= Cycle of a holomorphic map
{title2=$R(w_j)=w_{j+1}$}
= Cycles of a holomorphic map
{synonym}
= Holomorphic cycle
{synonym}
An exact $n$-cycle consists of $n$ distinct points $w_0,\ldots,w_{n-1}$ with $R(w_j)=w_{j+1}$, indices read modulo $n$. Every point has exact period $n$. The return multiplier is $\prod_{j=0}^{n-1}R'(w_j)$, computed in local charts if necessary; cyclically changing the starting point leaves it unchanged. Thus an attracting, superattracting or repelling return is of the same type at every point of the cycle. If a period-$n$ root coincides with a lower-period point at a parameter limit, multiplicity in the equation $R^n(z)=z$ need not represent distinct points of an exact cycle.
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