On the boundary of attractors with non-void interior

Lau, Ka-Sing; Xu, You

doi:10.1090/S0002-9939-99-05303-4

On the boundary of attractors with non-void interior
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by Ka-Sing Lau and You Xu PDF

Proc. Amer. Math. Soc. 128 (2000), 1761-1768 Request permission

Abstract:

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.

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