ИСТИНА

Thermodynamic formalism is a domain of the theory of dynamical systems related to some ideas of statistical physics. A starting material for developing thermodynamic formalism is a map T of some measurable space X and a measurable real function f on X. In the case under consideration, T is a Markov shift in the space X of the double-side infinite sequences x on an infinite alpphabet V, and f(x) jnly depends on two coordinates of x. One can describe thermodynamic formalism of countable state symbolic Markov chains either in terms of infinite matrices with nonnegative entries or in terms of infinite loaded graphs. Here the latter way is used. A loaded graph (LG) is the pair (G,W), where G=(V,E) is a directed graph with vertex set V=V(G) and edge set E=E(G), while W is a positive function on E. There are four classes of LGs: stable positive, unstable positive, null-recurrent, and transient. In the talk, a motivation for such a terminology was presented and alternative characterization of some classes of LGs was given. For stable and unstable positive LGs, there exists a translation invariant probability measure on the space of infinite paths of G that maximizes the difference between the Kolmogorov--Sinai entropy of the shift transformation and the mean energy determined by W. The points of maximum of this functional are said to be equilibrium measures (or equilibrium states). In our situation there is only one equilibrium measure. The problem on asymptotic behavior of the equilibrium measures corresponding to an increasing sequence of finite subgraphs of G is rather old. It is solved for all classes of LGs with one exception: when the LG of interest is unstable positive.