Volume 10 Issue 4
The McMillan Theorem for Colored Branching Processes and Dimensions of Random Fractals
Victor Bakhtin
1Department of Mathematics, IT and Landscape Architecture, John Paul II Catholic University of Lublin, Konstantynuv Str. 1H, 20-708 Lublin, Poland
2Department of Mechanics and Mathematics, Belarusian State University, Nezavisimosti Ave. 4, 220030 Minsk, Belarus
Abstract
For the simplest colored branching process, we prove an analog to the McMillan theorem and calculate the Hausdorff dimensions of random fractals defined in terms of the limit behavior of empirical measures generated by finite genetic lines. In this setting, the role of Shannon’s entropy is played by the Kullback–Leibler divergence, and the Hausdorff dimensions are computed by means of the so-called Billingsley–Kullback entropy, defined in the paper.
Keywords:colored branching process; random fractal; Hausdorff dimension; spectral potential; Kullback action; Billingsley–Kullback entropy; basin of a probability measure; maximal dimension principle