Chaotic behavior countable topological Markov chains with meromorphic zeta function

Сountable topological Markov chains (TMC) are considered. It is assumed that any power of the transition matrix of TMC has finite trace and thus, for TMC, the dynamical Artin-Mazur zeta function is well-defined. Furthermore, it is assumed that the following two conditions are satisfied: 1) the radius of convergence of zeta functions for subsystems of TMC corresponding to submatrices with sufficiently large indexes is greater than $r(A),$ the radius of convergence of zeta function of original TMC, and 2) zeta function is meromorphic in a disk of radius greater than $r(A).$ These conditions are natural because they take place for countable TMC which are the symbolic models of one-dimensional piecewise-monotone maps with positive topological entropy. We show that under these conditions, the transition matrix of irreducible TMC is $r(A)$-positive and, as a consequence, zeta function of TMC has simple poles on the circle $|z|=r(A)$ of the complex plane, and so, TMC has principal ergodic properties of finite TMC (in particular, there exists a unique measure of maximal entropy)