Break-up of resonant invariant circles in perturbations of the geodesic circular billiard on surfaces of constant curvature

TL;DR

This paper investigates how invariant circles in geodesic circular billiards on surfaces of constant curvature break up under perturbations, extending planar results to curved surfaces.

Contribution

It generalizes Ramírez-Ros's planar billiard results to billiards on curved surfaces, demonstrating the non-persistence of invariant circles under perturbations.

Findings

Invariant circles do not persist under perturbations on curved surfaces.

Results extend planar billiard theory to constant curvature surfaces.

Supports the universality of invariant circle breakdown in billiards.

Abstract

We study the non-persistence of horizontal invariant circles for geodesically convex perturbations of the geodesic circular billiard on surfaces of constant curvature and show that the result obtained by Ram\'irez-Ros for the planar case, remains true for billiards on surfaces with constant curvature.