Editing Mandelbrot set (section)

===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"/>