Dynamical Zeta Functions for Typical Extensions of Full Shifts

We consider a family of isometric extensions of the full shift onpsymbols (forpa prime) parametrized by a probability space. Using Heath-Brownʹs work on the Artin conjecture, it is shown that for all but two primespthe 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.