Ward, Thomas B. Almost all \(S\)-integer dynamical systems have many periodic points. (English) Zbl 0915.58081 Ergodic Theory Dyn. Syst. 18, No. 2, 471-486 (1998). Summary: We show that for almost every ergodic \(S\)-integer dynamical system the radius of convergence of the dynamical zeta function is no longer than \(\exp(-{1\over 2} h_{\text{top}})< 1\). In the arithmetic case almost every zeta function is irrational.We conjecture that for almost every ergodic \(S\)-integer dynamical system the radius of convergence of the zeta function is exactly \(\exp(-h_{\text{top}})< 1\) and the zeta function is irrational.In an important geometric case (the \(S\)-integer systems corresponding to isometric extensions of the full \(p\)-shift or, more generally, linear algebraic cellular automata on the full \(p\)-shift) we show that the conjecture holds with the possible exception of at most two primes \(p\).Finally, we explicitly describe the structure of \(S\)-integer dynamical systems as isometric extensions of (quasi-)hyperbolic dynamical systems. Cited in 1 ReviewCited in 5 Documents MSC: 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics 37D99 Dynamical systems with hyperbolic behavior 37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. Keywords:ergodic \(S\)-integer dynamical system; dynamical zeta function; radius of convergence; isometric extensions; (quasi-)hyperbolic dynamical systems PDFBibTeX XMLCite \textit{T. B. Ward}, Ergodic Theory Dyn. Syst. 18, No. 2, 471--486 (1998; Zbl 0915.58081) Full Text: DOI Online Encyclopedia of Integer Sequences: a(n) = 2^(n-2^ord_2(n)) (or 2^(n-A006519(n))). Number of orbits of length n in a map whose periodic points come from A059991.