ALMOST-ALL S-INTEGER DYNAMICAL-SYSTEMS HAVE MANY PERIODIC POINTS

We show that for almost every ergodic S-integer dynamical system the r adius of convergence of the dynamical zeta function is no larger than exp(-1/2 h(top)) < 1. In the arithmetic case almost every zeta functio n is irrational. We conjecture that for almost every ergodic S-integer dynamical system the radius of convergence of the zeta function is ex actly exp(-h(top)) < 1 and the zeta function is irrational. In an impo rtant geometric case (the S-integer systems corresponding to isometric extensions of the full p-shift or, more generally, linear algebraic c ellular automata on the full p-shift) we show that the conjecture hold s with the possible exception of at most two primes p. Finally, we exp licitly describe the structure of S-integer dynamical systems as isome tric extensions of (quasi-)hyperbolic dynamical systems.