Geodesic
Calculation of hausdorff dimensions of basins of ergodic measures in encoding spaces
Journal of the Belarusian State University. Mathematics and Informatics, Tome 3 (2017), pp. 11-18.

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In the article we consider spaces $X^{\mathbb{N}}$ of sequences of elements of finite alphabet $X$ (encoding spaces) and ergodic measures on them, basins of ergodic measures and Hausdorff dimensions of such basins with respect to ultrametrics defined by a product of coefficients of unit interval $\theta(x), x\in X$. We call a basin of ergodic measure a set of points of the encoding space which define empiric measures by means of shift map, which limit (in a weak topology generated by continuous functions) is the ergodic measure. The methods of Billingsley and Young are used, which connects Hausdorff dimension and a pointwise dimension of some measure on the space, as well as Shannon – McMillan – Breiman theorem to obtain a lower bound of the dimension of a basin, and a partial analogue of McMillan theorem to obtain the upper bound. The goal of the article is to obtain a formula which can help us to calculate the Hausdorff dimension via entropy of the ergodic measure and a coefficient defined by the ultrametrics.
Keywords: Hausdorff dimension; basin of an ergodic measure; entropy. 
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P. N. Varabei. Calculation of hausdorff dimensions of basins of ergodic measures in encoding spaces. Journal of the Belarusian State University. Mathematics and Informatics, Tome 3 (2017), pp. 11-18. http://geodesic.mathdoc.fr/item/BGUMI_2017_3_a1/

[1] G. Edgar, “Measure, Topology, and Fractal Geometry”, New York: Springer, 2008 | MR

[2] K. Falconer, “Fractal Geometry. Mathematical foundations and applications”, Chichester: Wiley, 2003 | MR | Zbl

[3] Ya. Pesin, “Teoriya razmernosti i dinamicheskie sistemy”, Sovremennyi vzglyad i prilozheniya, 2002, Minsk : Institut kompyuternykh issledovanii

[4] P. Billingsley, “Hausdorff dimension in probability theory”, Ill. J. Math, 1960, 187–209 | MR | Zbl

[5] P. Billingsley, “Hausdorff dimension in probability theory”, Ill. J. Math, 1961, 291–298 | MR

[6] L. S. Young, “Dimension, entropy and Lyapunov exponents”, Ergod. Theory Dyn. Syst, 1982, 109–124 | DOI | MR | Zbl

[7] V. I. Bakhtin, “The McMillan theorem for colored branching processes and dimensions of random fractals”, Entropy, 16, # 12 (2014), 6624–6653 | DOI | MR | Zbl

[8] P. Billingsli, “Ergodicheskaya teoriya i informatsiya”, Mir, 1969 | MR

[9] M. Brin, G. Stuck, “Introduction to dynamical systems”, Cambridge: Cambridge University Press, 2002 | MR | Zbl

[10] P. H. Algoet, T. M. Cover, “A sandwich proof of the Shannon – McMillan – Breiman theorem”, The Ann. Probab, 16, # 2 (1988), 899–909 | MR | Zbl