Editing Mandelbrot set (section)

==Other properties==

===Main cardioid and period bulbs===
<!--[[Douady rabbit]] links directly here.-->[[File:Mandelbrot Set – Periodicities coloured.png|right|thumb|Periods of hyperbolic components]]

The ''main [[cardioid]]'' is the period 1 continent.<ref>{{Cite book |last1=Brucks |first1=Karen M. |url=https://books.google.com/books?id=p-amwZp0R-0C |title=Topics from One-Dimensional Dynamics |last2=Bruin |first2=Henk |date=2004-06-28 |publisher=Cambridge University Press |isbn=978-0-521-54766-6 |pages=264 |language=en}}</ref> It is the region of parameters <math>c</math> for which the map <math>f_c(z) = z^2 + c</math> has an [[Periodic points of complex quadratic mappings|attracting fixed point]].<ref>{{Cite book |last=Devaney |first=Robert |url=https://books.google.com/books?id=YEIPEAAAQBAJ |title=An Introduction To Chaotic Dynamical Systems |date=2018-03-09 |publisher=CRC Press |isbn=978-0-429-97085-6 |pages=147 |language=en}}</ref> It consists of all parameters of the form <math> c(\mu) := \frac\mu2\left(1-\frac\mu2\right)</math> for some <math>\mu</math> in the [[open unit disk]].<ref name=":5">{{Cite book |last1=Ivancevic |first1=Vladimir G. |url=https://books.google.com/books?id=mbtCAAAAQBAJ |title=High-Dimensional Chaotic and Attractor Systems: A Comprehensive Introduction |last2=Ivancevic |first2=Tijana T. |date=2007-02-06 |publisher=Springer Science & Business Media |isbn=978-1-4020-5456-3 |pages=492–493 |language=en}}</ref>{{close paraphrasing inline|date=March 2025}}

To the left of the main cardioid, attached to it at the point <math>c=-3/4</math>, a circular bulb, the ''period-2 bulb'' is visible.<ref name=":5" />{{close paraphrasing inline|date=March 2025}} The bulb consists of <math>c</math> for which <math>f_c</math> has an [[Periodic points of complex quadratic mappings|attracting cycle of period 2]]. It is the filled circle of radius 1/4 centered around −1.<ref name=":5" />{{close paraphrasing inline|date=March 2025}}

[[File:Animated cycle.gif|left|thumb|Attracting cycle in 2/5-bulb plotted over [[Julia set]] (animation)]]
More generally, for every positive integer <math>q>2</math>, there are <math>\phi(q)</math> circular bulbs tangent to the main cardioid called ''period-q bulbs'' (where <math>\phi</math> denotes the [[Euler's totient function|Euler phi function]]), which consist of parameters <math>c</math> for which <math>f_c</math> has an attracting cycle of period <math>q</math>.{{Citation needed|date=March 2025}} More specifically, for each primitive <math>q</math>th root of unity <math>r=e^{2\pi i\frac{p}{q}}</math> (where <math>0<\frac{p}{q}<1</math>), there is one period-q bulb called the <math>\frac{p}{q}</math> bulb, which is tangent to the main cardioid at the parameter <math> c_{\frac{p}{q}} := c(r) = \frac{r}2\left(1-\frac{r}2\right),</math> and which contains parameters with <math>q</math>-cycles having combinatorial rotation number <math>\frac{p}{q}</math>.<ref>{{Cite book |last1=Devaney |first1=Robert L. |url=https://books.google.com/books?id=4XrHCQAAQBAJ |title=Complex Dynamical Systems: The Mathematics Behind the Mandelbrot and Julia Sets: The Mathematics Behind the Mandelbrot and Julia Sets |last2=Branner |first2=Bodil |date=1994 |publisher=American Mathematical Soc. |isbn=978-0-8218-0290-8 |pages=18–19 |language=en}}</ref> More precisely, the <math>q</math> periodic [[Classification of Fatou components|Fatou components]] containing the attracting cycle all touch at a common point (commonly called the ''<math>\alpha</math>-fixed point''). If we label these components <math>U_0,\dots,U_{q-1}</math> in counterclockwise orientation, then <math>f_c</math> maps the component <math>U_j</math> to the component <math>U_{j+p\,(\operatorname{mod} q)}</math>.<ref name=":5" />{{close paraphrasing inline|date=March 2025}}

[[File:Juliacycles1.png|right|thumb|Attracting cycles and [[Julia set]]s for parameters in the 1/2, 3/7, 2/5, 1/3, 1/4, and 1/5 bulbs]]

The change of behavior occurring at <math>c_{\frac{p}{q}}</math> is known as a [[bifurcation theory|bifurcation]]: the attracting fixed point "collides" with a repelling period-''q'' cycle. As we pass through the bifurcation parameter into the <math>\tfrac{p}{q}</math>-bulb, the attracting fixed point turns into a repelling fixed point (the <math>\alpha</math>-fixed point), and the period-''q'' cycle becomes attracting.<ref name=":5" />{{close paraphrasing inline|date=March 2025}}

===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}}.

===Local connectivity===
It is conjectured that the Mandelbrot set is [[locally connected]]. This conjecture is known as ''MLC'' (for ''Mandelbrot locally connected''). By the work of [[Adrien Douady]] and [[John H. Hubbard]], this conjecture would result in a simple abstract "pinched disk" model of the Mandelbrot set. In particular, it would imply the important ''hyperbolicity conjecture'' mentioned above.{{Citation needed|date=July 2023}}

The work of [[Jean-Christophe Yoccoz]] established local connectivity of the Mandelbrot set at all finitely [[Renormalization|renormalizable]] parameters; that is, roughly speaking those contained only in finitely many small Mandelbrot copies.<ref name="yoccoz">{{cite book
 | last = Hubbard | first = J. H.
 | contribution = Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz
 | contribution-url = https://pi.math.cornell.edu/~hubbard/Yoccoz.pdf
 | location = Houston, TX
 | mr = 1215974
 | pages = 467–511
 | publisher = Publish or Perish
 | title = Topological methods in modern mathematics (Stony Brook, NY, 1991)
 | year = 1993}}. Hubbard cites as his source a 1989 unpublished manuscript of Yoccoz.</ref> Since then, local connectivity has been proved at many other points of <math>M</math>, but the full conjecture is still open.

===Self-similarity===
[[File:Self-Similarity-Zoom.gif|right|thumb|[[Self-similarity]] in the Mandelbrot set shown by zooming in on a round feature while panning in the negative-''x'' direction. The display center pans left from the fifth to the seventh round feature (−1.4002,&nbsp;0) to (−1.4011,&nbsp;0) while the view magnifies by a factor of 21.78 to approximate the square of the [[Feigenbaum constants|Feigenbaum ratio]].]]
The Mandelbrot set is [[self-similar]] under magnification in the neighborhoods of the [[Misiurewicz point]]s. It is also conjectured to be self-similar around generalized [[Feigenbaum point]]s (e.g., −1.401155 or −0.1528&nbsp;+&nbsp;1.0397''i''), in the sense of converging to a limit set.<ref>{{cite journal | last1 = Lei | year = 1990 | title = Similarity between the Mandelbrot set and Julia Sets | url = http://projecteuclid.org/euclid.cmp/1104201823| journal = Communications in Mathematical Physics | volume = 134 | issue = 3| pages = 587–617 | doi=10.1007/bf02098448| bibcode = 1990CMaPh.134..587L| s2cid = 122439436 }}</ref><ref>{{cite book |author=J. Milnor |chapter=Self-Similarity and Hairiness in the Mandelbrot Set |editor=M. C. Tangora |location=New York |pages=211–257 |title=Computers in Geometry and Topology |url=https://books.google.com/books?id=wuVJAQAAIAAJ |year=1989|publisher=Taylor & Francis|isbn=9780824780319 }})</ref> The Mandelbrot set in general is quasi-self-similar, as small slightly different versions of itself can be found at arbitrarily small scales. These copies of the Mandelbrot set are all slightly different, mostly because of the thin threads connecting them to the main body of the set.<ref>{{Cite web |title=Mandelbrot Viewer |url=https://math.hws.edu/eck/js/mandelbrot/MB.html |access-date=2025-03-01 |website=math.hws.edu}}</ref>

===Further results===
The [[Hausdorff dimension]] of the [[boundary (topology)|boundary]] of the Mandelbrot set equals 2 as determined by a result of [[Mitsuhiro Shishikura]].<ref name="shishikura"/> The fact that this is greater by a whole integer than its topological dimension, which is 1, reflects the extreme [[fractal]] nature of the Mandelbrot set boundary. Roughly speaking, Shishikura's result states that the Mandelbrot set boundary is so "wiggly" that it locally [[space-filling curve|fills space]] as efficiently as a two-dimensional planar region. Curves with Hausdorff dimension 2, despite being (topologically) 1-dimensional, are oftentimes capable of having nonzero area (more formally, a nonzero planar [[Lebesgue measure]]). Whether this is the case for the Mandelbrot set boundary is an unsolved problem.{{Citation needed|date=July 2023}}

It has been shown that the generalized Mandelbrot set in higher-dimensional hypercomplex number spaces (i.e. when the power <math>\alpha</math> of the iterated variable <math>z</math> tends to infinity) is convergent to the unit (<math>\alpha</math>−1)-sphere.<ref>{{cite journal|last1=Katunin|first1=Andrzej|last2=Fedio|first2=Kamil|title=On a Visualization of the Convergence of the Boundary of Generalized Mandelbrot Set to (n-1)-Sphere|url=https://reader.digitarium.pcss.pl/Content/295117/JAMCM_2015_1_6-Katunin_Fedio.pdf|access-date=18 May 2022|date=2015|journal=Journal of Applied Mathematics and Computational Mechanics|volume=14|issue=1|pages=63–69|doi=10.17512/jamcm.2015.1.06}}</ref>

In the [[Blum–Shub–Smale machine|Blum–Shub–Smale]] model of [[real computation]], the Mandelbrot set is not computable, but its complement is [[Recursively enumerable set|computably enumerable]]. Many simple objects (e.g., the graph of exponentiation) are also not computable in the BSS model. At present, it is unknown whether the Mandelbrot set is computable in models of real computation based on [[computable analysis]], which correspond more closely to the intuitive notion of "plotting the set by a computer". Hertling has shown that the Mandelbrot set is computable in this model if the hyperbolicity conjecture is true.{{Citation needed|date=July 2023}}

===Relationship with Julia sets===
[[File:Julia Mandelbrot Relationship.png|thumb|A mosaic made by matching Julia sets to their values of c on the complex plane. The Mandelbrot set is a map of connected Julia sets.]]
As a consequence of the definition of the Mandelbrot set, there is a close correspondence between the [[geometry]] of the Mandelbrot set at a given point and the structure of the corresponding [[Julia set]]. For instance, a value of c belongs to the Mandelbrot set if and only if the corresponding Julia set is connected. Thus, the Mandelbrot set may be seen as a map of the connected Julia sets.<ref>{{cite web |last=Sims |first=Karl |title=Understanding Julia and Mandelbrot Sets |url=https://www.karlsims.com/julia.html |website=karlsims.com |access-date=January 27, 2025}}</ref>{{Better source needed|date=January 2025}}

This principle is exploited in virtually all deep results on the Mandelbrot set. For example, Shishikura proved that, for a dense set of parameters in the boundary of the Mandelbrot set, the Julia set has [[Hausdorff dimension]] two, and then transfers this information to the parameter plane.<ref name="shishikura">{{cite journal
 | last = Shishikura | first = Mitsuhiro
 | arxiv = math.DS/9201282
 | doi = 10.2307/121009
 | issue = 2
 | journal = Annals of Mathematics
 | mr = 1626737
 | pages = 225–267
 | series = Second Series
 | title = The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets
 | volume = 147
 | year = 1998| jstor = 121009
 | s2cid = 14847943
 }}.</ref> Similarly, Yoccoz first proved the local connectivity of Julia sets, before establishing it for the Mandelbrot set at the corresponding parameters.<ref name="yoccoz"/>