高维Bedford-McMullen子移位的临界Hausdorff测度

中文摘要 AI 辅助

设$\Lambda=\operatorname{diag}(m_1,\ldots,m_d)$为对角扩张整数矩阵，满足$m_1\ge\cdots\ge m_d\ge2$且至少有两个不同的对角元，并设$X\subset\mathcal A^{\mathbb N}$为在$\mathcal A=\prod_{r=1}^d\{0,1,\ldots,m_r-1\}$上具有弱规范性质的子移位。令$K=R(X)$为相应的自仿集。记$\gamma=\dim_{\mathrm H} K$，我们证明了一个尖锐的临界测度二分法：\begin{align*} \dim_{\mathrm H} K=\dim_{\mathrm B} K \Longleftrightarrow 0<\mathcal H^\gamma(K)<\infty,\quad \dim_{\mathrm H} K<\dim_{\mathrm B} K \Longleftrightarrow \mathcal H^\gamma(K)=\infty. \end{align*}在第二种情形下，$\mathcal H^\gamma$在$K$上不是$\sigma$-有限的。这解决了Peres于1994年对平面系统提出的猜想，并将其推广到更高维度。对于sofic子移位，我们证明了盒维数的存在性，并表明维数相等意味着临界Hausdorff测度为正且$\sigma$-有限，尽管它可能是无限的。我们还构造了两个满足$\dim_{\mathrm H} K<\dim_{\mathrm B} K$的例子：在一个例子中临界Hausdorff测度为正且有限，而在另一个例子中它是无限的但$\sigma$-有限。

英文摘要

Let $Λ=\operatorname{diag}(m_1,\ldots,m_d)$ be a diagonal expanding integer matrix with $m_1\ge\cdots\ge m_d\ge2$ and at least two distinct diagonal entries, and let $X\subset\mathcal A^{\mathbb N}$ be a subshift with weak specification over $\mathcal A=\prod_{r=1}^d\{0,1,\ldots,m_r-1\}$. Let $K=R(X)$ be the corresponding self-affine set. Writing $γ=\dim_{\mathrm H} K$, we prove a sharp critical-measure dichotomy: \begin{align*} \dim_{\mathrm H} K=\dim_{\mathrm B} K \Longleftrightarrow 0<\mathcal H^γ(K)<\infty,\quad \dim_{\mathrm H} K<\dim_{\mathrm B} K \Longleftrightarrow \mathcal H^γ(K)=\infty. \end{align*} In the second case, $\mathcal H^γ$ is not $σ$-finite on $K$. This resolves Peres' 1994 conjecture for planar systems and extends to higher dimensions. For sofic subshifts, we prove the existence of the box dimension and show that dimension equality implies that the critical Hausdorff measure is positive and $σ$-finite, although it may be infinite. We also construct two examples with $\dim_{\mathrm H} K<\dim_{\mathrm B} K$: in one example the critical Hausdorff measure is positive and finite, whereas in the other it is infinite but $σ$-finite.

发表机构

• Nanjing University（南京大学）

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40 pages