Projections of self-similar sets with no separation condition

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 ${\mathcal{H}}^t(g(K))=0$.