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Topic: Fatou set

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	Embedded Julia Set, Mu-Ency at MROB
		The size of an embedded Julia set is proportional to the size of its influencing mu-molecule, and inversely proportional to its distance from the influencing mu-molecule, or distance from other larger-size embedded Julia Sets (whichever distance is less).
		The shape of the Embedded Julia Set is very close to (but never exactly the same as) the Julia set whose Parameter whose position relative to the Continent corresponds to the embedded Julia set's position relative to the influencing island.
		The second component is a set of Filaments whose shape is very similar to the filaments adorning the influencing island, which in turn resemble the filament of which the influencing island is a part.
	www.mrob.com /pub/muency/embeddedjuliaset.html (1165 words)

	Take a BrainSip (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-06)
		In mathematics, a Fatou set is defined informally as the largest open set of points on which iterations of a given map (or collection of maps) have relatively tame long-term behaviour, in the sense that points that start close together stay close together.
		A map's Fatou set is the complement of its Julia set.
		Less commonly, the term Fatou set or Fatou dust is sometimes used to refer to a Julia set that is not connected.
	fatou-set.mestskadoprava.sk.cob-web.org:8888 (104 words)

	Chaos Theory
		Fatou also studied the motion of a planet in a resistant medium with the intention of explaining how twin stars would form with the capture of one moving in the atmosphere of the other.
		The boundary of this set is a Julia set.
		A set of points whose forward orbits move toward the same limit point is called a basin of attraction, and this limit point is known as the set's attractor.
	www.emayzine.com /infoage/math/math1.htm (13247 words)

	Cantor set (Site not responding. Last check: 2007-11-06)
		The Cantor set is the prototype of a fractal.
		Since the Cantor set is the complement of a union of open sets, it itself is a closed subset of the reals, and therefore a complete metric space.
		The Cantor set is the set of all points on the Koch curve that intersect the original horizontal line segment.
	cantor-set.kiwiki.homeip.net (1391 words)

	Features of Mandelbrot and Julia Sets
		A set of points is connected if, for any two points in the set, there is at least one path consisting entirely of points in the set, which leads from one point to the other.
		A Julia Set depends on an iteration exactly like that for the Mandelbrot Set except that the initial value of z is the complex number representing the point whose membership is to be tested, and c is a parameter of the set.
		That is, the Julia set corresponding to a point c in the boundary of the Mandelbrot set will have as part or most of its shape an infinite repetition of the shape of the Mandelbrot set near c.
	www.laetusinpraesens.org /docs00s/cardrep4.php (10603 words)

	Math 189A: Complex Dynamics (Site not responding. Last check: 2007-11-06)
		On the Fatou set, the dynamics of the rational map is regular, and iterates of nearby starting points have the same longterm behaviour.
		On the Julia set, the dynamics is chaotic, nearby points are driven apart by iteration, and longterm behaviour is sensitively dependent on the choice of initial point.
		For example, the repelling periodic points of a given rational mapping are dense in the Julia set, and the Julia set can also be generated as the set of accumulation points of the backwards orbit of a generic point in the Fatou set.
	www.math.hmc.edu /~ward/math189 (371 words)

	The Mandelbrot, Julia and Fatou sets (Site not responding. Last check: 2007-11-06)
		The Mandelbrot set (M) is the set of all points c on complex plane (parameter space) such that iterations z
		The Mandelbrot set is the fl region on this image.
		The Julia set is the boundary between colored and fl regions in the left image.
	www.ibiblio.org /e-notes/MSet/Mandelbrot.htm (312 words)

	Julia Sets (Site not responding. Last check: 2007-11-06)
		The boundary of this set is a Julia
		A set of points whose forward orbits approach the same limit point is called a basin of attraction, and this limit point is known as the set's attractor.
		A Julia set is a set of exceptional points that separate different basins of attraction.
	home.comcast.net /~davebowser/fractals/julia.html (894 words)

	Mandelbrot Set
		In the Mandelbrot set, nature (or is it mathematics) provides us with a powerful visual counterpart of the musical idea of 'theme and variation': the shapes are repeated everywhere, yet each repetition is somewhat different.
		Note important, as it is, the classification of Julia set in terms of disconnected sets, this still doesn't allow one to visualize the shape of the set of points, in the parameter space, for which the Julia set is connected.
		Figure 3: The buds of the Mandelbrot set corresponding to Julia sets that bound the basins of attraction (trapping sets) of periodic orbits.
	chaos.phy.ohiou.edu /~thomas/fractal/mandel.html (1324 words)

	[No title]
		Although Julia sets (and bifurcation diagrams) of transcendental functions can compete in their beauty and complexity very well with those of rational functions, this article is not illustrated with such pictures.
		The basic objects studied in iteration theory are the Fatou set $F=F(f)$ and the Julia set $J=J(f)$ of a meromorphic function $f$.
		Roughly speaking, the Fatou set is the set where the iterative behavior is relatively tame in the sense that points close to each other behave similarly, while the Julia set is the set where chaotic phenomena take place.
	www.ams.org /journals/bull/pre-1996-data/199329-2/Bergweiler (9297 words)

	HOLOMORPHIC DYNAMICS (Site not responding. Last check: 2007-11-06)
		Fatou and Julia: dynamics on the Riemann sphere.
		Repelling cycles are dense in the Julia set.
		Sullivan's classification of the connectivity components of the Fatou set.
	www.wisdom.weizmann.ac.il /courses/hol-dyn-03.html (241 words)

	Commentary: MAT335
		The set of points z_0 for which the functions {f^n(z)}, defined on a neighborhood of the point z_0, are regular is called the Fatou set, and the other points, where {f^n(z)} is not regular, is called the Julia set.
		In addition, they were able to deduce that the dynamics of f on the Fatou set is regular while the dynamics of f on the Julia set is (what we would now call) chaotic.
		The Julia set of N(z) is the boundary between the basins of attraction of the fixed points of N (the roots of f).
	www.sfu.ca /~rpyke/335/W00/28mar.html (1843 words)

	Abstracts of papers of Juan E. Rivera-Letelier
		We show furthermore that for rational maps with one critical point in the Julia set all the conditions above are equivalent to the usual Collet-Eckmann and backward Collet-Eckmann conditions; thus the latter ones are invariant by topological conjugacy in the unicritical setting.
		We prove that Julia sets have zero Lebesgue measure, when not equal to the whole sphere, and in the polynomial case every connected component of the Julia set is locally connected.
		On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets.
	www.math.sunysb.edu /~rivera/abs.html (1962 words)

	Bericht 99-4 des Mathematischen Seminars Kiel (Site not responding. Last check: 2007-11-06)
		Let $D$ be an unbounded invariant component of the Fatou set of a transcendental entire function $f$.
		Then the set $\Theta := \{ \theta \in \partial \dz \pl ; \lms \phi (r\theta) = \infty \}$ is closely related to the Julia set of the corresponding inner function $g:= \phi^{-1} \circ f \circ \phi$.
		In the first part of the paper we further develop the theory of Julia sets of inner functions and the dynamical behaviour on their Fatou sets.
	www.numerik.uni-kiel.de /reports/1999/99-4.html (149 words)

	Some Julia Sets
		The Julia set of f then is the set of all points of G, at which this sequence of iterated functions is not equicontinous.
		The Julia sets are white, the Fatou sets fl.
		Attracting fixed points always are in the Fatou set, and repelling fixed points in the Julia set.
	www.ijon.de /mathe/julia/some_julia_sets_1_en.html (1022 words)

	Julia Set, Mu-Ency at MROB
		General definition: A Julia set is a maximal set of points (on the complex plane) with the property that any member of the set will be replaced with some other member of the set (or perhaps the same value) when mapped by a function Z' = f(Z).
		Two Julia Sets (for values of C=0+0i and C=-2+0i) are very simple but the rest are fractals, and their boundaries have Hausdorff dimensions ranging all the way from 0.0 to 2.0.
		The topology of the Julia set can be determined by looking at the critical orbit, e.g.
	www.mrob.com /pub/muency/juliaset.html (334 words)

	Pierre Fatou - Wikipedia, the free encyclopedia
		Pierre Fatou was the first to define the Mandelbrot set.
		He entered the École Normale Supérieure in Paris in 1898 to study mathematics and graduated in 1901 when he was appointed an astronomy post in the Paris Observatory.
		Fatou wrote many papers developing a Fundamental theory of iteration in 1917, which he published in the December 1917 part of Comptes Rendus.
	en.wikipedia.org /wiki/Pierre_Fatou (245 words)

	Chaos and Fractals
		Julia set is the boundary of the basin of attraction of infinity, and it is either connected or totally disconnected.
		is the Julia set for this particular c, and the set M in parameter space is called the Mandelbrot set.
		A strange attractor is a set that attracts an open neighborhood, but is not the union of sinks.
	pirate.shu.edu /~wachsmut/Workshops/Camden (736 words)

	An explanation of Julia and Mandelbrot fractals
		The whole plane excluding the Julia set is known as the Fatou set.
		In this case A is an attractor, and the set of all points whose orbit is attracted to A is called the basin of A. In the third diagram above, the point A is highlighted and the basin of A fills the whole of the interior set.
		The Mandelbrot sets will be found to consist of an infinite number of joined bodies, in each of which the corresponding Julias contain an attractor with a period specific to that body.
	www.felicite-parmentier.freeserve.co.uk /page3.htm (1214 words)

	Structural stability in holomorphic dynamics (Spring 2003) (Site not responding. Last check: 2007-11-06)
		The study of iterations of rational mappings of the Riemann sphere was started at the end of 19-th - beginning of 20th century, in the works of Fatou and Julia.
		2) the Julia set, which is its complement and is the closure of the set of repelling periodic points (see [L]).
		R.Mane, P.Sad and D.Sullivan have proved in their joint paper [M] that each hyperbolic mapping is structurally stable on its Julia set: the mapping and its small perturbation (in the class of rational mappings of the same degree) are conjugated on their Julia sets by a homeomorphism}.
	www.mccme.ru /ium/s03/stabholdyn.html (376 words)

	Sekino’s Fractal Gallery (Site not responding. Last check: 2007-11-06)
		The divergence scheme leaves the Mandelbrot set stuck with the canvas color since none of the parameters in the set are attached to escaping sequences.
		Since it was published in 1980, the Mandelbrot set, or M set for short, became so popular that zillions of digital artists, mathematicians, and computer programmers and hackers have explored around it and shown their fractal images on a variety of objects including web pages, posters, book covers, T-shirts, and coffee mugs.
		The gatekeepers are painted by the convergence scheme with various shades of gold and the red background by the divergence scheme; thus, the Julia set in this pattern is depicted by the outlines of the gatekeepers.
	www.willamette.edu /~sekino/fractal (2754 words)

	More About Fractals
		Afterwards, Fatou and Julia continued to elaborate on the discovery and exploration of fractals.
		In a complex plane, the various points are associated with a Julia set, meaning that the Mandelbrot set acts as an "index" for the Julia set.
		If the set is to be connected, the value of C should be coming from the inside of the Mandelbrot set.
	freda.auyeung.net /fractals/facts.htm (372 words)

	Mandelbrot and Julia Set Overview
		Points in the Fatou set tend to stick together; that is, points close to each other will follow similar paths, drawing closer to either infinity or zero.
		Some Julia sets may consist of many disconnected points (called "dust sets"); others from larger "solid" areas that seem all connected; some form thin, wiggly lines that are all connected but do not outline any shapes ("dendritic" types).
		However, because the Mandelbrot set is an amalgamation of all Julia sets, the detail changes drastically depending on where you zoom in.
	www.gpf-comics.com /dl/mandel/java/docs/mandel.html (2489 words)

	Animations in Complex Dynamics (Site not responding. Last check: 2007-11-06)
		The Julia set divides the Riemann sphere into the basin of attraction of 0, the Baker domain, and all preimages of the basin and of the Baker domain.
		In this animation, the Julia set is shown in white on the Riemann sphere.
		The following two animations show sequences of Julia sets of the polynomials for increasing degrees d before reaching a picture of the limit Julia set, that is to say, the Julia set of the limit function f.
	www.math.uiuc.edu /~aimo/anim.html (401 words)

	fatoubandcases.nb
		If two of the critical values of a Weierstrass elliptic function lie in one component of the Fatou set then the Fatou set will have a toral band.
		The points in the Julia set are colored red.
		The points colored teal are in the Julia set.
	www.dickinson.edu /~koss/toralbandcases/index.html (388 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-06)
		is an element of the so-called Fatou set
		equals the closure of the set of repelling periodic points (cf.
		Sullivan has given an exhaustive classification of the elements of the Fatou set with respect to their dynamics.
	eom.springer.de /J/j054390.htm (212 words)

	Susan Goldstine's Research Interests
		Whenever we are given a function f from a set X to itself, we may view the function as generating a dynamical system by considering the effect of repeatedly applying the function f to the set X. One way to understand the resulting system is to ask what can be said about the orbit
		form the filled Julia set of f, a set whose shape is determined by the attracting or repelling nature of the preperiodic points of f in C.
		In this case, the set of points with non-zero p-adic local height is precisely the analytic component of the Fatou set containing
	www.amherst.edu /~sgoldstine/research/technical.html (1000 words)

	Papers by Robert Benedetto
		A wavelet theory is developed on G using coset representatives of the discrete quotient of the dual of G by the annihilator of H to circumvent this limitation.
		Although the Haar and Shannon wavelets are naturally antipodal in the Euclidean setting, it is observed that their analogues for G are equivalent.
		Using the theory of complex dynamics as a model, we define the Fatou and Julia sets of such a function and study their properties.
	www.cs.amherst.edu /~rlb/papers (1537 words)

	Atlas: On the Set of non Normality in Iteration Theory by P. Bhattacharyya (Site not responding. Last check: 2007-11-06)
		The set of points of the complex plane where the sequence of iterates of f is not normal (in the sense of Montel)is the Julia set and its complement is the Fatou set.
		The Julia set is closed and has no interior points unless it is the whole plane.
		In this paper we prove some theorems giving sufficient conditions for the Julia set of a function will be the whole plane.Apart from the rational and entire transcendental class of functions considered earlier we also consider the class of transcendental meromorphic functions considered by recently by Baker, Kotus and Lu.
	atlas-conferences.com /cgi-bin/abstract/caky-11 (209 words)

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