具有有理仿射关系的β变换的公共不变测度
Common invariant measures for beta-transformations with rational affine relations
- School of Mathematics and Big Data, Neijiang Normal University（内江师范学院数学与大数据学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文分类了具有有理仿射关系的两个β变换的公共不变概率测度，证明奇异不变测度仅为零处点质量，并确定了所有公共不变概率及排除线上的公共不动点。
AI中文摘要：
我们分类了两个β变换之间具有有理仿射关系时的公共不变概率测度。设β>1为无理数，且γ=aβ+c>1，其中a,c∈Q且a+c≠1。每个关于勒贝格测度奇异且对两个映射都不变的有限非负测度，都是零处点质量的倍数。不需要熵、遍历性或乘法独立性假设。证明构造了一个在无理旋转下不变的有限符号测度，并将其质量与原始测度的一阶矩联系起来。将此结果与已知的重合绝对连续不变概率的分类相结合，给出了该族中所有公共不变概率。唯一额外的测度是已知二次族中相邻基的混合，与一个公共绝对连续概率混合。我们还确定了排除线a+c=1上的所有公共不动点。
英文摘要：
We classify common invariant probabilities for two beta-transformations with a rational affine relation between their bases. Let $β>1$ be irrational and $γ=aβ+c>1$, where $a,c\in\mathbb Q$ and $a+c\ne1$. Every finite nonnegative measure that is singular with respect to Lebesgue measure and invariant under both maps is a multiple of the point mass at zero. No entropy, ergodicity or multiplicative independence assumption is needed. The proof constructs a finite signed measure invariant under an irrational rotation and relates its mass to the first moment of the original measure. Combining this result with the known classification of coincident absolutely continuous invariant probabilities gives all common invariant probabilities in this family. The only additional measures are mixtures with a common absolutely continuous probability for adjacent bases in a known quadratic family. We also determine all common fixed points on the excluded line $a+c=1$.
