Pointwise periodic self-map ($\forall x\exists n_x\ge 1:f^{n_x}(x)=x$)

A pointwise periodic self-map returns each point to itself after a positive number of iterates which may depend on the point. It is a bijection: each finite cycle supplies predecessors; equality of two images can be undone by an iterate with an exponent one below a common multiple of the two periods. The set consequently decomposes into disjoint finite permutation cycles. Pointwise periodicity does not imply that one iterate is the identity on an infinite set.

A pointwise periodic self-map has a uniform positive period exactly when its finite cycle lengths are bounded. A uniform period is divisible by every length; conversely, bounded lengths divide the least common multiple of $1,\dots ,M$ for some bound $M$. Arbitrarily many bounded-length cycles are allowed. Every such map on a finite set has a uniform period. On any infinite set containing a countably infinite subset, cycles of unbounded finite lengths and fixed points elsewhere give a counterexample. Thus, with the usual axiom of choice assumption, the sets on which every pointwise periodic self-map has a uniform period are exactly finite sets.