On the existence of invariant absolutely continuous probability measures for $C^1$ expanding maps of the circle

We prove that for any given modulus of continuity ω 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 ω-continuous, and also (uncountably many) $C^1$ uniformly expanding maps of the circle whose derivatives have ω 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.