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Quadratic polynomials

A very popular complex dynamical system is given by the family of complex quadratic polynomials, a special case of rational maps. Such quadratic polynomials can be expressed as

where c is a complex parameter. Fix some R > 0 {\displaystyle R>0} large enough that R 2 − R ≥ | c |. {\displaystyle R^{2}-R\geq |c|.} (For example, if c is in the Mandelbrot set, then | c | ≤ 2, {\displaystyle |c|\leq 2,} so we may simply let R = 2. {\displaystyle R=2~.}) Then the filled Julia set for this system is the subset of the complex plane given by

where f c n (z) {\displaystyle f_{c}^{n}(z)} is the n th iterate of f c (z). {\displaystyle f_{c}(z).} The Julia set J (f c) {\displaystyle J(f_{c})} of this function is the boundary of K (f c) {\displaystyle K(f_{c})}.

The parameter plane of quadratic polynomials – that is, the plane of possible c values – gives rise to the famous Mandelbrot set. Indeed, the Mandelbrot set is defined as the set of all c such that J (f c) {\displaystyle J(f_{c})} is connected. For parameters outside the Mandelbrot set, the Julia set is a Cantor space: in this case it is sometimes referred to as Fatou dust.

In many cases, the Julia set of c looks like the Mandelbrot set in sufficiently small neighborhoods of c. This is true, in particular, for so-called Misiurewicz parameters, i.e. parameters c for which the critical point is pre-periodic. For instance:

In other words, the Julia sets J (f c) {\displaystyle J(f_{c})} are locally similar around Misiurewicz points.