scavo@spencer.cs.uoregon.edu (Tom Scavo) (08/07/90)
In article <aqyvtb.iwm@wang.com> wdr@wang.com (William Ricker) writes: >nathant@pogo.WV.TEK.COM (Nathan Tenny) writes: > > >>Devaney has done some work on Julia sets of transcendental functions; >>there's stuff on it in both _Introduction to Chaotic Dynamical Systems_ >>and the _Science of Fractal Images_. I presume he has papers on the >>subject running around, too, but I don't know where... >>-- > >Devaney and a couple of his grad students (one now teaching fractals >as a math elective at Bentley, a Mass. regional MBA factory) will >have papers appearing "soon" in several of the MAA magazines. >Eventually, I intend to post a report here & in sci.math on their >short course at the New England MAA meetings, to answer the >now-expired question of whether Julia set images are artifacts of >machine precision. (Answer: not for practical purposes, the Bentley >prof whose name I'm forgetting proved as her Thesis that one could >prove certain correctness properties about a Julia set image. Oh yes, Devaney does have papers on the dynamics of complex trancendental functions. See for example the richly illustrated Devaney, Robert L. Chaotic Bursts in nonlinear dynamical systems. _Science_ 235 (1987), pp.342-5. in addition to the books mentioned above. For a most elementary introduction, see ch.11 of his newest _Chaos,_Fractals,_and_ Dynamics_ from Addison-Wesley which even has a very simple algorithm that plots the Julia set for a complex sine function. As I recall however, the originator of this thread was interested in rational functions. Newton's method gives rise to rational functions with fascinating dynamical behavior. See for example Curry, J., L. Garnett, and D. Sullivan. On the iteration of a rational function: computer experiments with Newton's method. _Comm_Math_Physics_ 91 (1983), pp.267-277. or Blanchard, Paul. Complex analytic dynamics on the Riemann sphere. _Bulletin_of_the_AMS_ 11 (July 1984), pp.85-141. which, btw, has numerous excellent references on this subject. The mysterious graduate student alluded to above is Lynne Durkin who recently lectured over her work on the complex exponential function at a short course given at Boston University. She did indeed sketch a proof that what one sees in the pictures is the truth. Presumably the details could be had in her thesis. She showed marvelous slides and videos, btw. Does anyone have any more information on the forthcoming articles in the "MAA magazines"? Tom Scavo <scavo@cs.uoregon.edu> ---------