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((Unreferenced|date=December 2009))

In one-dimensional [[complex dynamics]], the '''connectedness [[locus (mathematics)|locus]]''' of a parameterized family of one-variable [[Holomorphic function|holomorphic functions]] is a [[subset]] of the parameter space which consists of those parameters for which the corresponding [[Julia set]] is [[Connected space|connected]].

==Examples==
Without doubt, the most famous connectedness locus is the [[Mandelbrot set]], which arises from the family of [[complex quadratic polynomial]]s : 
:<math>f_c(z) = z^2+c\,</math>

The connectedness loci of the higher-degree unicritical families,
:<math> z\mapsto z^d+c\,</math>
(where <math>d\geq 3\,</math>) are often called '[[Multibrot set]]s'. 

For these families, the [[bifurcation locus]] is the boundary of the connectedness locus. This is no longer true in settings, such as the full parameter space of cubic polynomials, where there is more than one free [[critical point (mathematics)|critical point]]. For these families, even maps with disconnected Julia sets may display nontrivial dynamics. Hence here the connectedness locus is generally of less interest.

==External links==
*((cite journal |last1=Epstein |first1=Adam |last2=Yampolsky |first2=Michael |title=Geography of the cubic connectedness locus: Intertwining surgery |journal=Annales Scientifiques de l'École Normale Supérieure |date=March 1999 |volume=32 |issue=2 |pages=151–185 |doi=10.1016/S0012-9593(99)80013-5 |arxiv=math/9608213 |s2cid=18035406 ))

[[Category:Complex analysis]]
[[Category:Fractals]]
((DEFAULTSORT:Connectedness Locus))


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