### Chaotic behavior countable topological Markov chains with meromorphic zeta function

#### M. I. Malkin^{1}

Annotation | 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). |
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Keywords | topological Markov chains, zeta function, topological entropy |

^{1}Associate Professor of Department of differential equations and mathematical Analysis, Nizhny Novgorod State University, Nizhny Novgorod; malkin@unn.ru.

**Citation**: M. I. Malkin, "[Chaotic behavior countable topological Markov chains with meromorphic zeta function]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,16:4 (2014) 41–49 (In Russian)