Uniform perfectness of self-affine sets

Abstract

Let $f_i(x)=A_ix+b_i\ (1\le i\le n)$ be affine maps of Euclidean space $\mathbb {R}^N$ with each $A_i$ nonsingular and each $f_i$ contractive. We prove that the self-affine set $K$ of $\{f_1,\dots , f_n\}$ is uniformly perfect if it is not a singleton.