In one-dimensional complex dynamics, the connectedness locus in a parameter space of polynomials or rational functions consists of those parameters for which the corresponding Julia set is connected. Complex dynamics is the study of dynamical systems for which the phase space is a complex manifold. ... Julia sets, described by Gaston Julia, are fractal shapes defined on the complex number plane. ...

Without doubt, the most famous connectedness locus is the Mandelbrot set, which arises from the family A rendering of the Mandelbrot set In mathematics, the Mandelbrot set is defined as the connectedness locus of the family of complex quadratic polynomials. ...

of quadratic polynomials. The connectedness loci of the higher-degree unicritical families,

(where ) are often called 'Multibrot sets'.

Connectedness loci are generally of less importance for families, such as the full parameter space of cubic polynomials, for which there is more than one free critical point. In these families, even maps with disconnected Julia sets may display nontrivial dynamics. In mathematics, a critical point (or critical number) is a point on the domain of a function where the derivative is equal to zero. ...