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

TL;DR

This paper links the complexity of two-dimensional subshifts to periodicity and applies this to Furstenberg's conjecture, establishing that any counterexample must have high complexity, thus advancing understanding of invariant measures.

Contribution

It introduces a novel connection between subshift complexity and Furstenberg's conjecture, providing lower bounds on complexity for potential counterexamples.

Findings

Any counterexample to Furstenberg's conjecture must have nontrivial complexity.

The results relate the complexity of 2D subshifts to their periodicity properties.

Provides new insights into invariant measures under multiplicative transformations.

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