Projections of self-similar sets with no separation condition

TL;DR

This paper studies how the Hausdorff dimension and measure of self-similar sets change under linear and smooth maps without separation conditions, revealing conditions for dimension preservation and measure zero results.

Contribution

It extends known results on self-similar sets by removing separation conditions and analyzing the effects of the orthogonal parts of the defining maps.

Findings

Linear images of self-similar sets with finite orthogonal group are graph directed attractors.

Dimension drops can occur under certain projections.

Under dense orthogonal groups, the Hausdorff measure of images can be zero.

Abstract

We investigate how the Hausdorff dimension and measure of a self-similar set $K\subseteq\mathbb{R}^{d}$ behave under linear images. This depends on the nature of the group $\mathcal{T}$ generated by the orthogonal parts of the defining maps of $K$. We show that if $\mathcal{T}$ is finite then every linear image of $K$ is a graph directed attractor and there exists at least one projection of $K$ such that the dimension drops under the image of the projection. In general, with no restrictions on $\mathcal{T}$ we establish that $\mathcal{H}^{t}(L\circ O(K))=\mathcal{H}^{t}(L(K))$ for every element $O$ of the closure of $\mathcal{T}$, where $L$ is a linear map and $t=\dim_{H}K$. We also prove that for disjoint subsets $A$ and \textbf{$B$} of $K$ we have that $\mathcal{H}^{t}(L(A)\cap L(B))=0$. Hochman and Shmerkin showed that if $\mathcal{T}$ is dense in $SO(d,\mathbb{R})$ and the strong separation condition is satisfied then $\dim_{H}(g(K))=\min\{\dim_{H}K,l\} $ where $g$ is a continuously differentiable map of rank $l$. We deduce the same result without any separation condition and we generalize a result of Ero$\breve{\mathrm{g}}$lu by obtaining that $\mathcal{H}^{t}(g(K))=0$.