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

TL;DR

This paper constructs numerous $C^1$ expanding maps of the circle with invariant measures that are absolutely continuous and possess prescribed regularity, revealing complex behaviors in invariant measure existence.

Contribution

It demonstrates the existence of uncountably many $C^1$ expanding circle maps with invariant measures of specific regularity, including those with unbounded distortion.

Findings

Existence of maps with invariant measures having prescribed modulus of continuity.

Construction of maps preserving Lebesgue measure with optimal derivative regularity.

Many maps exhibit unbounded distortion, including those preserving Lebesgue measure.

Abstract

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.