On the Lipschitz equivalence of self-affine sets

Let $A\in M_d(\mathbb Z)$ be an expanding integer matrix 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 $K=A^{-1}(K+{\mathcal D})$. 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.