In one-dimensional
complex dynamics
Complex dynamics, or holomorphic dynamics, is the study of dynamical systems obtained by Iterated function, iterating a complex analytic mapping. This article focuses on the case of algebraic dynamics, where a polynomial or rational function is it ...
, the connectedness
locus of a parameterized family of one-variable
holomorphic functions is a
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of the parameter space which consists of those parameters for which the corresponding
Julia set
In complex dynamics, the Julia set and the Classification of Fatou components, Fatou set are two complement set, complementary sets (Julia "laces" and Fatou "dusts") defined from a function (mathematics), function. Informally, the Fatou set of ...
is
connected.
Examples
Without doubt, the most famous connectedness locus is the
Mandelbrot set
The Mandelbrot set () is a two-dimensional set (mathematics), set that is defined in the complex plane as the complex numbers c for which the function f_c(z)=z^2+c does not Stability theory, diverge to infinity when Iteration, iterated starting ...
, which arises from the family of
complex quadratic polynomial
A complex quadratic polynomial is a quadratic polynomial whose coefficients and variable (mathematics), variable are complex numbers.
Properties
Quadratic polynomials have the following properties, regardless of the form:
*It is a unicritical pol ...
s :
:
The connectedness loci of the higher-degree unicritical families,
:
(where
) are often called '
Multibrot sets'.
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. For these families, even maps with disconnected Julia sets may display nontrivial dynamics. Hence here the connectedness locus is generally of less interest.
References
External links
*
Complex analysis
Fractals
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