NetNews Usenet Archive 1992 #23/NN_1992_23.iso/spool/sci/fractals/148

Internet Message Format | 1992-10-15 | 1.7 KB

Path: sparky!uunet!ogicse!news.u.washington.edu!nntp.uoregon.edu!cie.uoregon.edu!scavo
From: [e-mail] (Tom Scavo)
Newsgroups: sci.fractals
Subject: Re: Mandelbrot Set
Message-ID: <[e-mail]>
Date: 15 Oct 92 16:28:38 GMT
Article-I.D.: nntp.1992Oct15.162838.24280
References: <[e-mail]> <[e-mail]> <[e-mail]>
Sender: [e-mail]
Organization: University of Oregon Campus Information Exchange
Lines: 29

In article <[e-mail]> [e-mail] (Rob Limbert) writes:
>
>In article <[e-mail]>, [e-mail] (Tom Scavo) writes:
>|> ...There is a Julia set
>|> for each  c  and this set lives in the complex plane.  Fix
>|> c  and iterate  z -> z^2 + c  for every  z  you can think of.
>|> For  c  in the Mandelbrot set, the Julia set is connected.
>|> For  c  not in  M , the Julia set is Cantor dust.
>|>
>
>   In fact, isn't it possible to define the Mandelbrot set as the set of
>all c whose associated Julia set is connected? Something like:
>
>                 M = {c : J is connected}
>                           c

I suppose you could (and many authors do), but it seems more
like a theorem than a definition to me.  It's certainly not the
way  M  came about historically, nor is it calculated this way.

How would you define the bifurcation set (of which the  M  set
is a special case) for some other iteration, like say the cubic
or the complex sine?  Is it known that  J_c  is connected when
c  is in the bifurcation set of the cubic (a 4-d object), for
instance?

--
Tom Scavo
[e-mail]