# On the Lipschitz equivalence of self-affine sets

## TL;DR

This paper investigates the Lipschitz equivalence of self-affine sets generated by expanding integer matrices and digit sets, establishing conditions under which these sets are Lipschitz equivalent based on their $w$-Hausdorff dimension.

## Contribution

It introduces a pseudo-norm based hyperbolic graph construction and extends known self-similar set results to self-affine sets under the open set condition.

## Key findings

- Self-affine sets can be identified with hyperbolic boundaries.
- Lipschitz equivalence is characterized by equal $w$-Hausdorff dimensions.
- Extends results from self-similar to 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.