Full-support invariant measures

Definitions and mathematical statement

Let $\mathbb F_2=\{0,1\}$ with addition modulo two, and define the Ledrappier three-dot system \[ X=\{x\in\mathbb F_2^{\mathbb Z^2}: x_{i,j}+x_{i+1,j}+x_{i,j+1}=0 \text{ for all }(i,j)\in\mathbb Z^2\}. \] It is a compact abelian group under coordinatewise addition and the product topology. Let $\alpha^{(a,b)}x$ have coordinate $(\alpha^{(a,b)}x)_{i,j}=x_{i+a,j+b}$, and let $m_X$ be normalized Haar probability on the compact group $X$.

The question asks whether $m_X$ is the only Borel probability $\mu$ on $X$ which has all three properties: $\alpha^v_*\mu=\mu$ for every $v\in\mathbb Z^2$; the whole $\mathbb Z^2$ action is ergodic for $\mu$; and $\operatorname{supp}\mu=X$. Ergodicity means that a measurable set invariant modulo $\mu$ under every $\alpha^v$ has measure zero or one. Full support means every nonempty relatively open cylinder in $X$ has positive measure.

No ergodicity of each individual generator, and no positive entropy assumption, is added. Haar probability is distinguished by invariance under all additions by elements of $X$; a competing measure is required to be invariant only under the spatial shifts. This explicit system is the $X$ of source Question 14.2; it resolves the catalogue's imprecise reference to the preceding commuting-endomorphism discussion.

Short English statement

Is Haar measure the sole ergodic, fully supported, shift-invariant probability on the binary three-dot system?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600018 (AMR-045-0018), 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 14.2. Original source indexed by AMR-045-0018. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.