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

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

Date: Mon, 01 Nov 2004 16:55:28 GMT

-------- Original Message --------
Subject: Re: two questions on the Mandelbrot set
Date: 1 Nov 2004 03:49:37 -0800
From: lasse_rempe@yahoo.de
Organization: http://groups.google.com
Newsgroups: sci.math
References: <Oshhd.4605$t45.712860@weber.videotron.net>

David Bernier wrote in message news:...
> Suppose we let M denote the Mandelbrot set.
> > (1) How many connected components does the interior of M have?
> > (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) ?
> > David Bernier

The answer to (1) is easy: there are infinitely many components of the
interior of M, since there are infinitely many HYPERBOLIC components
(i.e., those corresponding to parameters with an attracting cycle).
However, what we don't know is whether these are the only components
of the interior of M.

(2) is equivalent to the famous MLC conjecture. This is problem the
most famous open problem in holomorphic dynamics; a positive answer
would in particular imply that all components of the interior of M are
hyperbolic.

Lasse 

---
(lasse@remove.for.spam.maths.warwick.ac.uk)
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Respectfully, Roger L. Bagula
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Relevant Pages

  • Re: two questions on the Mandelbrot set
    ... > How many connected components does the interior of M have? ... since there are infinitely many HYPERBOLIC components ... is equivalent to the famous MLC conjecture. ... most famous open problem in holomorphic dynamics; ...
    (sci.math)
  • two questions on the Mandelbrot set
    ... Suppose we let M denote the Mandelbrot set. ... How many connected components does the interior of M have? ... David Bernier ...
    (sci.math)
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