Calculation of Hausdorff dimensions of basins of ergodic measures in encoding spaces

  • Pavel N. Varabei Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus

Abstract

In the article we consider spaces XN 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 θ(x), x ∈ 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. 

Author Biography

Pavel N. Varabei, Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus

postgraduate student at the department of nonlinear analysis and analytical economics, faculty of mechanics and mathematics

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Published
2018-01-25
Keywords: Hausdorff dimension, basin of an ergodic measure, entropy
Supporting Agencies The author would like to thank professor V. I. Bakhtin for the formulation of the problem, constructive criticism and productive discussion of the manuscript
How to Cite
Varabei, P. N. (2018). Calculation of Hausdorff dimensions of basins of ergodic measures in encoding spaces. Journal of the Belarusian State University. Mathematics and Informatics, 3, 11-18. Retrieved from https://journals.bsu.by/index.php/mathematics/article/view/753
Issue
No 3 (2017): Journal of the Belarusian State University. Mathematics and Informatics
Section
Real, Complex and Functional Analysis

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