The McMillan theorem for colored branching processes and dimensions of random fractals

TL;DR

This paper extends the McMillan theorem to colored branching processes, linking entropy concepts to fractal dimensions, and provides formulas for calculating the Hausdorff dimensions of associated random fractals.

Contribution

It introduces an analog of the McMillan theorem for colored branching processes and computes Hausdorff dimensions using a new entropy measure based on Kullback-Leibler divergence.

Findings

Established the McMillan theorem analog for colored branching processes

Derived formulas for Hausdorff dimensions of random fractals

Connected entropy measures with fractal dimension calculations

Abstract

For simplest colored branching processes we prove an analog to the McMillan theorem and calculate 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.