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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 2C 1 has positive measure for almost all self-similar sets. We prove that there are open sets of self-similar sets such that C 2C1 has arbitrarily small measure.


This is the author’s version of a work that was accepted for publication in Boletim da Sociedade Brasileira de Matemática - Bulletin/Brazilian Mathematical Society. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication.

A definitive version was subsequently published in Boletim da Sociedade Brasileira de Matemática - Bulletin/Brazilian Mathematical Society and can be found online at:

* At the time of publication J. J. P. Veerman was affiliated with Federal University of Pernambuco



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