Gibbs measures in a markovian context and dimension
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 $[0, 1\mathclose [$ and $[0, 1\mathclose [^2$. 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.