Gibbs measures in a Markovian context and dimension (Q2717731)

The authors consider a finite Markov chain with \(b\) possible states, the transition matrix \((p_{ij})\), and a family \((M_t)= (p^t_{ij})\) (where \(t\) is an exponent) indexed by a real parameter \(t\), which may be interpreted within thermodynamic framework as the variable conjugates to energy. Let \(c(t)\) be the logarithm of the Perron-Frobenius eigenvalue of the matrix \(M_t\). The authors deal with the paths \((i_0, i_1,\dots)\) of the chain for which the limit of \(n^{-1}\log (p_{i_0i_1} p_{i_1i_2}\cdots p_{i_{n-2}i_{n-1}})\) is finite. The derivative \(c'\) establishes a one-to-one mapping between the set of real numbers and the set of possible limits.NEWLINENEWLINENEWLINEThe authors derive a family of Gibbs measures \(\gamma_t\) supported for each \(t\) by the set of chain's paths whose limit is \(c'(t)\). As an application, it is shown how to compute the Hausdorff dimension of some subsets of \([0,1]\) and \([0,1)^2\). In some particular cases the authors recover the Shannon-McMillan-Breiman and Eggleston theorems. The proofs rely on the properties of nonnegative irreducible matrices and large deviations techniques from \textit{R. S. Ellis} [``Entropy, large deviations and statistical mechanics'' (1985; Zbl 0566.60097)].