Institute for Mathematical Physics Almost All S{integer Dynamical Systems Have Many Periodic Points Almost All S{integer Dynamical Systems Have Many Periodic Points

We show that for almost every ergodic S{integer dynamical system the radius of convergence of the dynamical zeta function is no larger than exp(? 1 2 htop) < 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(?htop) < 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. 1. Introduction The S{integer dynamical systems were introduced in 3], and the question of typical behaviour for one family of these systems was considered in 12]. We rst describe them: a complete description with references and examples is in 3]. They are an arithmetically natural class of isometric extensions of familiar maps like toral endomorphisms or algebraic cellular automata. Let k be an A {{eld (that is, an algebraic number eld or a rational function eld with positive characteristic)