  We prove that for any given modulus of continuity {\omega} there exist
(uncountably many) C1 uniformly expanding maps of the circle whose derivatives
have $C^1$ as an optimal modulus of continuity and which preserve an invariant
probability measure equivalent to Lebesgue whose density is
{\omega}-continuous, and also (uncountably many) $C^1$ uniformly expanding maps
of the circle whose derivatives have {\omega} as an optimal modulus of
continuity which preserve Lebesgue measure. Moreover, we show that many of
these maps, including those which preserve Lebesgue measure, have unbounded
distortion.
