Existence of Absolutely Continuous Invariant Measures for C¹ Expanding Circle Maps

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