发表机构
Department of Mathematics, University of Haifa at Oranim（奥拉尼姆海法大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究构造了二维环面上维数为$s$的遍历$\times p$-不变测度，其所有直线投影均保持维数且平面管质量满足$O(w^s)$界，同时证明正维数拟Bernoulli $T_p$-不变测度必存在维数下降投影。
AI 中文摘要
固定整数$p\geq2$且$0<s<1$。我们构造了二维环面$\mathbb{T}^2$上维数为$s$的遍历$\times p$-不变测度$\mu$，使得所有直线投影都保持维数，即便对应的投影迭代函数系（IFS）存在精确重叠的情况。实际上，我们的测度对每个宽度为$w$的平面管赋予的质量为$O(w^s)$。该结果在端点$s=1$处是紧的，这已由Pyörälä、Shmerkin、Suomala和Wu（2025）证明。相比之下，我们证明了每个具有正维数的拟Bernoulli $T_p$-不变测度都存在一个维数下降的投影。
英文摘要
Fix an integer $p\geq 2$ and $0<s<1$. We construct an ergodic $\times p$-invariant measure $μ$ on $\mathbb T^2$ of dimension $s$, such that every line projection preserves dimension, including when the corresponding projected IFS has exact overlaps. In fact, our measure assigns mass $O(w^s)$ to every planar tube of width $w$. The result is sharp at the endpoint $s=1$ as shown by Pyörälä, Shmerkin, Suomala and Wu (2025). In contrast, we show that every quasi-Bernoulli $T_p$-invariant measure with positive dimension admits a dimension-dropping projection.
