Hausdorff dimension of the arithmetic sum of self-similar sets
In his famous paper [Indiana Univ. Math. J. 30, 713--747 (1981; Zbl 0598.28011)] \textit{J. E. Hutchinson} proved that for each iterated function system (IFS) composed by self contractions on a complete metric space, there exists a unique invariant nonempty compact set, called the attractor. This is, generally, a fractal set. If, further, the contractions are similitudes, the attractor is a self-set and Hutchinson showed how its Hausdorff dimension can be obtained. In the present paper, the author considers the arithmetic sum of two self-similar sets generated by two iterated function systems composed by similitudes and he proves that this set is either a self-similar set or a unique attractor of some infinite iterated function system (Theorem 2.15). Moreover, under some conditions, the author evaluates the Hausdorff dimension of that sum.
scientific article; zbMATH DE number 6595322
scientific article