Intersecting self-similar Cantor sets

Abstract

We define a self-similar set as the (unique) invariant set of an iterated function system of certain contracting affine functions. A topology on them is obtained (essentially) by inducing theC 1-topology of the function space. We prove that the measure function is upper semi-continuous and give examples of discontinuities. We also show that the dimension is not upper semicontinuous. We exhibit a class of examples of self-similar sets of positive measure containing an open set. IfC 1 andC 2 are two self-similar setsC 1 andC 2 such that the sum of their dimensionsd(C 1)+d(C 2) is greater than one, it is known that the measure of the intersection setC 2−C 1 has positive measure for almost all self-similar sets. We prove that there are open sets of self-similar sets such thatC 2−C 1 has arbitrarily small measure.