Calculation of Hausdorff dimensions of basins of ergodic measures in encoding spaces
Keywords:
Hausdorff dimension, basin of an ergodic measure, entropyAbstract
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.
References
- Edgar G. Measure, Topology, and Fractal Geometry. New York : Springer, 2008.
- Falconer K. Fractal Geometry. Mathematical foundations and applications. Chichester : Wiley, 2003. DOI: 10.1002/0470013850.
- Pesin Y. [Dimension theory in dynamical systems. Contemporary views and applications]. Minsk : Institute of Computer Science, 2002 (in Russ.).
- Billingsley P. Hausdorff dimension in probability theory. I. Ill. J. Math. 1960. No. 4. P. 187–209.
- Billingsley P. Hausdorff dimension in probability theory. II. Ill. J. Math. 1961. No. 5. P. 291–298.
- Young L.-S. Dimension, entropy and Lyapunov exponents. Ergod. Theory Dyn. Syst. 1982. No. 2. P. 109 –124.
- Bakhtin V. I. The McMillan theorem for colored branching processes and dimensions of random fractals. Entropy. 2014. Vol. 16, No. 12. P. 6624 – 6653.
- Billingsley P. [Ergodic theory and information]. Moscow : Mir, 1969 (in Russ.).
- Brin M., Stuck G. Introduction to dynamical systems. Cambridge : Cambridge University Press, 2002.
- Algoet P. H., Cover T. M. A sandwich proof of the Shannon – McMillan – Breiman theorem. The Ann. Probab. 1988. Vol. 16, No. 2. P. 899 – 909.
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