Hyperbolic component

In the quadratic polynomial family $f_c(z)=z^2+c$, a hyperbolic component is a connected component of the set of parameters for which $f_c$ has an attracting periodic point. The finite critical orbit is then attracted to that cycle. Its exact period is constant throughout the component. The component is open because an attracting periodic point and its strict multiplier inequality persist under small parameter changes.

The main cardioid has an attracting periodic point of period one; the period-two bulb has an attracting exact two-cycle. Their periodic-orbit multipliers give the explicit coordinates $c=\lambda /2-{\lambda }^2/4$ and $\lambda =4(c+1)$, respectively, with $|\lambda |<1$. More generally the attracting-cycle multiplier gives a conformal coordinate on each quadratic polynomial hyperbolic component.

This is a definition within the specified parameter family. For general rational maps, the existence of one attracting cycle alone does not require every critical orbit to approach an attracting cycle, so it does not by itself establish global hyperbolicity.^[ref]

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.