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For simplest colored branching processes we prove an analog to the McMillantheorem and calculate Hausdorff dimensions of random fractals defined in termsof the limit behavior of empirical measures generated by finite genetic lines.In this setting the role of Shannon's entropy is played by theKullback--Leibler divergence and the Hausdorff dimensions are computed by meansof the so-called Billingsley--Kullback entropy, defined in the paper.

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