Invariant measures and subsystems

Definitions and mathematical statement

Consider the compact binary configuration space \[ X=\{x\in(\mathbb Z/2\mathbb Z)^{\mathbb Z^2}: x_{i,j}+x_{i+1,j}+x_{i,j+1}=0\pmod2 \text{ for every }(i,j)\}. \] For $v\in\mathbb Z^2$, the shift $\alpha^v$ translates coordinates by $v$. The problem has two classification targets. First, describe all Borel probabilities $\mu$ on $X$ satisfying $\mu((\alpha^v)^{-1}E)=\mu(E)$ for every Borel $E$ and every $v$. Second, describe all nonempty closed sets $Y\subset X$ with $\alpha^vY=Y$ for every $v\in\mathbb Z^2$.

Such a $Y$ is called a subsystem, with the restricted shift action. It need not be a subgroup of the compact group $X$, and it need not have a finite forbidden-pattern presentation. Likewise, the probabilities need not be Haar measures on invariant subgroups, need not be ergodic, and need not have full support. A measure classification should in particular identify the ergodic possibilities; general invariant measures then include their ergodic decompositions.

The defining parity rule fixes the ambient system throughout, so the question asks for its invariant probability simplex and its closed invariant subsets, rather than for arbitrary binary planar shifts. Finite orbits and proper subsystems are included. Source Problem 14.3 concerns this Ledrappier three-dot system, despite the catalogue's broad reference to the Furstenberg discussion earlier in that section.

Short English statement

Classify all invariant probabilities and closed invariant subsets of the Ledrappier three-dot shift.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600019 (AMR-045-0019), 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 14.3. Original source indexed by AMR-045-0019. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.