Local Scaling and Dimension Distortion of Generalized Cantor Functions

Abstract

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.