[Fwd: Re: two questions on the Mandelbrot set]

From: Roger Bagula (tftn_at_earthlink.net)
Date: 11/01/04 

Date: Mon, 01 Nov 2004 16:56:34 GMT

-------- Original Message --------
Subject: Re: two questions on the Mandelbrot set
Date: Mon, 01 Nov 2004 07:30:03 -0500
From: A N Niel <anniel@nym.alias.net.invalid>
Reply-To: A N Niel <anniel@nym.alias.net.invalid>
Organization: RPG
Newsgroups: sci.math
References: <Oshhd.4605$t45.712860@weber.videotron.net>

In article , David Bernier
 wrote:

> Suppose we let M denote the Mandelbrot set.
> > (1) How many connected components does the interior of M have?

infinitely many

> > (2) If bd(M) is the boundary of M, and S^1 = {(x,y) | x^2 + y^2 = 1},
> does there exist a continuous f: S^1 -> C (the complex plane),
> such that f(S^1) = bd(M) ?

The continuous images of S^1 are exactly the compact, arcwise-connected
sets. So your answer is "yes". More interesting would be an
analytic map from the exterior of M to {z in C | |z| > 1}
that extends continuously to the boundaries. The existence
of this would require bd(M) locally connected, which has not yet
been proved.

> > David Bernier 

-- 
Respectfully, Roger L. Bagula
tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
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