Stability of hyperbolic groups acting on their boundaries

A hyperbolic group acts by homeomorphisms on its Gromov boundary. We use a dynamical coding of boundary points to show that such actions are topologically stable in the dynamical sense: any nearby action is semi-conjugate to (and an extension of) the standard boundary action. This result was previously known in the special case that the boundary is a topological sphere. Our proof here is independent and gives additional information about the semiconjugacy in that case. Our techniques also give a new proof of global stability when the boundary is a circle.