High-order persistence of resonant caustics in perturbed circular billiards

We find necessary and sufficient conditions for high-order persistence of resonant caustics in perturbed circular billiards. The main tool is a perturbation theory based on the Bialy-Mironov generating function for convex billiards. All resonant caustics with period $q$ persist up to order $\lceil q/n \rceil -1$ under any polynomial deformation of the circle of degree $n$.

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