2013/07/10 by Farkas, Ábel
#28A78 #28A80 #37C45 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1307.2841
We investigate how the Hausdorff dimension and measure of a self-similar set K⊆ℝd 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 Ht(L∘ O(K))=Ht(L(K)) for every element O of the closure of T, where L is a linear map and t=dimHK. We also prove that for disjoint subsets A and B of K we have that Ht(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\dimHK,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\breveglu by obtaining that Ht(g(K))=0.