Iterated Function Systems and the Local Dimension of Measures

Abstract

Given an iterated function system $\mathcal{S}$ in $\mathbb{R}^d$, with full support and such that the rotation in it fixes the hypercube $[−1/2, 1/2]^d$, then S satisfies the weak separation condition if and only if it satisfies the generalized finite-type condition. With this in mind, we extend the notion of net intervals from $\mathbb{R}$ to $\mathbb{R}^d$. We also use net intervals to calculate the local dimension of a self-similar measure with the finite-type condition and full support. We study the local dimension of the convolution of two measures. We give conditions for bounding the local dimension of the convolution on the basis of the local dimension of one of them. Moreover, we give a formula for the local dimension of some special points in the support of the convolution. We study the local dimension of the addition of two measures. We give an exact formula for the lower local dimension of the addition based on the local dimension of the two added measures. We give an upper bound to the upper local dimension of the addition of two measures. We explore the special case where the two measures satisfy the convex additive finite-type condition that we introduce. We introduce the notion of graph iterated function system. We show that we can always associate a self-similar measure to the graph iterated function system.