= Uniform period criterion for pointwise periodic maps
{title2=$f^N=\operatorname{id}\iff\text{all cycle lengths divide }N$}
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 $1,\ldots,M$ 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>.
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