Geometry in the Furstenberg Conjecture

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.