Invariants of the entropy type of non-transitive countable topological Markov chains

Countable topological Markov chains (TMC) ((non-transitive, in general) are considered. It is assumed that any power of the transition matrix of TMC has a finite trace, so the dynamical Artin-Mazur zeta function of TMC is well defined. It is also assumed that the radius of convergence for zeta-function of subsystems of TMC which correspond to submatrices with indexes sufficiently big is less than the radius $r(A)$ for zeta fuction of the initial TMC with matrix $A$. These conditions are natural because they are satisfied for countable TMC being symbolic models of one-dimensional piecewise monotone maps with positive topological entropy. We show in the paper that under these condition, several invariants of the entropy type (such as the radius of convergence $r(A)$, logarithm of the entropy $-\log h_{top}(A)$ and some others) actually coincide.