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.