More from Lee Skinner (we are still working on why he can't post):
Bill,
I have no problem with anything that your references state: that there are
no islands within the main Mandelbrot lake or in any of the surrounding
midgets. But my conjecture similarly states just the opposite: That there
are no islands in the surrounding space outside the Mandelbrot Set, i.e.,
the tendrils do not enclose any of that space. So, just viewing the
Mandelbrot Set as black on a white background (no other colors), there are
no islands in either the black or white regions.
Lee
On 1/31/2024 10:09 AM, Bill Jemison wrote:
I may have misspoken - or more likely don't understand. Rather than saying
the entire Mset is a single line, I probably should have said that the
boundary line (or shoreline) should be a single line if MLC holds. There is
no mention of multiple boundary lines or shorelines that I remember.
Let's say that the fractal is the lake. My understanding of MLC is that
there can be no islands in the M-lake, since that would bring into play
more than one shoreline. Here is a quote from the article:
***************
In the Orsay notes, Douady and Hubbard proved several major theorems that
were motivated by the computer images they’d seen. *They showed that the
Mandelbrot set was connected — that you can draw a line from any point in
the set to any other without lifting your pencil*. Mandelbrot had initially
suspected the opposite: His first images of the set looked like one big
island with lots of little ones floating in a sea around it. But later,
after seeing higher-resolution pictures — including ones that used color to
illustrate how quickly equations outside the set flew off to infinity —
Mandelbrot changed his guess.* It became clear that those little islands
were all connected by very thin tendrils.* The introduction of color “is a
very mundane thing, but it’s important,” said Søren Eilers of the
University of Copenhagen.
****************
And here is another:
*****************
Density of hyperbolicity deals with the Mandelbrot set’s interior. But* MLC
would also enable mathematicians to assign an address to every point on the
set’s boundary.* “It gives a name to every dot. And then, once you have
been able to name every dot of the boundary of the Mandelbrot set, you can
hope to really understand it completely,” Hubbard said.
******************
It seems to me that naming every point on the set's boundary strongly
implies that there is indeed only one boundary. If there is an island in
the fractal lake (or outside it?) that would introduce another unbroken
boundary.
Obviously, I am struggling to understand some of the concepts and theories,
but am fascinated nonetheless.
