describe a proof of (1) and partial results concerning (2).

This is joint work with Mark Spivakovsky (Toulouse) and Serge Tabachnikov (Penn State).

References:

* https://arxiv.org/abs/2112.07852

* https://arxiv.org/abs/2406.11074.

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