In algebraic geometry, in the sixties, Weil proposed to look at the action of the Frobenius automorphism on a variety over a finite field, and apply a Lefschetz formula in some suitable cohomology, in order to study the property of the zeta function associated to the variety (the so called Weil conjectures). The program was carried by Grothendieck and Deligne who were each awarded with a Fields medal for their work.

So reduction mod p of algebraic systems gives you interesting examples of "finite" dynamical systems, and the Frobenius automorphism appears only after reduction. I am sure that there are many people on MO that can provide more details, and perhaps discuss more recent developments, on that subject.

There is of course a strong parallel with dynamical zeta functions and their rationality in the field of hyperbolic dynamical systems.

EDIT: I guess that reduction mod p is also a motivation for studying p-adic dynamics. I think there was some attempt to show that the Mandlebrot set is locally connected by iterating the quadratic map the over the p-adic numbers.

In algebraic geometry, in the sixties, Weil proposed to look at the action of the Frobenius automorphism on a variety over a finite field, and apply a Lefschetz formula in some suitable cohomology, in order to study the property of the zeta function associated to the variety (the so called Weil conjectures). The program was carried by Grothendieck and Deligne who were each awarded with a Fields medal for their work.

So reduction mod p of algebraic systems gives you interesting examples of "finite" dynamical systems, and the Frobenius automorphism appears only after reduction. I am sure that there are many people on MO that can provide more details, and perhaps discuss more recent developments, on that subject.

There is of course a strong parallel with dynamical zeta functions and their rationality in the field of hyperbolic dynamical systems.

EDIT: I guess that reduction mod p is also a motivation for studying p-adic dynamics. I think there was some attempt to show that the Mandlebrot set is locally connected by iterating the quadratic map over the p-adic numbers.

In algebraic geometry, in the sixties, it was proposed to look at the action of the Frobenius automorphism on a variety over a finite field, and apply a Lefschetz formula in some suitable cohomology, in order to study the property of the zeta function associated to the variety. The program was carried by Grothendieck and Deligne who were each awarded with a Fields medal for their work.

So reduction mod p of algebraic systems gives you interesting examples of "finite" dynamical systems, and the Frobenius automorphism appears only after reduction. I am sure that there are many people on MO that can provide more details, and perhaps discuss more recent developments, on that subject.

There is of course a strong parallel with dynamical zeta functions and their rationality in the field of hyperbolic dynamical systems.

In algebraic geometry, in the sixties, Weil proposed to look at the action of the Frobenius automorphism on a variety over a finite field, and apply a Lefschetz formula in some suitable cohomology, in order to study the property of the zeta function associated to the variety (the so called Weil conjectures). The program was carried by Grothendieck and Deligne who were each awarded with a Fields medal for their work.

So reduction mod p of algebraic systems gives you interesting examples of "finite" dynamical systems, and the Frobenius automorphism appears only after reduction. I am sure that there are many people on MO that can provide more details, and perhaps discuss more recent developments, on that subject.

There is of course a strong parallel with dynamical zeta functions and their rationality in the field of hyperbolic dynamical systems.

EDIT: I guess that reduction mod p is also a motivation for studying p-adic dynamics. I think there was some attempt to show that the Mandlebrot set is locally connected by iterating the quadratic map the over the p-adic numbers.