  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 $x\mapsto px \pmod 1$ and $x\mapsto qx \pmod 1$, showing that any
potential counterexample has a nontrivial lower bound on its complexity.
