JournalsjfgVol. 3 , No. 1pp. 33–74

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

  • Ariel Rapaport

    The Hebrew University of Jerusalem, Israel
On the Hausdorff and packing measures of slices of dynamically defined sets cover
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Abstract

Let 1m<n1 \le m < n be integers, and let KRnK \subset \mathbb{R}^{n} be a self-similar set satisfying the strong separation condition, and with dim K=s>mK = s > m. We study the a.s. values of the sms-m-dimensional Hausdorff and packing measures of KVK \cap V, where VV is a typical nmn-m-dimensional affine subspace.

For 0<ρ<120 <\rho < \frac{1}{2} let Cρ[0,1]C_{\rho} \subset[0,1] be the attractor of the IFS {fρ,1,fρ,2}\{f_{\rho,1},f_{\rho,2}\}, where fρ,1(t)=ρtf_{\rho,1}(t)=\rho\cdot t and fρ,2(t)=ρt+1ρf_{\rho,2}(t)=\rho\cdot t+1-\rho for each tRt \in \mathbb{R}. We show that for certain numbers 0<a,b<120 < a,b < \frac{1}{2}, for instance a=14a=\frac{1}{4} and b=13b=\frac{1}{3}, if K=Ca×CbK=C_{a} \times C_{b}, then typically we have Hsm(KV)=0\mathcal{H}^{s-m}(K\cap V)=0.

Cite this article

Ariel Rapaport, On the Hausdorff and packing measures of slices of dynamically defined sets. J. Fractal Geom. 3 (2016), no. 1 pp. 33–74

DOI 10.4171/JFG/29