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) parametrised by a probability space. Using Heath-Brown's work 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 points is infinite almost surely. This shows in particular that the dynamical zeta function is not algebraic almost surely.