Journal of Fractal Geometry


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Volume 3, Issue 1, 2016, pp. 33–74
DOI: 10.4171/JFG/29

Published online: 2016-05-09

On the Hausdorff and packing measures of slices of dynamically defined sets

Ariel Rapaport[1]

(1) The Hebrew University of Jerusalem, Israel

Let $1 \le m < n$ be integers, and let $K \subset \mathbb{R}^{n}$ be a self-similar set satisfying the strong separation condition, and with dim $K = s > m$. We study the a.s. values of the $s-m$-dimensional Hausdorff and packing measures of $K \cap V$, where $V$ is a typical $n-m$-dimensional affine subspace.

For $0 <\rho < \frac{1}{2}$ let $C_{\rho} \subset[0,1]$ be the attractor of the IFS $\{f_{\rho,1},f_{\rho,2}\}$, where $f_{\rho,1}(t)=\rho\cdot t$ and $f_{\rho,2}(t)=\rho\cdot t+1-\rho$ for each $t \in \mathbb{R}$. We show that for certain numbers $0 < a,b < \frac{1}{2}$, for instance $a=\frac{1}{4}$ and $b=\frac{1}{3}$, if $K=C_{a} \times C_{b}$, then typically we have $\mathcal{H}^{s-m}(K\cap V)=0$.

Keywords: Self-similar set, self-affine set, Hausdorff measures, packing measures

Rapaport Ariel: On the Hausdorff and packing measures of slices of dynamically defined sets. J. Fractal Geom. 3 (2016), 33-74. doi: 10.4171/JFG/29