Falconer's formula for the Hausdorff dimension of a self-affine set in R²

Abstract

Let A 1 , A 2 ,…, A k be a finite set of contractive, affine, invertible self-mappings of R 2 . A compact subset Λ of R 2 is said to be self-affine with affinities A 1 , A 2 ,…, A k if It is known [8] that for any such set of contractive affine mappings there is a unique (compact) SA set with these affinities. When the affine mappings A 1 , A 2 ,…, A k are similarity transformations, the set Λ is said to be self-similar . Self-similar sets are well understood, at least when the images A i (Λ) have ‘small’ overlap: there is a simple and explicit formula for the Hausdorff and box dimensions [ 12, 10 ]; these are always equal; and the δ-dimensional Hausdorff measure of such a set (where δ is the Hausdorff dimension) is always positive and finite.