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

TL;DR

This paper establishes an upper bound on the Assouad dimension of visible parts of self-similar sets with finite rotation groups, linking geometric projections to fractal dimensions and introducing the concept of penetrable parts.

Contribution

It introduces the concept of penetrable parts and provides new bounds on the Assouad dimension for visible parts of self-similar sets under specific conditions.

Findings

Visible parts have Assouad dimension bounded by the penetrable part.

In the planar case, visible parts are 1-dimensional if projections are unions of intervals.

Assouad dimension of visible parts is less than Hausdorff dimension if the projection has interior points.

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.