On dimensions of visible parts of self-similar sets with finite rotation groups

Abstract

We derive an upper bound for the Assouad dimension of visible parts of self-similar sets generated by iterated function systems with finite rotation groups and satisfying the open set condition. The bound is valid for all visible parts and it depends on the penetrable part of the set, which is a concept defined in this paper. As a corollary, we obtain in the planar case that if the projection is a finite or countable union of intervals then the visible part is 1-dimensional. We also prove that the Assouad dimension of a visible part is strictly smaller than the Hausdorff dimension of the set provided the projection contains interior points.

Changes to the previous version: Theorems 2.4 and 2.11 are proved assuming only the weak separation condition. Theorem 2.11 is proved using a simplified argument