The Hausdorff dimension of self-affine fractals

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.