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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]

Let $p$ be prime and defineThe recursion $P_{n+1}=P_n^2+c$ shows that $P_p$ has degree $2^{p-1}$. Moreover $P_p(c)=c+O(c^2)$ at zero, so $c=0$ is a simple root. Since the degree is greater than one, $P_p$ has a nonzero root $c_0$. At $c_0$, the critical point zero is periodic with period dividing $p$. It is not fixed because $c_0\ne 0$, so primality makes its exact period $p$. Thus $c_0$ is the center of a hyperbolic component of exact period $p$.

The multiplier map on this component covers the unit disc. Move to its boundary along parameters whose attracting-cycle multiplier tends to $-1$. Compactness of the Mandelbrot set gives a limiting parameter $c_{\ast }$. The periodic cycle persists with exact period $p$: at multiplier $-1$, every point is a simple root of $f_{c_{\ast }}^p(z)-z$, so no collision to a lower-period orbit occurs. Its multiplier is the root of unity $-1$, and hence it is a parabolic cycle after squaring the return map. We have produced a parabolic cycle of exact period $p$ for every prime $p$. Therefore the parabolic periods in the quadratic family form an infinite set.