Projections of self-similar sets with no separation condition

Abstract

We investigate how the Hausdorff dimension and measure of a self-similar set K ⊆ R d behave under linear images. This depends on the nature of the group T generated by the orthogonal parts of the defining maps of K. We show that if 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 T we establish that H t (L o O(K)) = H t (L(K)) for every element O of the closure of T, where L is a linear map and t = dim H K. We also prove that for disjoint subsets A and B of K we have that H t (L(A) ⋂ L(B)) = 0. Hochman and Shmerkin showed that if T is dense in SO(d,R) and the strong separation condition is satisfied, then dimH (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 Eroglu by obtaining that H t (g(K)) = 0.