Dynamical zeta functions for typical extensions of full shifts

We consider a family of isometric extensions of the full shift on p symbols (for p a prime) parametrized by a probability space. Using Heath-Brown's w ork on the Artin conjecture, it is shown that for all but two primes p the set of limit points of the growth rate of periodic paints is infinite almos t surely. This shows in particular that the dynamical zeta function is not algebraic almost surely. (C) 1999 Academic Press.