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 $\lceil q\mathrm{/}n\rceil -1$ under any polynomial deformation of the circle of degree $n$.