Equilibrium states with incomplete supports and periodic trajectories

The relationship between analytic properties of the Artin–Mazur–Ruelle zeta function and the structure of the state of equilibrium states for a topological Markov chain is studied for a class of functions constant on a system of cylinder sets. The convergence of discrete invariant measures to equilibrium states is studied. Special attention is paid to the case in which the uniqueness condition is violated. Dynamic Ruelle–Smale zeta functions are considered, as well as the distribution laws for the number of periodic trajectories of special flows corresponding to topological Markov chains and to positive functions of this class.