Free ergodic ${\mathbb{Z}}^2$-systems and complexity

Abstract

Using results relating the complexity of a two dimensional subshift to its periodicity, we obtain an application to the well-known conjecture of Furstenberg on a Borel probability measure on $[0,1)$ which is invariant under both $x\mapsto px(\mathrm{mod}1)$ and $x\mapsto qx(\mathrm{mod}1)$, showing that any potential counterexample has a nontrivial lower bound on its complexity.