page_title: Julia set
section_title: Quadratic polynomials
hierarchical_section_title: Julia set / Quadratic polynomials
caption_attribution_description: English: Collection of Julia sets laid out in a grid such that the centre of each image corresponds to the same position in the complex plane as the value of the set. When laid out like this the overall image resembles the Mandelbrot set. Here there are 100x100 julia sets.
context_page_description: In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values.
  Thus the behavior of the function on the Fatou set is "regular", while on the Julia set its behavior is "chaotic".
  The Julia set of a function f is commonly denoted J, and the Fatou set is denoted F. These sets are named after the French mathematicians Gaston Julia and Pierre Fatou whose work began the study of complex dynamics during the early 20th century.
context_section_description: 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  large enough that . (For example, if  is in the Mandelbrot set, then , so we may simply let .) Then the filled Julia set for this system is the subset of the complex plane given by
  where  is the nth iterate of . The Julia set  of this function is the boundary of .
  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  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:
  At c = i, the shorter, front toe of the forefoot, the Julia set looks like a branched lightning bolt.
  At c = −2, the tip of the long spiky tail, the Julia set is a straight line segment.
  In other words, the Julia sets  are locally similar around Misiurewicz points.
caption: Collection of Julia sets laid out in a 100 × 100 grid such that the center of each image corresponds to the same position in the complex plane as the value of the set. When laid out like this, the overall image resembles the Mandelbrot set.
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