On the Hausdorff dimension of certain self-affine sets
Volume 152 / 2002
Abstract
A subset $E$ of ${\mathbb R}^n$ is called self-affine with respect to a collection $\{\phi _1 , \ldots ,\phi _t\}$ of affinities if $E$ is the union of the sets $\phi _1(E), \dots, \phi _t(E)$. For $S \subset {\mathbb R}^n$ let ${\mit\Phi} (S) = \bigcup _{1\leq j \leq t } \phi _j (S)$. If ${\mit\Phi} (S) \subset S$ let $E_{{\mit\Phi}}(S)$ denote $\bigcap _{k\geq 0}{\mit\Phi} ^k(S)$. For given ${\mit\Phi}$ consisting of contracting “pseudo-dilations” (affinities which preserve the directions of the coordinate axes) and subject to further mild technical restrictions we show that there exist self-affine sets $E_{{\mit\Phi}}(S)$ of each Hausdorff dimension between zero and a positive number depending on ${\mit\Phi}$. We also investigate in detail the special class of cases in ${\mathbb R }^2$, where the images of a fixed square under some maps $\phi _1 , \ldots , \phi _t$ are some vertical and some horizontal rectangles of sides $1$ and $2$. This investigation is made possible by an extension of the method of calculating Hausdorff dimension developed by P. Billingsley.
