The McMillan Theorem for Colored Branching Processes and Dimensions of Random Fractals

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...

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Main Author:	Victor Bakhtin
Format:	Article
Language:	English
Published:	MDPI AG 2014-12-01
Series:	Entropy
Subjects:	colored branching process random fractal Hausdorff dimension spectral potential Kullback action Billingsley–Kullback entropy basin of a probability measure maximal dimension principle
Online Access:	http://www.mdpi.com/1099-4300/16/12/6624

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spelling	doaj-b4fdb876a0ec48f3b0ba728bece1dc4f2020-11-24T21:03:53ZengMDPI AGEntropy1099-43002014-12-0116126624665310.3390/e16126624e16126624The McMillan Theorem for Colored Branching Processes and Dimensions of Random FractalsVictor Bakhtin0Department of Mathematics, IT and Landscape Architecture, John Paul II Catholic University of Lublin, Konstantynuv Str. 1H, 20-708 Lublin, PolandFor 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.http://www.mdpi.com/1099-4300/16/12/6624colored branching processrandom fractalHausdorff dimensionspectral potentialKullback actionBillingsley–Kullback entropybasin of a probability measuremaximal dimension principle
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Victor Bakhtin
spellingShingle	Victor Bakhtin The McMillan Theorem for Colored Branching Processes and Dimensions of Random Fractals Entropy colored branching process random fractal Hausdorff dimension spectral potential Kullback action Billingsley–Kullback entropy basin of a probability measure maximal dimension principle
author_facet	Victor Bakhtin
author_sort	Victor Bakhtin
title	The McMillan Theorem for Colored Branching Processes and Dimensions of Random Fractals
title_short	The McMillan Theorem for Colored Branching Processes and Dimensions of Random Fractals
title_full	The McMillan Theorem for Colored Branching Processes and Dimensions of Random Fractals
title_fullStr	The McMillan Theorem for Colored Branching Processes and Dimensions of Random Fractals
title_full_unstemmed	The McMillan Theorem for Colored Branching Processes and Dimensions of Random Fractals
title_sort	mcmillan theorem for colored branching processes and dimensions of random fractals
publisher	MDPI AG
series	Entropy
issn	1099-4300
publishDate	2014-12-01
description	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.
topic	colored branching process random fractal Hausdorff dimension spectral potential Kullback action Billingsley–Kullback entropy basin of a probability measure maximal dimension principle
url	http://www.mdpi.com/1099-4300/16/12/6624
work_keys_str_mv	AT victorbakhtin themcmillantheoremforcoloredbranchingprocessesanddimensionsofrandomfractals AT victorbakhtin mcmillantheoremforcoloredbranchingprocessesanddimensionsofrandomfractals
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The McMillan Theorem for Colored Branching Processes and Dimensions of Random Fractals

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