The interview is I think from March 23, 2010 - at least that's what it says , and the conjecture he mentions is nothing to do with connectedness AFAIK - rather he says it's concerning two alternative definitions of the Set. See from 8.30 to 10.40 in the "Full Interview". See around 9.40 into the "full interview". On 27 Jul 2011, at 03:28, Fred lunnon wrote:

Extract from http://en.wikipedia.org/wiki/Mandelbrot_set

<< Douady and Hubbard have shown that the Mandelbrot set is connected. In fact, they constructed an explicit conformal isomorphism between the complement of the Mandelbrot set and the complement of the closed unit disk. Mandelbrot had originally conjectured that the Mandelbrot set is disconnected. This conjecture was based on computer pictures generated by programs which are unable to detect the thin filaments connecting different parts of M. Upon further experiments, he revised his conjecture, deciding that M should be connected.

...

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It looks to me as if that interview is rather ancient. In particular, Mandelbrot died in October last year.

WFL

On 7/27/11, David Makin <makinmagic@tiscali.co.uk> wrote:

...

In this interview:

http://bigthink.com/benoitmandelbrot

He mentions two forms for the Mandelbrot that are believed to be equivalent but are not proved to be so. He doesn't give the details - can anyone either explain in more detail or point me at the details ? I've tried looking on the web but failed to find the conjecture the he mentions...

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This is odd. I've been studying the Mandelbrot set off and on for a frighteningly long time (26 years). I listened to the interview and read the transcript (which has many errors, but is at bigthink.com/ideas/19207 ) and did some Google searches in case maybe I was missing something. Just to clarify the question, Mandelbrot said the following (I have corrected several errors in the transcript): *The conjecture itself consists in two definitions of Mandelbrot set - two alternative definitions which are too technical to describe without a blackboard, but which are both very simple and which I assumed naively to be equivalent. Why did I assume so? Because in the pictures I could not see any difference. Obtaining pictures in one way or another way, I couldn't tell them apart. Therefore, I assumed they were identical and I went on studying this beast. I found that, again, many interesting observations of which most were very confirmed by many other very, very skilled mathematicians. But the idea that these two conditions, definitions, are identical is still open. So there are two definitions of Mandelbrot set, the usual one and another one, and they may theoretically be different. People are getting close, but have not proven it completely.* The best I can do at connecting this statement to truth is as follows: The only thing that has been worked on extensively by "many very skilled mathematicians" who are "getting closer, but have not proven it completely" is the MLC conjecture, namely that the Mandelbrot set is locally connected. The Mandelbrot set is normally defined by the "z=0, iterate z'=z^2+c, see if it remains bounded" definition which can be implemented as a computer program. This is done by defining a c value for each pixel on the screen and computing the z values numerically. Using a modern programming model (like OpenCL), each pixel performs its own iteration, all pixels iterate in parallel, and if you have infinite patience (-: all pixels destined to "escape" will eventually do so. The Mandelbrot set is also sometimes defined as (using the wording on the Wikipedia page) "the limit set of a sequence of plane algebraic curves, the Mandelbrot curves, of the general type known as polynomial lemniscates". These curves are: L1: |c|=2, L2: |c^2+c|=2, L3: |(c^2+c)^2+c|=2, and so on. They form a set of "contour lines" which are seen in many early computer images, including some in Mandelbrot's 1986 book. They first few are illustrated (with formulas) here: http://en.wikipedia.org/wiki/File:Lemniscates5.png A Mandelbrot set image can be produced by plotting these curves, which needs to be done not by scanning pixels but by "walking" along the curve, using a numerical method similar to root-finding. So my guess at making sense of Mandelbrot's "conjecture" statement is that, the "limit of lemniscates" definition might imply MLC. In other words, if it can be shown that the lemniscates L1, L2, L3, ... converge on the boundary of the Mandelbrot set, then perhaps such a proof would also prove MLC. If so, then the MLC conjecture is equivalent to a conjecture that the two ways of drawing the Mandelbrot set (pixel scanning and tracing lemniscates) yield the same image, and Mandelbrot's statements would make sense. It's a stretch, but it's the best I can do. - Robert Munafo On Tue, Jul 26, 2011 at 21:18, David Makin <makinmagic@tiscali.co.uk> wrote:

In this interview:

http://bigthink.com/benoitmandelbrot

He mentions two forms for the Mandelbrot that are believed to be equivalent but are not proved to be so. He doesn't give the details - can anyone either explain in more detail or point me at the details ? I've tried looking on the web but failed to find the conjecture the he mentions...

bye Dave

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On Thu, Jul 28, 2011 at 1:36 AM, Robert Munafo <mrob27@gmail.com> wrote:

The Mandelbrot set is also sometimes defined as (using the wording on the Wikipedia page) "the limit set of a sequence of plane algebraic curves, the Mandelbrot curves, of the general type known as polynomial lemniscates". These curves are:

  L1: |c|=2,   L2: |c^2+c|=2,   L3: |(c^2+c)^2+c|=2,

and so on. They form a set of "contour lines" which are seen in many early computer images, including some in Mandelbrot's 1986 book. They first few are illustrated (with formulas) here: http://en.wikipedia.org/wiki/File:Lemniscates5.png

So my guess at making sense of Mandelbrot's "conjecture" statement is that, the "limit of lemniscates" definition might imply MLC. In other words, if it can be shown that the lemniscates L1, L2, L3, ... converge on the boundary of the Mandelbrot set, then perhaps such a proof would also prove MLC. If so, then the MLC conjecture is equivalent to a conjecture that the two ways of drawing the Mandelbrot set (pixel scanning and tracing lemniscates) yield the same image, and Mandelbrot's statements would make sense.

Except that the statement that the leminscates converge to the boundary of the mandelbrot set is easily seen to be true. If a point is outside of Ln, then one of the iterates has norm > 2, and further iteration will increase the norm, so the point is not in the Mandelbrot set. And if a point is inside Ln for every n, then all the iterates lie within L1, so the iterates don't diverge, and the point is in the mandelbrot set. Here is a half-remembered, and possibly mangled, conjecture that two definitions both define the Mandelbrot set. Let f_c be the map f_c(z) = z^2 + c. And let f_c^n be the nth iterate of f_c. Among the points in the Mandelbrot set, we have at least the following points: A. Fixed points of iterates; Those points c for which f_c^n(0) = 0 for some n. B. Points in the basin of attraction of fixed points of iterates: Suppose there exist z and n for which f_c^n(z) = z, and |f_c^n'(z)| < 1. Then for points w sufficiently near z, the sequence f_c^(n*m)(w) converges to z. If 0 is sufficiently near z, then c is in the Mandelbrot set. More succinctly, these are the values c for which there exists an n such that lim_(m->infinity) f_c^nm(0) exists. C. The Mandelbrot set is closed, (it is the intersection of the closed sets |f_c^n(0)| < 100), so any set in the closure of A and B is in the Mandelbrot set. There could be another category I'm leaving out, but I think the unproved conjecture is that this is all there is, that is, that every point in the Mandelbrot set is in one of these 3 categories. But I don't see a good way to plot the points in category B, so I can't reconcile this with Mandelbrot's claim that the conjecture seemed to be true experimentally. Andy