There are no $\sigma$-finite absolutely continuous invariant measures for multicritical circle maps

TL;DR

This paper proves that multicritical circle maps without periodic orbits cannot have an infinite, absolutely continuous invariant measure, confirming that such maps only admit a unique singular invariant probability measure.

Contribution

It demonstrates that no $\sigma$-finite absolutely continuous invariant measure exists for multicritical circle maps without periodic orbits, using Katznelson's criterion.

Findings

No $\sigma$-finite absolutely continuous invariant measure exists

Multicritical circle maps only admit a unique singular invariant probability measure

The result applies to maps without periodic orbits

Abstract

It is well-known that every multicritical circle map without periodic orbits admits a unique invariant Borel probability measure which is purely singular with respect to Lebesgue measure. Can such a map leave invariant an infinite, $\sigma$-finite invariant measure which is absolutely continuous with respect to Lebesgue measure? In this paper, using an old criterion due to Katznelson, we show that the answer to this question is no.