Galton-Watson Iterated Function Systems

Iterated Function Systems (IFS) are interesting parametric models for generating fractal sets and functions. The general idea is to compress, deform and translate a given set or function with a collection of operators and to iterate the procedure. Under weak assumptions, IFS possess a unique fixed point which is in general fractal. IFS were introduced in a deterministic context, then were generalized to the random setting on abstract spaces in the early 80’s. Their adaptation to random signals was carried out by Hutchinson and Rüschendorff [9] by considering random operators. This study extends their model with not only random operators but also a random underlying construction tree. We show that the corresponding IFS converges under certain hypothesis to a unique fractal fixed point. Properties of the fixed point are also described. Submitted to: J. Phys. A: Math. Theor. PACS numbers: PACS-key : 02.50.-r Probability theory, stochastic processes, and statistics, 05.45.Df Fractals Galton-Watson Iterated Function Systems 2 Figure 1. Shadow of a shape drawn in the sand of a Queensland beach at dusk. Credit G. Decrouez.