广义康托函数的局部标度与维数畸变

中文摘要 AI 辅助

设μ是ℝ上与概率权重向量p关联的自相似康托测度，令K=suppμ，F表示μ的分布函数。我们刻画了K中点x的局部标度指数\\(\lim_{\substack{y\to x, y\in K}} \frac{\log |F(y)-F(x)|}{\log |y-x|}\\)存在且取给定值的情况，该刻画基于累积对数质量与几何尺度之比的收敛性，以及左右端点序列的次线性增长。不同于经典三分康托情形，本文方法适用于任意收缩比和概率权重。作为应用，我们构造了K中具有满豪斯多夫测度的子集M，使得F在M上的局部标度指数为\\({h(\mathbf q,\mathbf p)}/{\chi(\mathbf q)}\\)，并对每个A⊂M建立了精确维数畸变公式\\(\dim_{\mathrm H} F(A) = \frac{\chi(\mathbf q)}{h(\mathbf q,\mathbf p)} \dim_{\mathrm H} A\\)，其中q是自然概率向量，χ(q)是对应的李雅普诺夫指数，h(q,p)是q相对于p的交叉熵。最后，我们给出了一个三分支示例以说明所得结果。

英文摘要

Let \(μ\) be a self-similar Cantor measure on \(\mathbb R\) associated with a probability weight vector \(\mathbf p\), let \(K=\operatorname{supp}μ\), and let \(F\) denote the distribution function of \(μ\). We characterize the points \(x\in K\) at which the local scaling exponent \[ \lim_{\substack{y\to x, y\in K}} \frac{\log |F(y)-F(x)|}{\log |y-x|} \] exists and assumes a prescribed value. The characterization is formulated in terms of the convergence of the ratio between the accumulated logarithmic mass and geometric scales, together with the sublinear growth of the endpoint runs. Unlike the classical ternary case, our approach applies to arbitrary contraction ratios and probability weights. As an application, we construct a subset \(M\subset K\) of full Hausdorff measure on which the local scaling exponent of \(F\) is ${h(\mathbf q,\mathbf p)}/{χ(\mathbf q)}$ and establish the exact dimension-distortion formula \[ \dim_{\mathrm H} F(A) = \frac{χ(\mathbf q)}{h(\mathbf q,\mathbf p)} \dim_{\mathrm H} A \] for every \(A\subset M\), where \(\mathbf q\) is the natural probability vector, \(χ(\mathbf q)\) is the corresponding Lyapunov exponent, and \(h(\mathbf q,\mathbf p)\) is the cross-entropy of \(\mathbf q\) relative to \(\mathbf p\). A three-branch example illustrates the results.

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14 pages, 1 figure