Julia Set

The Fatou set $F(f)\,$ of $f\,$ is the Complement of the Julia set: that is, the set of points which exhibit 'stable' behavior. Thus on $F(f)\,$ , the behavior of $f\,$ is 'regular', while on $J(f)\,$ , it is ' Chaotic '. These sets are named in honor of the French mathematicians Gaston Julia and Pierre Fatou , who initiated the theory of Complex Dynamics in the early 20th century. FORMAL DEFINITION Let : $f:X o X\,$ be an analytic self-map of a Riemann Surface $X\,$ . We will assume that $X\,$ is either the Riemann sphere, the Complex Plane , or the once-punctured complex plane, as the other cases do not give rise to interesting dynamics. (Such maps are Completely Classified .) We will be considering $f\,$ as a Discrete Dynamical System on the Phase Space $X\,$ , so we are interested in the behavior of the Iterates $f^n\,$ of $f\,$ (that is, the $n\,$ -fold compositions of $f\,$ with itself). The Fatou set of $f\,$ consists of all points $z\in X\,$ such that the family of iterates : $(f^n)_{n\in\mathbb{N}}$ forms a Normal Family in the sense of Montel when restricted to some open neighborhood of $z\,$ . The Julia set of $f\,$ is the complement of the Fatou set in $X\,$ . EQUIVALENT DESCRIPTIONS OF THE JULIA SET $J(f)\,$ is the smallest closed set containing at least three points which is completely invariant under $f\,$ . $J(f)\,$ is the Closure of the set of repelling Periodic Point s. For all but at most two points $z\in X\,$ , the Julia set is the set of limit points of the full backwards orbit $\bigcup_n f^{-n}(z)$ . (This suggests a simple algorithm for plotting Julia sets, see below.) If $f\,$ is an Entire Function - in particular, when $f\,$ is a Polynomial , then $J(f)\,$ is the Boundary of the set of points which converge to infinity under iteration. If $f\,$ is a polynomial, then $J(f)\,$ is the boundary of the Filled Julia Set ; that is, those points whose orbits under $f\,$ remain bounded. PROPERTIES OF THE JULIA SET AND FATOU SET The Julia set and the Fatou set of $f$ are both Completely Invariant under $f$ , i.e. $\ f^{-1}(J(f)) = f(J(f)) = J(f)$ and $\ f^{-1}(F(f)) = f(F(f)) = F(f)$ . Beardon, ''Iteration of Rational Functions'', Theorem 3.2.4 RATIONAL MAPS There has been extensive research on the Fatou set and Julia set of iterated Rational Functions , known as rational maps. For example, it is known that the Fatou set of a rational map has either 0,1,2 or infinitely many Components .Beardon, ''Iteration of Rational Functions'', Theorem 5.6.2 Each component of the Fatou set of a rational map can be classified into one of Four Different Classes .Beardon, Theorem 7.1.1 QUADRATIC POLYNOMIALS A very popular complex dynamical system is given by the family of Quadratic Polynomials , a special case of Rational Map s. The Quadratic Polynomials can be expressed as : $f_c(z) = z^2 + c\,$ (where $c\,$ is a complex parameter).
	Image:Time Escape Julia Set From Coordinate (phi-2, Phi-1)jpgJulia Set For F<sub>c</sub>, C	(φ&minus2)+(φ&minus1)i =-04+06i
	Image:Time Escape Julia Set From Coordinate (0285, 0)jpgJulia Set For F<sub>c</sub>, C	0285+0i
	Image:Julia Set (highres 01)jpgJulia Set For F<sub>c</sub>, C	0285+001i
	Image:Julia Set Camp3jpgJulia Set For F<sub>c</sub>, C	045+01428i
	Image:Julia Set Camp1jpgJulia Set For F<sub>c</sub>, C	-070176-03842i
	Image:Julia Set Camp2jpgJulia Set For F<sub>c</sub>, C	-0835-02321i
	Image:Julia Set Camp4 Hi RezpngJulia Set For F<sub>c</sub>, C	-08+0156i

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Julia Set