|
Journal of the Belarusian State University. Mathematics and Informatics, 2017, Volume 3, Pages 11–18
(Mi bgumi140)
|
|
|
|
Real, Complex and Functional analysis
Calculation of hausdorff dimensions of basins of ergodic measures in encoding spaces
P. N. Varabei Belarusian State University, 4 Niezaliežnasci Аvenue, Minsk 220030, Belarus
Abstract:
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.
Citation:
P. N. Varabei, “Calculation of hausdorff dimensions of basins of ergodic measures in encoding spaces”, Journal of the Belarusian State University. Mathematics and Informatics, 3 (2017), 11–18
Linking options:
https://www.mathnet.ru/eng/bgumi140 https://www.mathnet.ru/eng/bgumi/v3/p11
|
Statistics & downloads: |
Abstract page: | 38 | Full-text PDF : | 12 | References: | 11 |
|