Furstenberg times-p-times-q problem

Definitions and mathematical statement

Write $\mathbb T=\mathbb R/\mathbb Z$ for the circle with its Borel $\sigma$-algebra, and let $m$ be normalized Lebesgue (Haar) measure. Fix integers $p,q>1$ which are multiplicatively independent, meaning $p^r\ne q^s$ for all positive integers $r,s$. Define the continuous endomorphisms $M_p(x)=px\pmod1$ and $M_q(x)=qx\pmod1$.

Suppose $\mu$ is a Borel probability on $\mathbb T$ satisfying \[ \mu(M_p^{-1}E)=\mu(E)=\mu(M_q^{-1}E) \qquad\text{for every Borel }E\subset\mathbb T, \] and $\mu(\{x\})=0$ for every $x\in\mathbb T$. Must $\mu=m$? Equivalently, can there be a nonatomic probability other than Haar measure invariant under both multiplications?

Invariance is expressed by inverse images because $M_p$ and $M_q$ are not one-to-one maps of the circle. The problem puts no absolute continuity or full-support assumption on $\mu$ and no lower bound on its entropy. It also does not assume ergodicity for either individual map. The nonatomic hypothesis excludes measures supported on finite rational orbits and all probability measures assigning positive mass to a point. Multiplicative independence excludes the immediate situation in which both maps are powers of the same multiplication map. Both the integers and the measure are universally quantified in the uniqueness formulation; a counterexample for any one admissible pair would answer the existence question affirmatively.

Short English statement

Is Lebesgue measure the only nonatomic probability on the circle invariant under multiplication by two multiplicatively independent integers?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600017 (AMR-045-0017), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Boyle - Open Problems in Symbolic Dynamics (2008), Problem/Question/Conjecture 14.1. Original source indexed by AMR-045-0017. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.