Iterated Function Systems and Control Languages

Valuations — morphisms from (Σ ; ;e) to ((0;∞); ;1) — are a generalization of Bernoulli morphisms introduced in [10]. Here, we show how to generalize the notion of entropy (of a language) in order to obtain new formulae to determine the Hausdorff dimension of fractal sets (also in Euclidean spaces) especially defined via regular (ω-)languages. By doing this, we can sharpen and generalize earlier results [1, 14, 15, 26, 38] in two ways: firstly, we treat the case where the underlying basic iterated function system contains non-contractive mappings, and secondly, we obtain results valid for non-regular languages as well. A preliminary version appeared in: “Mathematical Foundations of Computer Science” (L. Brim, J. Gruska and J. Zlatuska, eds.), Lecture Notes in Comput. Sci. No. 1450, Springer–Verlag, Berlin 1998, pp. 740 – 750. †email: [email protected] ‡email: [email protected] 2 H. Fernau and L. Staiger