Editing Mandelbrot set (section)

===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 Hubbard cites as his source a 1989 unpublished manuscript of Yoccoz.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.