The Hausdorff dimension of self-affine fractals

Abstract

Abstract If T is a linear transformation on ℝ n with singular values α 1 ≥ α 2 ≥ … ≥ α n , the singular value function ø s is defined by where m is the smallest integer greater than or equal to s . Let T 1 , …, T k be contractive linear transformations on ℝ n . Let where the sum is over all finite sequences (i 1 , …, i r ) with 1 ≤ i j ≤ k. Then for almost all (a 1 , …, a k ) ∈ ℝ nk , the unique non-empty compact set F satisfying has Hausdorff dimension min{ d, n }. Moreover the ‘box counting’ dimension of F is almost surely equal to this number.