Hi.

Why is it that the Mandelbrot set contains small "midget" copies of
itself in itself, anyway? What is the mechanism that creates midgets,
exactly? Midgets appear along the real axis spike (a piece of
ornamentation) that lies along the negative real axis after the period
doubling cascade ends (around the Feigenbaum point, ~ -1.40155) but
before the divergence to infinity beyond c = -2. I've noticed that on
a "bifurcation diagram" of the real axis, we see the doubling
bifurcations go up to that point, and then afterwards the diagram
becomes wildly chaotic (after we've moved out of the main body of the
set and into ornamentation -- the real axis spike), with breaks of
various sizes ("islands of stability" in the chaos) at different
points corresponding to the locations of midgets (in fact they _are_
the midgets -- or, well, the parts that intersect the real line).
Could it perhaps be that whatever phenomenon causes these "islands" is
also the same mechanism that generates midgets when applied to the
complex plane?

mike3 a écrit :





Hi.

Why is it that the Mandelbrot set contains small "midget" copies of
itself in itself, anyway? What is the mechanism that creates midgets,
exactly? Midgets appear along the real axis spike (a piece of
ornamentation) that lies along the negative real axis after the period
doubling cascade ends (around the Feigenbaum point, ~ -1.40155) but
before the divergence to infinity beyond c = -2. I've noticed that on
a "bifurcation diagram" of the real axis, we see the doubling
bifurcations go up to that point, and then afterwards the diagram
becomes wildly chaotic (after we've moved out of the main body of the
set and into ornamentation -- the real axis spike), with breaks of
various sizes ("islands of stability" in the chaos) at different
points corresponding to the locations of midgets (in fact they _are_
the midgets -- or, well, the parts that intersect the real line).
Could it perhaps be that whatever phenomenon causes these "islands" is
also the same mechanism that generates midgets when applied to the
complex plane?

Yes, it is the same mechanism. This is a classical theorem saying that M
is universal (McMullen, C. "The Mandelbrot set is universal," pages 1-17
of The Mandelbrot Set, Theme and Variations, Tan Lei, ed., Cambridge
University Press (2000).); here is a small text by Douady hinting at the
reason :http://www.math.binghamton.edu/topics/mandel/mandel_why.html

On Nov 2, 11:15 pm, Denis Feldmann <denis.feldmann.asuppri...@club-
internet.fr> wrote:
mike3 a écrit :




Hmm. Can't get the paper you referenced, though, as I don't have
access to a university library (I need to know how to get rich so I'll
be able to.). Anyway, I read the website, and it was interesting.
Is that also why Mandelbrot midgets may often appear in other
iterative maps?

I also noticed on this page for the logistic map,
another quadratic map:

http://mathworld.wolfram.com/LogisticMap.html

"The period doubling bifurcations come faster and faster (8, 16,
32, ...), then suddenly break off. Beyond a certain point known as the
accumulation point, periodicity gives way to chaos, as illustrated
below. In the middle of the complexity, a window suddenly appears with
a regular period like 3 or 7 as a result of mode locking."

And a midget appears right at the spot in the complex
graph where that happens. So then does this phenomenon
of "mode locking" also have something to do with the
appearance of midgets in the Mandelbrot set?

mike3 a écrit :

On Nov 2, 11:15 pm, Denis Feldmann <denis.feldmann.asuppri...@club-
internet.fr> wrote:
mike3 a écrit :

Hmm. Can't get the paper you referenced, though, as I don't have
access to a university library (I need to know how to get rich so I'll
be able to.). Anyway, I read the website, and it was interesting.
Is that also why Mandelbrot midgets may often appear in other
iterative maps?

Yes







I also noticed on this page for the logistic map,
another quadratic map:

http://mathworld.wolfram.com/LogisticMap.html

"The period doubling bifurcations come faster and faster (8, 16,
32, ...), then suddenly break off. Beyond a certain point known as the
accumulation point, periodicity gives way to chaos, as illustrated
below. In the middle of the complexity, a window suddenly appears with
a regular period like 3 or 7 as a result of mode locking."

And a midget appears right at the spot in the complex
graph where that happens. So then does this phenomenon
of "mode locking" also have something to do with the
appearance of midgets in the Mandelbrot set?

Yes. You may be interested by this site :http://www.ibiblio.org/e-notes/MSet/Contents.htm

There are there a lot of interesting results, and a few proofs...

On Nov 3, 11:30 pm, Denis Feldmann <denis.feldmann.asuppri...@club-
internet.fr> wrote:
mike3 a écrit :

On Nov 2, 11:15 pm, Denis Feldmann <denis.feldmann.asuppri...@club-
internet.fr> wrote:
mike3 a écrit :
Hmm. Can't get the paper you referenced, though, as I don't have
access to a university library (I need to know how to get rich so I'll
be able to.). Anyway, I read the website, and it was interesting.
Is that also why Mandelbrot midgets may often appear in other
iterative maps?
Yes







I also noticed on this page for the logistic map,
another quadratic map:
http://mathworld.wolfram.com/LogisticMap.html
"The period doubling bifurcations come faster and faster (8, 16,
32, ...), then suddenly break off. Beyond a certain point known as the
accumulation point, periodicity gives way to chaos, as illustrated
below. In the middle of the complexity, a window suddenly appears with
a regular period like 3 or 7 as a result of mode locking."
And a midget appears right at the spot in the complex
graph where that happens. So then does this phenomenon
of "mode locking" also have something to do with the
appearance of midgets in the Mandelbrot set?
Yes. You may be interested by this site :http://www.ibiblio.org/e-notes/MSet/Contents.htm

There are there a lot of interesting results, and a few proofs...



Interesting stuff. It is kind of convenient that a
piece of ornamentation -- the real axis spike, happens to
lie along the real axis, as it makes it easy to study.
Based on the behavior of the bifurcation diagram there,
would it make any sense then to consider _all_ ornamentation
as the complex analogue of the chaotic region of the
real bifurcation diagram,

the set (cardioid and all bulbs/sub-bulbs/sub-sub-bulbs/etc.)
as analogous to the "stable" region of the diagram? It
seems that with complex numbers, the very _shape_ of
the chaotic region has become incredibly intricate!

Also, has any research been done to determine the precise
nature of ornamentation (not bulbs, but ornamentation)?

Why does ornamentation seem to contain motifs that
seem related to the periods of the bulbs it adorns?
It is in the ornamentation, after all, that all the grand Complexity
is found...

All this is well known (you should really get that book by Tan Lei). The

mike3 a écrit :



On Nov 3, 11:30 pm, Denis Feldmann <denis.feldmann.asuppri...@club-
internet.fr> wrote:
mike3 a écrit :

On Nov 2, 11:15 pm, Denis Feldmann <denis.feldmann.asuppri...@club-
internet.fr> wrote:
mike3 a écrit :
Hmm. Can't get the paper you referenced, though, as I don't have
access to a university library (I need to know how to get rich so I'll
be able to.). Anyway, I read the website, and it was interesting.
Is that also why Mandelbrot midgets may often appear in other
iterative maps?
Yes

I also noticed on this page for the logistic map,
another quadratic map:
http://mathworld.wolfram.com/LogisticMap.html
"The period doubling bifurcations come faster and faster (8, 16,
32, ...), then suddenly break off. Beyond a certain point known as the
accumulation point, periodicity gives way to chaos, as illustrated
below. In the middle of the complexity, a window suddenly appears with
a regular period like 3 or 7 as a result of mode locking."
And a midget appears right at the spot in the complex
graph where that happens. So then does this phenomenon
of "mode locking" also have something to do with the
appearance of midgets in the Mandelbrot set?
Yes. You may be interested by this site :http://www.ibiblio.org/e-notes/MSet/Contents.htm

There are there a lot of interesting results, and a few proofs...

Interesting stuff. It is kind of convenient that a
piece of ornamentation -- the real axis spike, happens to
lie along the real axis, as it makes it easy to study.
Based on the behavior of the bifurcation diagram there,
would it make any sense then to consider _all_ ornamentation
as the complex analogue of the chaotic region of the
real bifurcation diagram,

Yes, and it can even be proved, quite easily, for things like, say, the
branch ending at i.

and the remaining parts of

the set (cardioid and all bulbs/sub-bulbs/sub-sub-bulbs/etc.)
as analogous to the "stable" region of the diagram? It
seems that with complex numbers, the very _shape_ of
the chaotic region has become incredibly intricate!

Also, has any research been done to determine the precise
nature of ornamentation (not bulbs, but ornamentation)?

Yes, a lot.

Why does ornamentation seem to contain motifs that
seem related to the periods of the bulbs it adorns?
It is in the ornamentation, after all, that all the grand Complexity
is found...

All this is well known (you should really get that book by Tan Lei). The
mùain tool is the correspondance between M and the Julia sets. What is
still mysterious is , say, the locally connected nature of M, or the
presence (or not) of other components that the midget M's ...

Here is a serious article on the Web which may clarify things for you :http://www.u-cergy.fr/rech/pages/tan/papers/similarity.ps(the ps format
may be somewhat annoying, I admit)

On Nov 4, 4:03 am, Denis Feldmann <denis.feldmann.asuppri...@club-
internet.fr> wrote:
mike3 a écrit :


Yes. You may be interested by this site :http://www.ibiblio.org/e-notes/MSet/Contents.htm
There are there a lot of interesting results, and a few proofs...
Interesting stuff. It is kind of convenient that a
piece of ornamentation -- the real axis spike, happens to
lie along the real axis, as it makes it easy to study.
Based on the behavior of the bifurcation diagram there,
would it make any sense then to consider _all_ ornamentation
as the complex analogue of the chaotic region of the
real bifurcation diagram,
Yes, and it can even be proved, quite easily, for things like, say, the
branch ending at i.


How does one do that, even though that branch does not lie
along an "easy" path like how that real axis spike lies along
the path it does?

and the remaining parts of

the set (cardioid and all bulbs/sub-bulbs/sub-sub-bulbs/etc.)
as analogous to the "stable" region of the diagram? It
seems that with complex numbers, the very _shape_ of
the chaotic region has become incredibly intricate!
Also, has any research been done to determine the precise
nature of ornamentation (not bulbs, but ornamentation)?
Yes, a lot.


What is the stuff (ornamentation), then?

Why does ornamentation seem to contain motifs that
seem related to the periods of the bulbs it adorns?
It is in the ornamentation, after all, that all the grand Complexity
is found...
All this is well known (you should really get that book by Tan Lei). The
mùain tool is the correspondance between M and the Julia sets. What is
still mysterious is , say, the locally connected nature of M, or the
presence (or not) of other components that the midget M's ...

Here is a serious article on the Web which may clarify things for you :http://www.u-cergy.fr/rech/pages/tan/papers/similarity.ps(the ps format
may be somewhat annoying, I admit)

Hmm. So then is it like bits and pieces of the shapes of Julia sets
all
amalgamated together?

why the Julia sets look the way they do.

exactly
the period of the bulb influences the ornamentation the way it does.
Why that p-1 motif appears all throughout. What mechanism is
responsible for creating this?

What is meant by other components than midgets, anyway? You mean
like if, somewhere, deep in the Mandelbrot set, there lurks a
bizarre, frightening blob that looks like nothing like a midget?

Exactly (well, it could be a M_3 set, for instance (this is what results

mike3 a écrit :





On Nov 4, 4:03 am, Denis Feldmann <denis.feldmann.asuppri...@club-
internet.fr> wrote:
mike3 a écrit :

Yes. You may be interested by this site :http://www.ibiblio.org/e-notes/MSet/Contents.htm
There are there a lot of interesting results, and a few proofs...
Interesting stuff. It is kind of convenient that a
piece of ornamentation -- the real axis spike, happens to
lie along the real axis, as it makes it easy to study.
Based on the behavior of the bifurcation diagram there,
would it make any sense then to consider _all_ ornamentation
as the complex analogue of the chaotic region of the
real bifurcation diagram,
Yes, and it can even be proved, quite easily, for things like, say, the
branch ending at i.

How does one do that, even though that branch does not lie
along an "easy" path like how that real axis spike lies along
the path it does?

The main idea here is conformal transformations : the fact that
(locally) there is a multiplication by a (complex) constant leaving the
set invariant explains the spirals, for instance.

and the remaining parts of

the set (cardioid and all bulbs/sub-bulbs/sub-sub-bulbs/etc.)
as analogous to the "stable" region of the diagram? It
seems that with complex numbers, the very _shape_ of
the chaotic region has become incredibly intricate!
Also, has any research been done to determine the precise
nature of ornamentation (not bulbs, but ornamentation)?
Yes, a lot.

What is the stuff (ornamentation), then?

Mostly, images of the real axis by (multivalued) functions...

Why does ornamentation seem to contain motifs that
seem related to the periods of the bulbs it adorns?
It is in the ornamentation, after all, that all the grand Complexity
is found...
All this is well known (you should really get that book by Tan Lei). The
mùain tool is the correspondance between M and the Julia sets. What is
still mysterious is , say, the locally connected nature of M, or the
presence (or not) of other components that the midget M's ...

Here is a serious article on the Web which may clarify things for you :http://www.u-cergy.fr/rech/pages/tan/papers/similarity.ps(theps format
may be somewhat annoying, I admit)

Hmm. So then is it like bits and pieces of the shapes of Julia sets
all
amalgamated together?

Yes You got it

Which then of course raises the question of

why the Julia sets look the way they do.

Much easier : those are really self similar, and the mapping is
*exactly* z->z^2+c

But I'd like to know why

exactly
the period of the bulb influences the ornamentation the way it does.
Why that p-1 motif appears all throughout. What mechanism is
responsible for creating this?

This is harder (the idea is that roots of unity lurks there, but it is
not so easy to see why : a lot to do with, for instance, the doubling
cascade in the Feigenbaum sequences)

What is meant by other components than midgets, anyway? You mean
like if, somewhere, deep in theMandelbrotset, there lurks a
bizarre, frightening blob that looks like nothing like a midget?

Exactly (well, it could be a M_3 set, for instance (this is what results
from the sequence z--> z^3+c ), but perhaps something really new. Otoh,
almost evry specialist believes this is not teh case...)

And for "frightening", have a look at http://www.ansible.co.uk/writing/c-b-faq.html(the BLIT's Faq)

On Nov 4, 11:56 am, Denis Feldmann <denis.feldmann.asuppri...@club-
internet.fr> wrote:
mike3 a écrit :





On Nov 4, 4:03 am, Denis Feldmann <denis.feldmann.asuppri...@club-
internet.fr> wrote:
mike3 a écrit :
Yes. You may be interested by this site :http://www.ibiblio.org/e-notes/MSet/Contents.htm
There are there a lot of interesting results, and a few proofs...
Interesting stuff. It is kind of convenient that a
piece of ornamentation -- the real axis spike, happens to
lie along the real axis, as it makes it easy to study.
Based on the behavior of the bifurcation diagram there,
would it make any sense then to consider _all_ ornamentation
as the complex analogue of the chaotic region of the
real bifurcation diagram,
Yes, and it can even be proved, quite easily, for things like, say, the
branch ending at i.
How does one do that, even though that branch does not lie
along an "easy" path like how that real axis spike lies along
the path it does?
The main idea here is conformal transformations : the fact that
(locally) there is a multiplication by a (complex) constant leaving the
set invariant explains the spirals, for instance.


How does that explain the spiral, anyway? Does it have
something to do with the fact that multiplication by a
complex is a dilation+rotation, which is how a spiral
is formed (spiraling can be described as repeated
multiplication of a starting point by a complex number --
repeated dilation and rotation. This can be made continuous
by the definition of exponentiation of a complex base:
(a + bi)^x = e^(x log(a + bi)).)?

and the remaining parts of
the set (cardioid and all bulbs/sub-bulbs/sub-sub-bulbs/etc.)
as analogous to the "stable" region of the diagram? It
seems that with complex numbers, the very _shape_ of
the chaotic region has become incredibly intricate!
Also, has any research been done to determine the precise
nature of ornamentation (not bulbs, but ornamentation)?
Yes, a lot.
What is the stuff (ornamentation), then?
Mostly, images of the real axis by (multivalued) functions...


So then does the number of values of these multivalued "functions"
have something to do with the number of ornaments? And hence
the connections to the period, etc. of the bulbs?

Yes

What are these multivalued functions, anyway, and can they
be expressed in an equation (even if it involves lots of "weird"
special functions that are not elementary, or even if not
in closed form at all but only in "open" form like infinite
sums/products)?

And also, does this mean then that one can think of all
the ornaments as just "evolutions" of the real axis spike,
which can be thought of as a "prototype" for all other
ornamentation? What about the candelabras in "Candelabra
Valley" as I call it (cent. of star & approx c = -0.562202621 +
0.6428171490i, zoom into a midget to see them)? What
is the mechanism that is creating them? What do they
represent?






Why does ornamentation seem to contain motifs that
seem related to the periods of the bulbs it adorns?
It is in the ornamentation, after all, that all the grand Complexity
is found...
All this is well known (you should really get that book by Tan Lei). The
mùain tool is the correspondance between M and the Julia sets. What is
still mysterious is , say, the locally connected nature of M, or the
presence (or not) of other components that the midget M's ...
Here is a serious article on the Web which may clarify things for you :http://www.u-cergy.fr/rech/pages/tan/papers/similarity.ps(theps format
may be somewhat annoying, I admit)
Hmm. So then is it like bits and pieces of the shapes of Julia sets
all
amalgamated together?
Yes You got it


I kind of wondered if that was the case, but wasn't
quite sure.

Which then of course raises the question of
why the Julia sets look the way they do.
Much easier : those are really self similar, and the mapping is
*exactly* z->z^2+c


So then what determines the base shape of the J-set?
Like, for example, the J-sets along the positive real axis
beyond c = 1/4 have a base shape that looks like this:
(use a fixed-width font to view and forgive if my crummy
ASCII drawing is not exacting):

/----\
/-- --\
/ \
| /\ /\ |
| | > < | |
\-/ \-/

(the spirals of course wind infinitely far)

(and of course in the actual J set this is formed by
miniature copies of itself, plus there's an identical
mirror-image copy below it.)

But why those two spirals, and what's the significance
of the points they are centered on? Those points
seem to have real parts equal to +/- 1/2. (Any
idea what the imaginary parts are given by?)

But I'd like to know why

exactly
the period of the bulb influences the ornamentation the way it does.
Why that p-1 motif appears all throughout. What mechanism is
responsible for creating this?
This is harder (the idea is that roots of unity lurks there, but it is
not so easy to see why : a lot to do with, for instance, the doubling
cascade in the Feigenbaum sequences)


But how can it actually be done? And roots of unity lurk
where? So somehow the doubling bifurcation cascade not
only determines the _location_ of the chaotic region
but also the _shape_ of the chaotic region too, apparently
depending in some way (does anybody know the details or
is this a "mystery") upon the initial period the
cascade starts from (ie. the period of the main bulb)?

I believe all those details are known. Look at the reference I gave, for

What is meant by other components than midgets, anyway? You mean
like if, somewhere, deep in theMandelbrotset, there lurks a
bizarre, frightening blob that looks like nothing like a midget?
Exactly (well, it could be a M_3 set, for instance (this is what results
from the sequence z--> z^3+c ), but perhaps something really new. Otoh,
almost evry specialist believes this is not teh case...)


Not the case that what? The something other than a midget could
appear?

And for "frightening", have a look at http://www.ansible.co.uk/writing/c-b-faq.html(the BLIT's Faq)

What the !!!! is that about???

mike3 a écrit :





On Nov 4, 11:56 am, Denis Feldmann <denis.feldmann.asuppri...@club-
internet.fr> wrote:
mike3 a écrit :

On Nov 4, 4:03 am, Denis Feldmann <denis.feldmann.asuppri...@club-
internet.fr> wrote:
mike3 a écrit :
Yes. You may be interested by this site :http://www.ibiblio.org/e-notes/MSet/Contents.htm
There are there a lot of interesting results, and a few proofs...
Interesting stuff. It is kind of convenient that a
piece of ornamentation -- the real axis spike, happens to
lie along the real axis, as it makes it easy to study.
Based on the behavior of the bifurcation diagram there,
would it make any sense then to consider _all_ ornamentation
as the complex analogue of the chaotic region of the
real bifurcation diagram,
Yes, and it can even be proved, quite easily, for things like, say, the
branch ending at i.
How does one do that, even though that branch does not lie
along an "easy" path like how that real axis spike lies along
the path it does?
The main idea here is conformal transformations : the fact that
(locally) there is a multiplication by a (complex) constant leaving the
set invariant explains the spirals, for instance.

How does that explain the spiral, anyway? Does it have
something to do with the fact that multiplication by a
complex is a dilation+rotation, which is how a spiral
is formed (spiraling can be described as repeated
multiplication of a starting point by a complex number --
repeated dilation and rotation. This can be made continuous
by the definition of exponentiation of a complex base:
(a + bi)^x = e^(x log(a + bi)).)?

Mostly, yes, but it doesn't have to be continuous (the "curves" are
never really complete, but just the locus of the midget's centers) ; it
is enough that part of the set be send to another part by a conformal
transform, locally a dilation+rotation (the corresponding complex number
being, of course, the derivative of the function)

and the remaining parts of
the set (cardioid and all bulbs/sub-bulbs/sub-sub-bulbs/etc.)
as analogous to the "stable" region of the diagram? It
seems that with complex numbers, the very _shape_ of
the chaotic region has become incredibly intricate!
Also, has any research been done to determine the precise
nature of ornamentation (not bulbs, but ornamentation)?
Yes, a lot.
What is the stuff (ornamentation), then?
Mostly, images of the real axis by (multivalued) functions...

So then does the number of values of these multivalued "functions"
have something to do with the number of ornaments? And hence
the connections to the period, etc. of the bulbs?

Yes

What are these multivalued functions, anyway, and can they
be expressed in an equation (even if it involves lots of "weird"
special functions that are not elementary, or even if not
in closed form at all but only in "open" form like infinite
sums/products)?

No, those are simply the inverse of polynomials : f^(-n)(z), where
f(z)=z^2+z (actually, you need only quadratic approximations of those
polynomials f^(n))...

Those are Misiurewicz points. They appear when the fixed point of the
Julia (or Mandelbrot) sequence (f^(n)(z)=g(z)=z) is repulsive (|g'(z)|>1)

The calculation is not really hard in your case, as here z^2+c=z has the
solutions z=1/2 +/- i sqrt(4c-1)...

But I'd like to know why

exactly
the period of the bulb influences the ornamentation the way it does.
Why that p-1 motif appears all throughout. What mechanism is
responsible for creating this?
This is harder (the idea is that roots of unity lurks there, but it is
not so easy to see why : a lot to do with, for instance, the doubling
cascade in the Feigenbaum sequences)

But how can it actually be done? And roots of unity lurk
where? So somehow the doubling bifurcation cascade not
only determines the _location_ of the chaotic region
but also the _shape_ of the chaotic region too, apparently
depending in some way (does anybody know the details or
is this a "mystery") upon the initial period the
cascade starts from (ie. the period of the main bulb)?

I believe all those details are known. Look at the reference I gave, for
instance the cosinus approximation for midgets near -2...

What is meant by other components than midgets, anyway? You mean
like if, somewhere, deep in theMandelbrotset, there lurks a
bizarre, frightening blob that looks like nothing like a midget?
Exactly (well, it could be a M_3 set, for instance (this is what results
from the sequence z--> z^3+c ), but perhaps something really new. Otoh,
almost evry specialist believes this is not teh case...)

Not the case that what? The something other than a midget could
appear?

Yes

And for "frightening", have a look athttp://www.ansible.co.uk/writing/c-b-faq.html(theBLIT's Faq)

What the !!!! is that about???

Well, you dont seem to want to enter the really deep math stuff.

So, for
a change, I suggested you look at some other kind od deep stuff (check
BLIT in the Wikipedia
(http://en.wikipedia.org/wiki/BLIT_%28short_story%29, orhttp://www.infinityplus.co.uk/stories/blit.htm) , or " motif of harmful
sensation" , athttp://en.wikipedia.org/wiki/Motif_of_harmful_sensation)

On Nov 4, 11:47 pm, Denis Feldmann <denis.feldmann.asuppri...@club-
internet.fr> wrote:
mike3 a écrit :



Mostly, yes, but it doesn't have to be continuous (the "curves" are
never really complete, but just the locus of the midget's centers) ; it
is enough that part of the set be send to another part by a conformal
transform, locally a dilation+rotation (the corresponding complex number
being, of course, the derivative of the function)


But what circumstances must align so that the points will follow such
rotational paths, anyway? Like it sees as one goes down the cusps
that separate one bulb from another, it seems the "top" ornament of
the bulb is progressively twisted or rotated.

What is the stuff (ornamentation), then?
Mostly, images of the real axis by (multivalued) functions...
So then does the number of values of these multivalued "functions"
have something to do with the number of ornaments? And hence
the connections to the period, etc. of the bulbs?

Yes


But if they are the inverses of the Mandelbrot polynomials, as
you mention below, and indeed each value corresponds to a different
ornament, then we should expect to see ornaments that are only
powers of 2 (since the degree of the polynomials increases 2, 4, 8,
etc.). Unless of course roots repeat and repeated roots appear
regardless of the input parameter ("c") to them.

What are these multivalued functions, anyway, and can they
be expressed in an equation (even if it involves lots of "weird"
special functions that are not elementary, or even if not
in closed form at all but only in "open" form like infinite
sums/products)?
No, those are simply the inverse of polynomials : f^(-n)(z), where
f(z)=z^2+z (actually, you need only quadratic approximations of those
polynomials f^(n))...


So then they could theoretically be written down in
closed form, but you might need to use special functions,
like various types of hypergeometric functions, for example,
to write down the ones when the degree exceeds 5 (although
the expressions are of tremendous complexity.) But you
said you only needed quadratic approximations, which
means you should be able to write it down using only simple
radicals, no?

snip
Those are Misiurewicz points. They appear when the fixed point of the
Julia (or Mandelbrot) sequence (f^(n)(z)=g(z)=z) is repulsive (|g'(z)|>1)

The calculation is not really hard in your case, as here z^2+c=z has the
solutions z=1/2 +/- i sqrt(4c-1)...


How do you prove that solving z^2 + c = z though will give you those
points?

But how can it actually be done? And roots of unity lurk
where? So somehow the doubling bifurcation cascade not
only determines the _location_ of the chaotic region
but also the _shape_ of the chaotic region too, apparently
depending in some way (does anybody know the details or
is this a "mystery") upon the initial period the
cascade starts from (ie. the period of the main bulb)?
I believe all those details are known. Look at the reference I gave, for
instance the cosinus approximation for midgets near -2...


Which reference? You gave several in this discussion.

What is meant by other components than midgets, anyway? You mean
like if, somewhere, deep in theMandelbrotset, there lurks a
bizarre, frightening blob that looks like nothing like a midget?
Exactly (well, it could be a M_3 set, for instance (this is what results
from the sequence z--> z^3+c ), but perhaps something really new. Otoh,
almost evry specialist believes this is not teh case...)
Not the case that what? The something other than a midget could
appear?
Yes


Why is that, anyway?

You also mentioned that there was indeed a connection between midget
appearance and "mode locking", just as for why islands of stability
appear on the bifurcation diagram's chaotic region.

And for "frightening", have a look athttp://www.ansible.co.uk/writing/c-b-faq.html(theBLIT's Faq)
What the !!!! is that about???
Well, you dont seem to want to enter the really deep math stuff.

Where are you getting that idea from?

paper you posted a link to, by the way.

So, for
a change, I suggested you look at some other kind od deep stuff (check
BLIT in the Wikipedia
(http://en.wikipedia.org/wiki/BLIT_%28short_story%29, orhttp://www.infinityplus.co.uk/stories/blit.htm) , or " motif of harmful
sensation" , athttp://en.wikipedia.org/wiki/Motif_of_harmful_sensation)

mike3 a écrit :





On Nov 4, 11:47 pm, Denis Feldmann <denis.feldmann.asuppri...@club-
internet.fr> wrote:
mike3 a écrit :

Mostly, yes, but it doesn't have to be continuous (the "curves" are
never really complete, but just the locus of the midget's centers) ; it
is enough that part of the set be send to another part by a conformal
transform, locally a dilation+rotation (the corresponding complex number
being, of course, the derivative of the function)

But what circumstances must align so that the points will follow such
rotational paths, anyway? Like it sees as one goes down the cusps
that separate one bulb from another, it seems the "top" ornament of
the bulb is progressively twisted or rotated.

I am sorry, but you have to go really into the math there (and I am not
sure you want to do the effort) . The answer here, for instance, is the
exact value of g'(c), where c is a Misiurewicz point and g =f^(n), with
f(z)= z^2+z.

What is the stuff (ornamentation), then?
Mostly, images of the real axis by (multivalued) functions...
So then does the number of values of these multivalued "functions"
have something to do with the number of ornaments? And hence
the connections to the period, etc. of the bulbs?
Yes

But if they are the inverses of the Mandelbrot polynomials, as
you mention below, and indeed each value corresponds to a different
ornament, then we should expect to see ornaments that are only
powers of 2 (since the degree of the polynomials increases 2, 4, 8,
etc.). Unless of course roots repeat and repeated roots appear
regardless of the input parameter ("c") to them.

No, the connection is not about roots of the polynomial, but with
periods of the sequence : you have to check solutions of z=f^n(z), those
are indeed in number 2^n, but correspond to period n (and branching
factor n)

What are these multivalued functions, anyway, and can they
be expressed in an equation (even if it involves lots of "weird"
special functions that are not elementary, or even if not
in closed form at all but only in "open" form like infinite
sums/products)?
No, those are simply the inverse of polynomials : f^(-n)(z), where
f(z)=z^2+z (actually, you need only quadratic approximations of those
polynomials f^(n))...

So then they could theoretically be written down in
closed form, but you might need to use special functions,
like various types of hypergeometric functions, for example,
to write down the ones when the degree exceeds 5 (although
the expressions are of tremendous complexity.) But you
said you only needed quadratic approximations, which
means you should be able to write it down using only simple
radicals, no?

No, because approximations means you will get nothing better that
writing their numerical value. Anyway, why would one like to get
explicit values ? Already, the "bulbs" are (except for the main cardioid
and the perfect circle on its left) approximations of cardioids and
circles, whose exact equation is algebraic, but of a quickly increasing
degree

snip
Those are Misiurewicz points. They appear when the fixed point of the
Julia (or Mandelbrot) sequence (f^(n)(z)=g(z)=z) is repulsive (|g'(z)|>1)

The calculation is not really hard in your case, as here z^2+c=z has the
solutions z=1/2 +/- i sqrt(4c-1)...

How do you prove that solving z^2 + c = z though will give you those
points?

Now, this is a new one. If you are interested in proofs, you need *to
learn the math*

Still ,the fact that this works perfectly (i.e. gives
the correct "grapic" result) should hint that this method is reasonably
correct :-)

And why do the repetitions of the base shape spiral there?

Easy, Julia sets are very simple (compared to M), and those are the
pre-images of the main spiral

But how can it actually be done? And roots of unity lurk
where? So somehow the doubling bifurcation cascade not
only determines the _location_ of the chaotic region
but also the _shape_ of the chaotic region too, apparently
depending in some way (does anybody know the details or
is this a "mystery") upon the initial period the
cascade starts from (ie. the period of the main bulb)?
I believe all those details are known. Look at the reference I gave, for
instance the cosinus approximation for midgets near -2...

Which reference? You gave several in this discussion.

This one :http://www.ibiblio.org/e-notes/MSet/Contents.htm,
especially (for cosinus approx) :http://www.ibiblio.org/e-notes/MSet/Cosine.htm
(the whole site contains, I believe, *all* the answers to your questions
(but few proofs, they give only references to *that*)

What is meant by other components than midgets, anyway? You mean
like if, somewhere, deep in theMandelbrotset, there lurks a
bizarre, frightening blob that looks like nothing like a midget?
Exactly (well, it could be a M_3 set, for instance (this is what results
from the sequence z--> z^3+c ), but perhaps something really new. Otoh,
almost evry specialist believes this is not teh case...)
Not the case that what? The something other than a midget could
appear?
Yes

Why is that, anyway?

Too hard to explain. Linked to the universality of M.



You also mentioned that there was indeed a connection between midget
appearance and "mode locking", just as for why islands of stability
appear on the bifurcation diagram's chaotic region.

Yes, now a start could behttp://www.ibiblio.org/e-notes/MSet/Crisis.htm



And for "frightening", have a look athttp://www.ansible.co.uk/writing/c-b-faq.html(theBLIT'sFaq)
What the !!!! is that about???
Well, you dont seem to want to enter the really deep math stuff.

Where are you getting that idea from?

Because your questions look as if you never even studied things like
conformal maps, or read some serious papers on M. If I am wrong, I am
sorry...

I looked at the math-heavy



paper you posted a link to, by the way.

So, for
a change, I suggested you look at some other kind od deep stuff (check
BLIT in the Wikipedia
(http://en.wikipedia.org/wiki/BLIT_%28short_story%29, orhttp://www.infinityplus.co.uk/stories/blit.htm) , or " motif of harmful
sensation" , athttp://en.wikipedia.org/wiki/Motif_of_harmful_sensation)

On Nov 6, 12:32 am, Denis Feldmann <denis.feldmann.asuppri...@club-
internet.fr> wrote:
mike3
I am sorry, but you have to go really into the math there (and I am not
sure you want to do the effort) . The answer here, for instance, is the
exact value of g'(c), where c is a Misiurewicz point and g =f^(n), with
f(z)= z^2+z.


Well, if you could tell me where I could get started, then I might go
and undertake the effort.

mike3 a écrit :

On Nov 6, 12:32 am, Denis Feldmann <denis.feldmann.asuppri...@club-
internet.fr> wrote:
mike3
I am sorry, but you have to go really into the math there (and I am not
sure you want to do the effort) . The answer here, for instance, is the
exact value of g'(c), where c is a Misiurewicz point and g =f^(n), with
f(z)= z^2+z.

Well, if you could tell me where I could get started, then I might go
and undertake the effort.

The paper by Tan Lei is probably too hard (but the book is not). Try
the references given athttp://en.wikipedia.org/wiki/Mandelbrot_set#References(especiallyhttp://citeseer.ist.psu.edu/cache/papers/cs/28564/http:zSzzSzwww.math...
, but even this may be quite hard...)