Assouad dimension and local structure of self-similar sets with overlaps in $\mathbb{R}^d$

TL;DR

This paper characterizes the Assouad dimension of self-similar sets with overlaps in , linking it to the dimensions of overlapping directions and projections, and extends results to graph directed sets.

Contribution

It provides a new formula for the Assouad dimension of overlapping self-similar sets, connecting it to the structure of overlaps and projections, and answers an open question.

Findings

Assouad dimension equals the sum of the dimension of overlapping directions and the projection dimension.

Conditions are given for the Assouad dimension to exceed 2.

The results extend to graph directed self-similar sets.

Abstract

For a self-similar set in $\mathbb{R}^d$ that is the attractor of an iterated function system that does not verify the weak separation property, Fraser, Henderson, Olson and Robinson showed that its Assouad dimension is at least $1$. In this paper, it is shown that the Assouad dimension of such a set is the sum of the dimension of the vector space spanned by the set of $\textit{overlapping directions}$ and the Assouad dimension of the orthogonal projection of the set self-similar set onto the orthogonal complement of that vector space. This result is applied to give sufficient conditions on the orthogonal parts of the similarities so that the self-similar set has Assouad dimension bigger than $2$, and also to answer a question posed by Farkas and Fraser. The result is also extended to the context of graph directed self-similar sets. The proof of the result relies on finding an appropriate weak tangent to the set. This tangent is used to describe partially the topological structure of self-similar sets which are both attractors of an iterated function system not satisfying the weak separation property and of an iterated functions system satisfying the open set condition.