Chothi, Vijay; Everest, Graham; Ward, Thomas B. \(S\)-integer dynamical systems: periodic points. (English) Zbl 0879.58037 J. Reine Angew. Math. 489, 99-132 (1997). We associate via duality a dynamical system to each pair \((R_S,\xi)\), where \(R_S\) is the ring of \(S\)-integers in an \(A\)-field \(k\), and \(\xi\) is an element of \(R_S \backslash \{0\}\). These dynamical systems include the circle doubling map, certain solenoidal and toral endomorphisms, full one- and two-sided shifts on prime power alphabets, and certain algebraic cellular automata.In the arithmetic case, we show that for \(S\) finite the systems have properties close to hyperbolic systems: the growth rate of periodic points exists and the periodic points are uniformly distributed with respect to Haar measure. The dynamical zeta function is in general irrational however. For \(S\) infinite the systems exhibit a wide range of behaviour. Using Heath-Brown’s work on the Artin conjecture, we exhibit examples in which \(S\) is infinite but the upper growth rate of periodic points is positive. Reviewer: Thomas B. Ward (Norwich, U.K.) Cited in 2 ReviewsCited in 17 Documents MSC: 37B99 Topological dynamics 37D99 Dynamical systems with hyperbolic behavior 54H20 Topological dynamics (MSC2010) 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics 37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. Keywords:ring of \(S\)-integers; dynamical system; hyperbolic systems; periodic points; dynamical zeta function PDFBibTeX XMLCite \textit{V. Chothi} et al., J. Reine Angew. Math. 489, 99--132 (1997; Zbl 0879.58037) Full Text: Crelle EuDML 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.