Stability of hyperbolic groups acting on their boundaries

A hyperbolic group \Gamma acts by homeomorphisms on its Gromov boundary \partial \Gamma . 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 \partial \Gamma is a topological sphere. Our proof here is independent and gives additional information about the semi-conjugacy in that case. Our techniques also give a new proof of global stability when \partial \Gamma = S^{1} .