On uniformly continuous endomorphisms of hyperbolic groups

We prove a generalization of the fellow traveller property for certain type quasi-geodesics and use it to present three equivalent geometric formulations bounded reduction that this is preservation coarse median. then provide an affirmative answer question from Araújo Silva as whether every nontrivial uniformly continuous endomorphism hyperbolic group with respect visual metric satisfies Hölder condition. remark these results combined work done by Paulin admitting extension completion has finitely generated fixed point subgroup.