This question is quite open-ended, but I will formulate several sub-questions that I'll try to make precise. It is about finite-state dynamical system: start with a finite set $X$, with say $n$ elements, and consider an arbitrary map $f:X\to X$. This is a dynamical system: one can iterate $f$ and look at what happens.
Of course, it is in some sense trivial: there are a certain number of periodic orbits (including fixed points), and all other points are attracted to exactly one of these periodic orbits. But I am convinced that behind this triviality, there are interesting questions that may not have been considered much.
I know one instance of an interesting question that has been asked, by Misiurewicz: it is about discretizations of the logistic map. The same kind of question can be asked for other continuous-space dynamical systems, of course, but the logistic map seems to have the good amount of simplicity and complicated behavior to make it a reasonable first case to consider (I am not implying that the conjecture stated there should be easy!).
My first sub-question is reasonably precise:
Is there any case of finite-state dynamical systems which have been considered in the literature, other than numerical simulation of continuous-state dynamical systems?
Of course, theoretical results on such numerical simulations would also be of interest to me, though I am even more interested in knowing what one can say interesting in general about finite-state dynamical systems. My second sub-question is less focused.
Can one deduce global dynamical properties of a finite-state dynamical system (number of periodic orbits, length of periodic orbits, size of their basin of attraction) from local properties (let us call a property local if it is defined for subsets of $X$ of at most $k$ elements, and can be checked by iterating at most $k$ times the map $f$, with $k$ independent of $n$). Non-trivial inequalities would be of course very good.
As an intermediate case, we could look at semi-local properties where $k$ is allowed to grow with $n$, but very slowly.
Last, I would also be interested in the typical behavior of a random finite-state dynamical system:
What can be said about the dynamical quantities ((number of periodic orbits, length of periodic orbits, size of their basin of attraction) of a finite-state dynamical system on $n$ points, drawn uniformly among all maps?
Added in edit: a relevant paper appeared today : Random cyclic dynamical systems by Michał Adamaszek, Henry Adams, Francis Motta, where randomness is on a subset of the phase space (there, the circle) where a continuous dynamical system (there, a rotation) is to be approximated. An applications to computational topology is given.
Best Answer
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.