TITLE: Finite-space dynamical systems
QUESTION [13 upvotes]: 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.

REPLY [3 votes]: As Vidit Nanda pointed out in the third answer, cellular automata (on lattices) or generalized cellular automata (GCAs, or graph dynamical systems) provide a large and interesting class of finite dynamical systems such that the global behavior emerges from iterating locally defined transition maps. 
I have an interesting GCA model for studying synchronization of coupled oscillators, such as synchronous blinking of fireflies. You can find examples and some global theorems in my recent paper: 
Lyu, Hanbaek. "Synchronization of finite-state pulse-coupled oscillators." Physica D: Nonlinear Phenomena 303 (2015): 28-38. (preprint http://arxiv.org/abs/1407.1103)
To give you a quick taste, the 4-color model on a simple graph $G=(V,E)$ is defined as follows: 
1) A 4-configuration is mapping $X:V\rightarrow \mathbb{Z}_{4}$
2) Transition map $\tau_{G}:T\mapsto T'$ is defined by
\begin{equation}          
        \tau_{G}(X)(v) = 
\begin{cases}
    X(v) & \text{if $X(v)\in \{2,3\}$ and $\exists$ a neighbor $u$ of $v$ with $X(u)=1$}\\
X(v)+1 & \text{otherwise}
\end{cases}
\end{equation}
In words, each vertex is an oscillator with 4 states, and at each iteration, every blinking vertex(i.e., of state 1) inhibits its hasty neighbors (i.e., of state 2 and 3) for one iteration. Similar model can be defined for any period $n$ with blinking state $b(n):=\lfloor \frac{n-1}{2}\rfloor$. 
As you have mentioned, every orbit must converge to exactly one periodic orbit. This may seem so trivial that there is nothing left to study, but it is not. There is a one special periodic orbit, called synchrony, where all vertices eventually have the same state. Then the question is the following: what are the conditions for the underlying graph $G$ and the initial configuration $X$ that guarantee the orbit eventually converges to synchrony? Each vertex tries to synchronize with their neighbors by a weak coupling, but yet it is highly unclear if this aggregation of local effort would really lead to global synchronization. I am currently studying this model on $\mathbb{Z}^{2}$. 
A related model is the Cyclic Cellular Automata(CCA) invented by David Griffeath. There are some rigorous results on clustering of 3-color CCA on $\mathbb{Z}$ by R. Fisch, where the author uses correspondence between the induced edge particle system and simple random walks to show clustering on a fixed finite interval with high probability. CCA on $\mathbb{Z}^{2}$  generally shows very complicated spiraling behavior, which is relevant to autowaves on excitable medium.