The main goal is to use Gibbs measures in a markovian matrices context and in a more general context, to compute the Hausdorff dimension of subsets of [0, 1[ and [0, 1[². We introduce a parameter t which could be interpreted within thermodynamic framework as the variable conjugate to energy. In some particular cases we recover the Shannon-McMillan-Breiman and Eggleston theorems. Our proofs are deeply rooted in the properties of non-negative irreducible matrices and large deviations techniques as introduced by Ellis.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm88-2-4,
author = {L. Farhane and G. Michon},
title = {Gibbs measures in a markovian context and dimension},
journal = {Colloquium Mathematicae},
volume = {89},
year = {2001},
pages = {215-223},
zbl = {0984.60040},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm88-2-4}
}
L. Farhane; G. Michon. Gibbs measures in a markovian context and dimension. Colloquium Mathematicae, Tome 89 (2001) pp. 215-223. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm88-2-4/