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.

中文翻译：

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Dyck 和 Motzkin 的轨道增长通过 Artin-Mazur Zeta 函数移动

对于离散动力系统，主轨道和默滕斯轨道计数函数以某种方式指示系统中闭合轨道的增长。这些函数类似于数论中素数的计数函数。在本文中，我们证明了计数函数对于某些类型的移位空间的渐近行为，称为 Dyck 和 Motzkin 移位。这是通过周期点数量的生成函数完成的，该函数称为 Artin-Mazur zeta 函数。该证明依赖于其 Artin-Mazur zeta 函数的亚纯扩展的性质，特别是扩展的分析性和非零性。