endofunctions

An endofunction on a set S is a function from S to itself. If S is finite with cardinality n, there are n^n endofunctions. Endofunctions on S are sometimes called endomaps of S; in category theory, an endomorphism is any morphism from an object to itself, but endofunctions emphasize the underlying set-theoretic case.

Examples include the identity function id_S, which fixes every element, and constant functions that map all elements to a fixed s0 in S. A permutation is also an endofunction, but most endofunctions are not invertible.

Functional graphs: Each endofunction f on S defines a directed graph on S in which every vertex has out-degree exactly 1. The graph decomposes into connected components, each consisting of a directed cycle with directed trees feeding into cycle vertices. A point x is fixed if f(x) = x, and more generally a point is periodic if f^k(x) = x for some k > 0.

Dynamics: Under iteration, the sequence x, f(x), f^2(x), … eventually becomes periodic on finite sets. The cycle lengths and the shape of the attached trees describe the dynamical behavior of f.

Algebraic viewpoint: The set End(S) of all endofunctions on S forms the transformation monoid T_S under composition, with identity id_S; elements need not be invertible, unlike permutations which form a subgroup of T_S.

When S is infinite, the number of endofunctions is |S|^|S|.