On the Hausdorff dimension of fibres
The unit square is partitioned into four congruent subsquares, and an arbitrary one of them is deleted. This operation will be applied to all remaining squares ad infinitum. The authors prove that for the resulting limit set \(F\) its fibre sets \(F_ x=\{y\in[0,1]\mid(x,y)\in F\}\) satisfy \(\dim(F_ x)\geq 1/2\) for almost all \(x\in[0,1]\) with respect to Lebesgue measure (dim denotes Hausdorff dimension). They also discuss the Hausdorff dimension behaviour of exceptional fibres. Finally, the relations of Hausdorff and Minkowski dimension of \(F_ x\) are considered.
scientific article; zbMATH DE number 32240