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Regular version of the site

Article

Galton-Watson iterated function systems

Journal of Physics A: Mathematical and Theoretical. 2009. Vol. 42. No. 9. P. 1-17.
Decrouez G. G., Amblard P. O., Brossier J. M., Jones O. D.

 

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 1980s. Their adaptation to random signals was carried out by Hutchinson and Rüschendorff 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.