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

P. N. Varabei

Journal of the Belarusian State University. Mathematics and Informatics, Tome 3 (2017), pp. 11-18

Résumé

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.

Détail

Keywords: Hausdorff dimension; basin of an ergodic measure; entropy.