On the Hausdorff dimension of certain self-affine sets

A. G. Abercrombie¹ ; R. Nair¹

¹ Mathematical Sciences University of Liverpool P.O. Box 147 Liverpool L69 3BX, U.K.

Studia Mathematica, Tome 152 (2002) no. 2, pp. 105-124

Résumé

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.

Détail

DOI : 10.4064/sm152-2-1

Keywords: subset mathbb called self affine respect collection phi ldots phi affinities union sets phi dots phi subset mathbb mit phi bigcup leq leq phi mit phi subset mit phi denote bigcap geq mit phi given mit phi consisting contracting pseudo dilations affinities which preserve directions coordinate axes subject further mild technical restrictions there exist self affine sets mit phi each hausdorff dimension between zero positive number depending mit phi investigate detail special class cases mathbb where images fixed square under maps phi ldots phi vertical horizontal rectangles sides investigation made possible extension method calculating hausdorff dimension developed nbsp billingsley