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

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.

Keywords

• hausdorff dimension
• basin of an ergodic measure
• entropy