Topological stability of relatively hyperbolic groups acting on their boundaries

We prove a topological stability result for the actions of hyperbolic groups on their Bowditch boundaries. More precisely, we show that a sufficiently small perturbation of the standard boundary action, if assumed on each parabolic subgroup to be a perturbation by semi-conjugacy, is in fact always globally semi-conjugate to the standard action. This proves a relative version of the main result of arXiv:2206.14914. The assumption of control on the perturbation of parabolics is necessary.