论文标题
Furstenberg猜想中的几何形状
Geometry in the Furstenberg Conjecture
论文作者
论文摘要
在本文中，我们展示了几何形状如何在Furstenberg猜想的研究中发挥作用（请参阅〜\ cite {f}）。令$ p> 1 $和$ q> 1 $是两个相对主要的正整数。我们证明，具有平衡几何形状的非原子$ p $ - 和$ q $ - invariant措施必须是Lebesgue度量。在证明中，我们不会假设该度量的奇特性。结果提供了一个直观的几何标准，可以证明Furstenberg猜想或构建反例。
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.
