Approximation of maximal measures for countable topological Markov chains with meromorphic zeta function

Countable topological Markov chains (TMC) are considered. It is assumed that powers of the transition matrix of TMC have finite traces and so, for TMC, the dynamical (Artin-Mazur) zeta-function is well defined. It is also assumed that the following two conditions are fulfilled: 1) the raduis of convergence of TMC associated with submatrices with big indexes is bigger than $r(A)$, the the raduis of convergence of the initial countable transition matrix $A$, 2) zeta-function of the TMC is meromorphic in a disc of radius bigger than $r(A)$. Such conditions are satisfied, in particular, for countable TMC being symbolic models of one dimensional piecewise monotonic maps with positive topological entropy. In the paper it is shown that under these condition, an irreducible TMC has a unique measure with maximal entropy, which can be approximated (in the weak topology) by maximal measures of finite TMCs as subsistems of the initial one