Editing Mandelbrot set (section)

==Basic properties==
The Mandelbrot set is a [[compact set]], since it is [[closed set|closed]] and contained in the [[closed disk]] of radius 2 centred on [[Origin (mathematics)|zero]]. A point <math>c</math> belongs to the Mandelbrot set if and only if <math>|z_n|\leq 2</math> for all <math>n\geq 0</math>. In other words, the [[absolute value]] of <math>z_n</math> must remain at or below 2 for <math>c</math> to be in the Mandelbrot set, <math>M</math>, and if that absolute value exceeds 2, the sequence will escape to infinity. Since <math>c=z_1</math>, it follows that <math>|c|\leq 2</math>, establishing that <math>c</math> will always be in the closed disk of radius 2 around the origin.<ref>{{cite web|url=https://mrob.com/pub/muency/escaperadius.html|title=Escape Radius|access-date=17 January 2024}}</ref>

[[File:Verhulst-Mandelbrot-Bifurcation.jpg|thumb|Correspondence between the Mandelbrot set and the [[bifurcation diagram]] of the [[Complex quadratic polynomial|quadratic map]]]]
[[File:Logistic Map Bifurcations Underneath Mandelbrot Set.gif|thumb|With <math>z_{n}</math> iterates plotted on the vertical axis, the Mandelbrot set can be seen to bifurcate at the period-2<sup>k</sup> components.]]
The [[intersection (set theory)|intersection]] of <math>M</math> with the real axis is the interval <math>\left[-2,\frac{1}{4}\right]</math>. The parameters along this interval can be put in [[one-to-one correspondence]] with those of the real [[logistic map|logistic family]],
:<math>x_{n+1} = r x_n(1-x_n),\quad r\in[1,4].</math>
The correspondence is given by

:<math>r = 1+\sqrt{1- 4 c},
\quad
c = \frac{r}{2}\left(1-\frac{r}{2}\right),
\quad
z_n = r\left(\frac{1}{2} - x_n\right).</math>

This gives a correspondence between the entire [[parameter space]] of the logistic family and that of the Mandelbrot set.<ref>{{Cite web |last=thatsmaths |date=2023-12-07 |title=The Logistic Map is hiding in the Mandelbrot Set |url=https://thatsmaths.com/2023/12/07/the-logistic-map-is-hiding-in-the-mandelbrot-set/ |access-date=2024-02-18 |website=ThatsMaths |language=en}}</ref>

Douady and Hubbard showed that the Mandelbrot set is [[connected space|connected]]. They constructed an explicit [[holomorphic function|conformal isomorphism]] between the complement of the Mandelbrot set and the complement of the [[closed unit disk]]. Mandelbrot had originally conjectured that the Mandelbrot set is [[Disconnected (topology)|disconnected]]. This conjecture was based on computer pictures generated by programs that are unable to detect the thin filaments connecting different parts of <math>M</math>. Upon further experiments, he revised his conjecture, deciding that <math>M</math> should be connected. A [[Topology|topological]] proof of the connectedness was discovered in 2001 by [[Jeremy Kahn]].<ref>{{cite web|url=http://www.math.brown.edu/~kahn/mconn.pdf|title=The Mandelbrot Set is Connected: a Topological Proof|last=Kahn|first=Jeremy|date=8 August 2001}}</ref>
[[File:Wakes near the period 1 continent in the Mandelbrot set.png|thumbnail|right|External rays of wakes near the period 1 continent in the Mandelbrot set]]
The dynamical formula for the [[uniformization theorem|uniformisation]] of the complement of the Mandelbrot set, arising from Douady and Hubbard's proof of the connectedness of <math>M</math>, gives rise to [[external ray]]s of the Mandelbrot set. These rays can be used to study the Mandelbrot set in combinatorial terms and form the backbone of the [[Jean-Christophe Yoccoz#Mathematical work|Yoccoz parapuzzle]].<ref>''The Mandelbrot set, theme and variations''. Tan, Lei. Cambridge University Press, 2000. {{isbn|978-0-521-77476-5}}. Section 2.1, "Yoccoz para-puzzles", [https://books.google.com/books?id=-a_DsYXquVkC&pg=PA121 p.&nbsp;121]</ref>

The [[boundary (topology)|boundary]] of the Mandelbrot set is the [[bifurcation locus]] of the family of quadratic polynomials. In other words, the boundary of the Mandelbrot set is the set of all parameters <math>c</math> for which the dynamics of the quadratic map <math>z_n=z_{n-1}^2+c</math> exhibits sensitive dependence on <math>c,</math> i.e. changes abruptly under arbitrarily small changes of <math>c.</math> It can be constructed as the limit set of a sequence of [[algebraic curves|plane algebraic curves]], the ''Mandelbrot curves'', of the general type known as [[polynomial lemniscate]]s. The Mandelbrot curves are defined by setting <math>p_0=z,\ p_{n+1}=p_n^2+z</math>, and then interpreting the set of points <math>|p_n(z)| = 2</math> in the complex plane as a curve in the real [[Cartesian coordinate system|Cartesian plane]] of degree <math>2^{n+1}</math>in ''x'' and ''y''.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Mandelbrot Set Lemniscate |url=https://mathworld.wolfram.com/MandelbrotSetLemniscate.html |access-date=2023-07-17 |website=Wolfram Mathworld |language=en}}</ref> Each curve <math>n > 0</math> is the mapping of an initial circle of radius 2 under <math>p_n</math>. These algebraic curves appear in images of the Mandelbrot set computed using the "escape time algorithm" mentioned below.