Self-similar and self-affine sets; measure of the intersection of two copies

TL;DR

This paper investigates the measure of intersections between self-similar or self-affine sets and their transformations, establishing stability, measure-topology relations, and extension properties under various assumptions.

Contribution

It provides new theorems characterizing intersection measures, stability, and invariance properties of self-similar and self-affine sets under affine transformations.

Findings

Existence of a uniform contraction constant for intersections

Characterization of positive measure intersections via interior points

Extension of measures to invariant measures on R^d

Abstract

Let K be a self-similar or self-affine set in R^d, let \mu be a self-similar or self-affine measure on it, and let G be the group of affine maps, similitudes, isometries or translations of R^d. Under various assumptions (such as separation conditions or we assume that the transformations are small perturbations or that K is a so called Sierpinski sponge) we prove theorems of the following types, which are closely related to each other; Non-stability: There exists a constant c<1 such that for every g\in G we have either \mu(K\cap g(K)) <c \mu(K) or K\subset g(K). Measure and topology: For every g\in G we have \mu(K\cap g(K)) > 0 \iff int_K (K\cap g(K)) is nonempty (where int_K is interior relative to K). Extension: The measure \mu has a G-invariant extension to R^d. Moreover, in many situations we characterize those g's for which \mu(K\cap g(K) > 0, and we also get results about those $g$'s for which $g(K)\su K$ or $g(K)\supset K$ holds.