Orbit growth of Dyck and Motzkin shifts via Artin-Mazur zeta function

For a discrete dynamical system, the prime orbit and Mertens' orbit counting functions indicate the growth of the closed orbits in the system in a certain way. These functions are analogous to the counting functions for primes in number theory. In this paper, we prove the asymptotic behaviours of the counting functions for certain types of shift spaces, which are called Dyck and Motzkin shifts. This is done via a generating function for the number of periodic points, which is called Artin-Mazur zeta function. The proof relies on the properties of the meromorphic extension for their Artin-Mazur zeta functions, specifically on the analiticity and non-vanishing property of the extension.