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

TL;DR

This paper proves that any deformation of a circle preserving specific rational caustics must be trivial, meaning only similarities are involved, thus showing rigidity in such geometric structures.

Contribution

It establishes a rigidity result for plane deformations of circles that preserve certain rational caustics, demonstrating these deformations are necessarily trivial.

Findings

Deformations preserving the $1/2$ and $1/(2l+1)$-rational caustics are trivial.

Such caustic-preserving deformations are limited to similarities.

The result applies to all positive integers $l$.

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).