Topology, Cantor sets, Difference sets, Measure theory
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 the C 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. If C 1 and C 2 are two self-similar sets C 1 and C 2 such that the sum of their dimensions d(C 1)+d(C 2) is greater than one, it is known that the measure of the intersection set C 2−C 1 has positive measure for almost all self-similar sets. We prove that there are open sets of self-similar sets such that C 2−C1 has arbitrarily small measure.
Veerman, J. J. P., "Intersecting Self-Similar Cantor Sets" (1995). Mathematics and Statistics Faculty Publications and Presentations. 146.