Fractals, Dimension, and Formal Languages

We consider classes of sets of r-adic expansions of reals speciied by means of the theory of formal languages or automata theory. It is shown how these speciications are used to calculate the Hausdorr dimension and Hausdorr measure of such sets. Since the appearence of Mandelbrot's 11 book \Fractals, Form, Chance and Dimen-sion" Fractal Geometry as a means providing a theory describing many of the seemingly complex patterns in nature and the sciences has become popular not only in the sciences (cf. Peitgen and Saupe 13), but also in computer science. Here Barnsley's 3 "Computa-tional Fractal Geometry" aims at a practical description of fractal patterns by so-called Iterative Function Systems (IFS). Besides IFS several other computational methods for the description (generation) of fractal images involving concepts of Automata or Formal Language Theory have been developed (see e.g. Berstel and Morcrette 4 , Berstel and Nait-Abdallah 5 , Culik and Dube 6; 7 , Prusinkiewicz et al. 14 , or Smith 15). These papers, however, deal mainly with the generation of fractal images by nite computational devices. On the other hand it is known from geometric measure theory (cf. Falconer 8) that the analysis, that is, the estimation of the fractal dimension or measure of even rather simply deenable sets is already a complicated task.