linguistic:paraphrasing, linguistic:translation
Hyperbolic components

Bulbs that are interior components of the Mandelbrot set in which the maps f c {\displaystyle f_{c}} have an attracting periodic cycle are called hyperbolic components.

It is conjectured that these are the only interior regions of M {\displaystyle M} and that they are dense in M {\displaystyle M}. This problem, known as density of hyperbolicity, is one of the most important open problems in complex dynamics. Hypothetical non-hyperbolic components of the Mandelbrot set are often referred to as "queer" or ghost components. For real quadratic polynomials, this question was proved in the 1990s independently by Lyubich and by Graczyk and Świątek. (Note that hyperbolic components intersecting the real axis correspond exactly to periodic windows in the Feigenbaum diagram. So this result states that such windows exist near every parameter in the diagram.)

Not every hyperbolic component can be reached by a sequence of direct bifurcations from the main cardioid of the Mandelbrot set. Such a component can be reached by a sequence of direct bifurcations from the main cardioid of a little Mandelbrot copy (see below).

Each of the hyperbolic components has a center, which is a point c such that the inner Fatou domain for f c (z) {\displaystyle f_{c}(z)} has a super-attracting cycle—that is, that the attraction is infinite. This means that the cycle contains the critical point 0, so that 0 is iterated back to itself after some iterations. Therefore, f c n (0) = 0 {\displaystyle f_{c}^{n}(0)=0} for some n. If we call this polynomial Q n (c) {\displaystyle Q^{n}(c)} (letting it depend on c instead of z), we have that Q n + 1 (c) = Q n (c) 2 + c {\displaystyle Q^{n+1}(c)=Q^{n}(c)^{2}+c} and that the degree of Q n (c) {\displaystyle Q^{n}(c)} is 2 n − 1 {\displaystyle 2^{n-1}}. Therefore, constructing the centers of the hyperbolic components is possible by successively solving the equations Q n (c) = 0, n = 1, 2, 3,... {\displaystyle Q^{n}(c)=0,n=1,2,3,...}. The number of new centers produced in each step is given by Sloane's OEIS: A000740.