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

Abstract

Let A1, A2,…,Ak be a finite set of contractive, affine, invertible self-mappings of R2. A compact subset Λ of R2 is said to be self-affine with affinitiesA1, A2,…,Ak ifIt 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 A1, A2,…,Ak are similarity transformations, the set Λ is said to be self-similar. Self-similar sets are well understood, at least when the images Ai(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.

A Class of Self-Affine Sets and Self-Affine Measures

“…Falconer, in a celebrated article [4], gave a variational formula for the Hausdorff and box dimensions for "almost all" self-affine sets under some assumptions. Later, Hueter and Lalley [12], and Solomyak [27] proved that Falconer's formula remains true under some weaker conditions. We focus on the L q spectrum and the Hausdorff and entropy dimensions of a selfaffine measure in this article.…”

Estimating singularity dimension

“…This could be deduced indirectly from equation (41) of [3]. However, it can also be seen easily directly by a simple geometric argument.…”

“…Next, we recall the result of Heuter-Lalley. Theorem 1.8 (Heuter-Lalley [3]). Under Hypotheses 1.7 we have that…”

A new class of exceptional self-affine fractals

“…Various authors have determined some conditions on F for the equality in (2) to hold; see [3][4][5][6][7]. However, very few classes of self-affine sets are known to be exceptional.…”