Ward, Thomas B. Dynamical zeta functions for typical extensions of full shifts. (English) Zbl 0956.37016 Finite Fields Appl. 5, No. 3, 232-239 (1999). The author considers the growth rate of the number of periodic points in a class of hyperbolic dynamical systems, which generalizes the shift on \(p\) symbols (\(p\) is a prime) in an arithmetical setting. It is shown that with respect to a probabilistic parametrization of valuation-theoretic significance, for all but two primes \(p\), the set of limit points of the growth rate is almost surely infinite, and the dynamical zeta function almost surely non-algebraic. The exclusion of two primes results from a connection with Artin’s conjecture on primitive roots. The minimalist presentation requires familiarity with the author’s previous work. Reviewer: Franco Vivaldi (London) Cited in 3 Documents MSC: 37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. 37D99 Dynamical systems with hyperbolic behavior 11R45 Density theorems 37B99 Topological dynamics Keywords:dynamical zeta functions; hyperbolic dynamical systems; Artin’s conjecture on primitive roots PDFBibTeX XMLCite \textit{T. B. Ward}, Finite Fields Appl. 5, No. 3, 232--239 (1999; Zbl 0956.37016) Full Text: DOI arXiv Link