Cycle of a holomorphic map ($R(w_j)=w_{j+1}$)

An exact $n$-cycle consists of $n$ distinct points $w_0,\dots ,w_{n-1}$ with $R(w_j)=w_{j+1}$, indices read modulo $n$. Every point has exact period $n$. The return multiplier is ${\prod }_{j=0}^{n-1}{R}^{\prime }(w_j)$, 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.