Parabolic cycle

A parabolic cycle is a periodic orbit whose multiplier is a root of unity; an iterate then has multiplier one and attracting petals.

A hyperbolic component in a parameter space consists of maps with a specified attracting cycle. Its multiplier gives a holomorphic coordinate on the component in the quadratic family, and root-of-unity boundary multipliers give parabolic parameters.

For every prime $p$, the polynomial $f_c^p(0)$ has a nonzero root, giving a superattracting cycle of exact period $p$. Moving in its hyperbolic component to a boundary point with cycle multiplier $-1$ gives a parabolic cycle of exact period $p$. Thus the set of parabolic periods in the quadratic family is infinite.