  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.
