We associate via duality a dynamical system to each pair (R_S,x) where R_S is the ring of S-integers in an A-field and x is a non-zero element of R_S. The maps produced form uncountable families of isometric extensions of familiar maps like the circle doubling map, toral endomorphisms in companion form, full shifts and algebraic cellular automata. Almost all of the examples (in the sense of probability) have a positive upper growth rate of periodic points and an irrational dynamical zeta function.
Early work in this area by Chothi, Everest, and Ward at the University of East Anglia is available at the url http://www.mth.uea.ac.uk/maths/maths-preprints-pure.html under ``Ergodic Theory''.