  We show that there are no nontrivial surjective uniformly asymptotically
regular mappings acting on a metric space and derive some consequences of this
fact. In particular, we prove that a jointly continuous left amenable or left
reversible semigroup generated by firmly nonexpansive mappings on a bounded
$\tau$-compact subset of a Banach space has a common fixed point, and give a
qualitative complement to the Markov-Kakutani theorem.
