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.
Categories: math.GT math.MG