Iterated Function Systems and Control Languages

Valuations—morphisms from (Σ*, ·, e) to ((0, ∞), ·, 1)—are a generalization of Bernoulli morphisms introduced by Eilenberg [“Automata, Languages, and Machines”, Academic Press, New York, 1974]. Here, we show how to generalize the notion of entropy (of a language) in order to obtain new formulas 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 in two ways: first, we treat the case where the underlying basic iterated function system contains noncontractive mappings and, second, we obtain results valid for nonregular languages as well.

Authors

• Henning Fernau
• Ludwig Staiger

Keywords

No keywords are indexed for this paper.