On the Lipschitz equivalence of self-affine sets

Abstract

Let $A$ be an expanding $d\times d$ matrix with integer entries and $\mathcal{D}\subset {\mathbb{Z}}^d$ be a finite digit set. Then the pair $(A,\mathcal{D})$ defines a unique integral self-affine set $K=A^{-1}(K+\mathcal{D})$. In this paper, by replacing the Euclidean norm with a pseudo-norm $w$ in terms of $A$, we construct a hyperbolic graph on $(A,\mathcal{D})$ and show that $K$ can be identified with the hyperbolic boundary. Moreover, if $(A,\mathcal{D})$ safisfies the open set condition, we also prove that two totally disconnected integral self-affine sets are Lipschitz equivalent if an only if they have the same $w$-Hausdorff dimension, that is, their digit sets have equal cardinality. We extends some well-known results in the self-similar sets to the self-affine sets.

arXiv comment16 pages, 3 figures