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