Editing Mandelbrot set (section)

==Formal definition==
The Mandelbrot set is the [[uncountable set]] of values of ''c'' in the [[complex plane]] for which the [[Orbit (dynamics)|orbit]] of the [[Complex quadratic polynomial#Critical point|critical point]] <math display="inline">z = 0</math> under [[Iterated function|iteration]] of the [[quadratic map]]

:<math>z \mapsto z^2 + c</math> <ref>{{Cite web |last=Weisstein |first=Eric W. |title=Mandelbrot Set |url=https://mathworld.wolfram.com/ |access-date=2024-01-24 |website=mathworld.wolfram.com |language=en}}</ref>

remains [[Bounded sequence|bounded]].<ref>{{cite web|url=http://math.bu.edu/DYSYS/explorer/def.html|title=Mandelbrot Set Explorer: Mathematical Glossary|access-date=7 October 2007}}</ref> Thus, a [[complex number]] ''c'' is a member of the Mandelbrot set if, when starting with <math>z_0 = 0</math> and applying the iteration repeatedly, the [[absolute value]] of <math>z_n</math> remains bounded for all <math>n > 0</math>.

For example, for ''c'' = 1, the [[sequence]] is 0, 1, 2, 5, 26, ..., which tends to [[infinity]], so 1 is not an element of the Mandelbrot set. On the other hand, for <math>c=-1</math>, the sequence is 0, −1, 0, −1, 0, ..., which is bounded, so −1 does belong to the set.

The Mandelbrot set can also be defined as the [[connectedness locus]] of the family of [[Quadratic equation|quadratic]] [[polynomial]]s <math>f(z) = z^2 + c</math>, the subset of the space of parameters <math>c</math> for which the [[Julia set]] of the corresponding polynomial forms a [[connected set]].<ref>{{Citation |last=Tiozzo |first=Giulio |title=Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set |date=2013-05-15 |arxiv=1305.3542 }}</ref> In the same way, the [[boundary (topology)|boundary]] of the Mandelbrot set can be defined as the [[bifurcation locus]] of this quadratic family, the subset of parameters near which the dynamic behavior of the polynomial (when it is [[Iterated function|iterated]] repeatedly) changes drastically.