Periodic point ($R^n(p)=p$)

A point $p$ is periodic for an iterated map $R$ if $R^n(p)=p$ for some positive integer $n$. Its exact period is the least such $n$, and the distinct points $p,R(p),\dots ,R^{n-1}(p)$ form its periodic orbit. For a holomorphic map its periodic-orbit multiplier is $(R^n{)}^{\prime }(p)$, computed in a local coordinate; at infinity use a reciprocal coordinate. The chain rule makes this the cyclic product of derivatives along the orbit, so it is the same at every point of that orbit.