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

localif 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.