Substitutions, branching processes and fractal sets

Abstract

Substitutions and branching processes provide analytical methods to describe many deterministic, respectively random sets. Fractal curves generated by substitutions are introduced, followed by a discussion of the determination of their Hausdorff dimension. We give a necessary and sufficient variant of the “open set condition”, and an application of this condition in the quasi-lattice case. Dynamical systems generated by substitutions are defined, and it is shown how this leads to an interpretation of many fractal curves as realizations of generalized random walks. We then define random substitutions, and mention their relation to branching processes. Finally we discuss a particular random fractal set known as Mandelbrot percolation.