On the boundary of attractors with non-void interior

Let $\left\{ f_i\right\} _{i=1}^N$ be a family of $N$ contractive mappings on $\mathbb{R}^{d\text{ }}$ such that the attractor $K$ has nonvoid interior. We show that if the $f_i$'s are injective, have non-vanishing Jacobian on $K$, and $f_i\left( K\right) \cap f_j\left( K\right)$ have zero Lebesgue measure for $i\neq j,$ then the boundary $\partial K$ of $K$ has measure zero. In addition if the $f_i$'s are affine maps, then the conclusion can be strengthened to $\dim _H\left( \partial K\right) <d$. These improve a result of Lagarias and Wang on self-affine tiles.