Pingree open problems — Hochman problem 2

Definitions and mathematical statement

Let $D:[0,1)\to[0,1)$ be $D(x)=2x-\lfloor2x\rfloor$. Fix a Borel probability $\mu$ with $D_*\mu=\mu$ which is ergodic: every measurable set invariant modulo $\mu$ has measure zero or one. Assume $0<h_\mu(D)<1$, with Kolmogorov--Sinai entropy measured using logarithms to base two. Regard $\mu$ also as a measure on $\mathbb R$ supported on $[0,1)$.

For real $a,b$, let $F_{a,b}(x)=ax+b$ and define its pushforward by $(F_{a,b*}\mu)(E)=\mu(F_{a,b}^{-1}E)$. In this question "non-singular" means that $\mu$ and $F_{a,b*}\mu$ are not mutually singular. Equivalently, there is a nonzero finite positive measure dominated by both; it does not mean that the two measures are equivalent on all of $\mathbb R$. The target is to characterize, in terms of the given invariant measure, \[ \mathcal A(\mu)=\{(a,b)\in\mathbb R^2: \mu\not\perp F_{a,b*}\mu\}. \] The description must address both the allowed slopes and the allowed translations. Coordinates are transformed in $\mathbb R$, without reducing $ax+b$ modulo one. The positive entropy hypothesis excludes atomic ergodic measures, so a constant affine map cannot qualify. A classification is sought uniformly over this class of measures, allowing the answer to depend on $\mu$; the question does not prescribe a particular Bernoulli digit distribution.

Short English statement

For an intermediate-entropy doubling-invariant measure, determine which affine changes of coordinates retain a nonzero overlapping measure component.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4400002 (AMR-043-0002), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open problems from the 3rd Pingree Workshop on Dynamical Systems. (2010), Mike Hochman, Problem 2. Original source indexed by AMR-043-0002. https://math.huji.ac.il/~mhochman/open-problems/pingree-open-problems.pdf.