robison@uiucdcsb.UUCP (05/14/87)
In the May 1987 issue of IEEE Computer Graphics, p.7, Figure 4 has the caption:
"Julia sets for Newton's method applied to polynomials of degree 3
(small spheres) and degree 4 (large spheres)."
What is a "Julia set"? Can someone point out a reference?
Arch D. Robison
University of Illinois at Urbana-Champaign
CSNET: robison@UIUC.CSNET
UUCP: {ihnp4,pur-ee,convex}!uiucdcs!robison
ARPA: robison@B.CS.UIUC.EDU (robison@UIUC.ARPA)epg@mcvax.cwi.nl (Ed Gronke) (05/15/87)
A Julia set , according to "An Introduction to Chaotic Dynamical Systems"
by Robert L. Devaney, Benjamin Cummings Publishing, 1986, ISBN # 0-8053-1601-9,
is defined by:
"Let P: C -> C be a polynomial. The Julia set of P, denoted by J(P), is the
closure of the set of repelling periodic points of P."
where C = the complex plane.
Also, a periodic point of perion n is repelling if the absolute value (modulus?)
of the derivative on the n iterate at the periodic point is > 0. i.e.,
n n
z = p (z ) and |(p )'(z )| > 1
0 0 0
-- Ed Gronke epg@cwi.nlscott@bu-cs.UUCP (05/15/87)
In <165500002@uiucdcsb>, robison@uiucdcsb.cs.uiuc.edu writes >In the May 1987 issue of IEEE Computer Graphics, p.7, Figure 4 has the caption: > > "Julia sets for Newton's method applied to polynomials of >degree 3 > (small spheres) and degree 4 (large spheres)." > >What is a "Julia set"? Can someone point out a reference? In this context, a Julia set is the boundary between the basins of attraction of the roots (and any other attracting periodic orbits). That is, all the points that do not eventually get sucked in to a root of the polynomial (or an attr. periodic orbit) are in the Julia set. [Notice that I keep throwing in attracting periodic orbits -- These are what cause Newton's method to fail for an open set of initial points] More generally, one definition of the Julia Set is "the closure of the repelling periodic points". A nice reference with lots of pictures is _The_Beauty_of_Fractals_ by H.O. Peitgen & P. Richter (Springer-Verlag). This whole business is part of the field of Complex Dynamics, in which there's a lot of mathematics going on lately. I haven't brought up any more mathematical references, (though there's lots), since you're obviously not familiar with the field. A name generally known in graphics circles is Mandlebrot (and the Mandlebrot set), which is intimitely related to Julia Sets. Hope that helped some.
platt@emory.UUCP (Dan Platt) (05/24/87)
In article <165500002@uiucdcsb> robison@uiucdcsb.UUCP writes: > >What is a "Julia set"? Can someone point out a reference? A Julia set is a fractal set which results from an iterative mapping in the complex domain. For a reference see: "An Introduction to Chaotic Dynamical Systems" by Robert L. Devaney published by Benjamin/Cummings.