Ax–Grothendieck and the Garden of Eden It's an obvious consequence of the pigeonhole principle that any injective function over finite sets is bijective. But there are some similar results in different areas of mathematics that apply to less-finite settings.
In algebraic geometry, the Ax–Grothendieck theorem states (if I have it correctly) that any injection from an algebraic variety over an algebraically closed field to itself is bijective; the standard proof involves some sort of local-global principle together with the same fact over finite fields.
In the theory of cellular automata, the Garden of Eden theorem states that any injective cellular automaton (over an integer grid of some fixed finite dimension, say) is bijective; the standard proof involves again the same fact for finite sets of cells together with a limiting argument that shows that for large enough bounded regions of an unbounded grid, the boundary of the region has negligible effect compared to the interior.
Is there some way of viewing these three injective-bijective statements (or others) as instances of a single more general phenomenon?
 A: Another example is a monorphisms (i.e. injective)  between finite dimentional vectorial spaces . I think taht  a  general key is the general concept of algebraic dependent (see matroid theory), free system and bases, this is a "generalization " of the usual vector spaces (linear algebra)  concept, for example you can generalize this "dependence theory" for  analyze the trascendence degree for field extension  .
If this "general theory of dependence" is applicable,  then a injective morphism maps a base (in the domain) to a free system (in the codomain) and a free system of the some cardinality of a base (all bases have the some cardinality, and we can call it the  dimention)   is a base, then a base generate the space and the map is surjective too.  
A: In the theory of von Neumann algebras, there is a similar phenomenon.
Let $M$ be a type $II_1$-factor. That is, it's an infinite dimensional von Neumann algebra with center $\mathbb C$, and with an (everywhere defined) trace $tr:M\to \mathbb C$.
Then there is a complete invariant of $M$-modules called the von Neumann dimension.
This invariant takes values in $\mathbb R_{\ge 0}\cup \{\infty\}$ and can take any value in that set. It has the property that any isometric map $H_1 \to H_2$ between modules of the same dimension is actually a unitary isomorphism (except if the von Neumann dimension is $\infty$, in which case, that's not true).
In particular, if $H$ is an $M$-module of finite von Neumann dimension, then any isometry (not assumed to be surjective) is actually a unitary isomorphism.
A: See the recent paper "On algebraic cellular automata", for a proof of how to derive the Garden of Eden theorem from the Ax-Grothendieck theorem. This is indeed in the spirit of Gromov's paper that Mohan mentioned in the comments. This paper is were he introduced Sofic groups. The story as I understand it is: The Ax-Grothendieck theorem tells us that every regular algebraic map from a complex algebraic variety to itself can not be a strict embedding. If one takes a power of the variety indexed by a group G, then the analogous result holds for G-equivariant pro-regular maps when the group G satisfies some conditions (can be approximated by "nice" groups). Gromov calls this property "surjunctive". The original GOE theorem is about $G=\mathbb{Z}^n$. It turns out that this works for any amenable group, and more generally for sofic groups, those groups that can be approximated by amenable groups.
