Non co-preservation of the $1/2$ & $1/(2l+1)$-rational caustics along deformations of circles

Abstract

For any given positive integer $l$, we prove that every plane deformation of a circle which preserves the $1/2$ and $1/(2l+1)$-rational caustics is trivial i.e. the deformation consists only of similarities (rescalings plus isometries).