Editing Mandelbrot set (section)

===Hyperbolic components===
Bulbs that are interior components of the Mandelbrot set in which the maps <math>f_c</math> have an attracting periodic cycle are called ''hyperbolic components''.<ref>{{cite thesis |last=Redona |first=Jeffrey Francis |title=The Mandelbrot set |year=1996 |type=Masters of Arts in Mathematics
|publisher=Theses Digitization Project |url=https://scholarworks.lib.csusb.edu/etd-project/1166}}</ref>

It is conjectured that these are the ''only'' [[Interior (topology)|interior regions]] of <math>M</math> and that they are [[dense set|dense]] in <math>M</math>. This problem, known as ''density of hyperbolicity'', is one of the most important open problems in [[complex dynamics]].<ref>{{cite arXiv|eprint=1709.09869 |author1=Anna Miriam Benini |title=A survey on MLC, Rigidity and related topics |year=2017 |class=math.DS }}</ref> Hypothetical non-hyperbolic components of the Mandelbrot set are often referred to as "queer" or ghost components.<ref>{{cite book |title=Exploring the Mandelbrot set. The Orsay Notes |first1=Adrien |last1=Douady |first2=John H. |last2=Hubbard |page=12 }}</ref><ref>{{cite thesis |first=Wolf |last=Jung |year=2002 |title=Homeomorphisms on Edges of the Mandelbrot Set |type=Doctoral thesis |publisher=[[RWTH Aachen University]] |id={{URN|nbn|de:hbz:82-opus-3719}} }}</ref> 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 [[Bifurcation diagram|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).
[[File:Centers8.png|thumb|Centers of 983 hyperbolic components of the Mandelbrot set.]]
Each of the hyperbolic components has a ''center'', which is a point ''c'' such that the inner Fatou domain for <math>f_c(z)</math> 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, <math>f_c^n(0) = 0</math> for some ''n''. If we call this polynomial <math>Q^{n}(c)</math> (letting it depend on ''c'' instead of ''z''), we have that <math>Q^{n+1}(c) = Q^{n}(c)^{2} + c</math> and that the degree of <math>Q^{n}(c)</math> is <math>2^{n-1}</math>. Therefore, constructing the centers of the hyperbolic components is possible by successively solving the equations <math>Q^{n}(c) = 0, n = 1, 2, 3, ...</math>.{{Citation needed|date=July 2023}} The number of new centers produced in each step is given by Sloane's {{oeis|A000740}}.