Periodic points of denumerable topological Markov chains

This paper considers the analytic properties of the Artin–Mazur–Ruelle and Ruelle–Smale zeta functions for denumerable topological Markov chains (abbreviated to TMC) and locally constant functions. The convergence of discrete invariant measures is investigated. An analogue of Chebyshev's asymptotic law for the distribution of prime numbers for periodic trajectories of a special flow constructed with respect to a TMC and a positive locally constant function is obtained.