Geometry in the Furstenberg Conjecture

TL;DR

This paper explores geometric methods in the study of the Furstenberg conjecture, demonstrating that certain invariant measures with balanced geometry must be Lebesgue, offering a new criterion for the conjecture.

Contribution

It introduces a geometric criterion for invariant measures that can be used to prove the Furstenberg conjecture or find counterexamples, without assuming ergodicity.

Findings

Non-atomic p- and q-invariant measures with balanced geometry are Lebesgue measures.

Provides a geometric approach to the Furstenberg conjecture.

Offers a criterion to either prove the conjecture or construct counterexamples.

Abstract

In this paper, we show how geometry plays in the study of the Furstenberg conjecture (refer to~\cite{F}). Let $p>1$ and $q>1$ be two relative prime positive integers. We prove that a non-atomic $p$- and $q$-invariant measure having balanced geometry must be the Lebesgue measure. In the proof, we will not assume the ergodicity of the measure. The result provides an intuitive geometric criterion to either prove the Furstenberg conjecture or construct a counter-example.