Julia set

In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets (Julia 'laces' and Fatou 'dusts') 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(f), and the Fatou set is denoted F(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.

Formal definition

Let f(z) be a complex rational function from the plane into itself, that is, , where p(z) and q(z) are complex polynomials. Then there is a finite number of open sets F₁, ..., F_r, that are left invariant by f(z) and are such that:

Set

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Julia

Julia is usually a woman's given name. It is a Latinate feminine form of the name Julius. Julius was a Roman family, derived from a founder Julus, the son of Aeneas and Creusa in Roman mythology, although the name's etymology may possibly derive from Greek ἴουλος (ioulos) "downy-[haired, bearded]" or alternatively from the name of the Roman god Jupiter.

Like its male counterpart, the given name Julia had been in use throughout Late Antiquity (e.g. Julia of Corsica) but became rare during the Middle Ages, and was revived only with the Italian Renaissance. It became common in the English-speaking world only in the 18th century. Today, it is frequently used throughout the world. It was the 10th most popular name for girls born in the United States in 2007 and the 88th most popular name for females in the 1990 census there. It has been among the top 150 names given to girls in the United States for the past 100 years. It was the 89th most popular name for girls born in England and Wales in 2007; the 94th most popular name for girls born in Scotland in 2007; the 13th most popular name for girls born in Spain in 2006; the fifth most popular name for girls born in Sweden in 2007; the 94th most popular name for girls born in Belgium in 2006; the 53rd most popular name for girls born in Norway in 2007; the 70th most popular name for girls born in Hungary in 2005; the 19th most popular name for girls born in British Columbia, Canada in 2006; the 9th most popular name for girls born in Germany in 2005; the 2nd most popular name for girls born in Poland in 2013 and the most popular name in Austria.