H\"older conditions for endomorphisms of hyperbolic groups

It is proved that an endomorphism $\phi$ of a hyperbolic group $G$ satisfies a H"older condition with respect to a visual metric if and only if $\phi$ is virtually injective and $G\phi$ is a quasiconvex subgroup of $G$. If $G$ is virtually free or torsion-free co-hopfian, then $\phi$ is uniformly continuous if and only if it it satisfies a H"older condition if and only if it is virtually injective. Lipschitz conditions are discussed for free group automorphisms.