That continuous iteration of quadratic maps thingy again.
mike3
I read this website, which talks about turning the quadratic map into
a continuous-time dynamical system (a flow):
http://www.xs4all.nl/~westy31/ContFract/Continuous_iteration_of_fractals.html
It mentions the problem of multivaluedness of the resulting "maps"
when one uses a simple power-series based iteration approach
(actually, the same matrix method that was discussed in my last thread
on this topic.), and suggests the use of a Riemann surface to get
around the problem. The reason this multivaluedness appears is because
the maps involved are not injective (they lose information).
Namely, the idea is that instead of working in the z-plane, that is, z
= x + yi, we instead work in a space of pairs u = (r, theta) (r is a
nonnegative real and theta is any real), which correspond to complex
numbers under the map Z(u) = re^(itheta), i.e. these pairs can be
thought of as complex numbers in polar form, but with the new twist
that the second item theta is allowed to have infinite range, and so
preserving information. For example, in this space, if u_1 = (r_1,
theta_1) and u_2 = (r_2, theta_2), then
u_1 * u_2 = (r_1 r_2, theta_1 + theta_2)
with no "mod 2pi". So if we "square" u, i.e.
u^2 = (r^2, 2 theta)
then we can recover u easily, as the resulting map is one-to-one
(injective).
However, the difficulty was in defining a suitable _addition_
operation: how do we define u_1 + u_2, without resorting to an
arctangent for which we do not know what the right value to choose?
I'm wondering, then, what _is_ the best addition function for the job?
Consider, for example, the function f(t) = z + e^(it), for some
complex z. For z far enough from the origin (|z| >= 1), the phase
angle never completes a full 360-degree (2pi-radian) circle. Yet if z
is small enough, it will complete a full circle, and so then the
result should be that the theta-parameter of our polarspace-addition
of the pairs (r, theta) for z and (1, t) for e^(it) should keep
increasing as t is increased without bound, and keep decreasing if t
is decreased without bound.
So how then would one add the pairs (r, theta) and (1, t) in this
space so as to preserve all that? (More generally, (r_1, theta_1) and
(r_2, theta_2).)
alainv...@gmail.com
> Hi.
>
> I read this website, which talks about turning the quadratic map into
> a continuous-time dynamical system (a flow):
>
Dear Mike,
I always wished getting out of these
heavy representations,
Regards,
Alain
Gottfried Helms
> Hi.
...
:-) exactly what I was speculating about over the last
days - the idea of preserving information seems
simply to be natural for the iteration-problem.
To stick at the residual of the rotationparameter phi
(mod 2 Pi) looks then similar to using modular (clock)
arithmetic instead of real arithmetic. It's just
using the last digit of a Pi-based numbersystem only...
How could one preserve the information? Yes, the
problem to keep it in the phi.parameter causes problems
for the addition-operation. A third parameter?
I had no good idea so far, so I disposed that subject
up to next rainy day... :-)
But, let's see, with what you'll be coming up -
Gottfried
mike3
wrote:
What is the problem with using these representations, anyway?
mike3
The addition is what is the big stumper here, though.
It just doesn't seem to have any obvious method that is
also "natural" enough.
mike3
> Dear Mike,
>
> I always wished getting out of these heavy representations,
>
> Regards,
> Alain
How about if one used a simpler Riemann surface, namely that of the
inverse
of z^2 + c, i.e. that of sqrt(z - c)? This is simpler as it has only 2
leaves.
Note that sqrt(z - c) sends points on the Riemann surface to points on
the
z-plane, and the map z^2 + c could then be thought of as sending
points on
the z-plane to points on the Riemann surface. The rub here would then
be:
1. finding a simple, easy representation for the Riemann surface's
points
2. finding a way to make the map z^2 + c send points on the Riemann
surface
*to other points on the Riemann surface* in the most "natural"
possible way
(as free of arbitrary choices as possible), while also remaining a
dynamical
system and capturing the dynamics of z^2 + c (i.e. the map must be
continuous,
and also if u is some point on the Riemann surface and Q(u) its image
under the
Riemann-surface-based "quadratic map", then C(u)^2 + c = C(Q(u)),
where C(u)
is the function sending points u on the Riemann surface to their
corresponding
points in C, the complex plane.).
It's 2 that is the rub here. Is there any way this is possible?
Mariano Suárez-Alvarez
> On Mar 7, 5:35 am, "alainvergh...@gmail.com" <alainvergh...@gmail.com>
> wrote:
>
> > On 7 mar, 11:15, mike3 <mike4...@yahoo.com> wrote:
> <snip>
> > Dear Mike,
>
> > I always wished getting out of these heavy representations,
>
> > Regards,
> > Alain
>
> How about if one used a simpler Riemann surface, namely that of the
> inverse
> of z^2 + c, i.e. that of sqrt(z - c)? This is simpler as it has only 2
> leaves.
> Note that sqrt(z - c) sends points on the Riemann surface to points on
> the
> z-plane, and the map z^2 + c could then be thought of as sending
> points on
> the z-plane to points on the Riemann surface. The rub here would then
> be:
>
> 1. finding a simple, easy representation for the Riemann surface's
> points
Let c be in C. Then the Riemann surface of f(z) = sqrt(c - z)
is simply the set
S = { (z, w) in C x C : w^2 + c = z }
The "complete function" corresponding to f (that is,
the "extension" of f to its Riemann surface) is the
function F : S --> C such that F(z, w) = w.
The covering p : S --> C is given by p(z, w) = z,
and it is (outside of the ramification point 0)
a two-sheeted covering.
-- m
mike3
But this doesn't solve the problem... how to represent the "dynamics"
of z^2 + c in such a way that the inverse map is single valued as
well.
This involves iteration (repeated application) of both z^2 + c and sqrt
(z - c).
amy666
good math of mariano.
but not so helpfull ?
> >
>
> But this doesn't solve the problem... how to
> represent the "dynamics"
> of z^2 + c in such a way that the inverse map is
> single valued as
> well.
> This involves iteration (repeated application) of
> both z^2 + c and sqrt
> (z - c).
indeed mike.
very recently however , i , the great tommy1729 :) , gave a new twist to iterations.
i considered extending the domain !!
w^2 + c = f(f(w))
f(w) maps complex w to an EXTENSION of the reals , different from C !
if also 2*w = f'(f(w)) * f'(w)
is satisfied , this might be what you look for !!
regards
the great :)
tommy1729
amy666
the real problem is perhaps very different.
im not uncomfortable with other numbers than " complex " but the issue is
you really want to solve the half iterate w^2 + c for complex number ??
if so , you cannot ignore the complex numbers !
also be aware that whatever you use needs a certain degree of algebraic closedness.
e.g.
if you need - anywhere in the computation - a number that satisfies ' number ' ^2 = -1 ,
you need a kind of ' i '
that might conflict with your desire of not using complex numbers ;
you number system might not contain an 'a' that satisfies 'a'^2 = -1
hence -> BIG TROUBLE.
if you use complex numbers and another number system , and those are suppose to be consistant with eachother and differentiable , then you are using an extension ...
which is basicly my own solution idea , see my other post in this thread.
regards
tommy1729
Mariano Suárez-Alvarez
I never claimed to solve the problem. In fact, I do
not even understand *what* the problem is.
I just wanted to show you that the Riemann surface
has a very simple description in this case.
-- m
mike3
OK. Hmm...
mike3
Thank you for the description of the representation of the
Riemann surface, though.
By the way, the "problem" is this. f(z) z^2 + c is not an injective
map, because z^2 is not an injective map. This means it has no proper
inverse map.
Therefore, one asks, "is there some domain D on which
we can define a bijective function G: D->D such that
c(G(u)) = f(c(u)), where c(u) is a surjective function c: D->C?"
(This function gives the complex number corresponding to a point in D.
Note that it itself need not be injective. f(z) on the z-plane loses
information, you know.) That way, iteration of G, i.e. G(u), G(G(u)),
G(G(G(u))), etc. provides a dynamical system that could possibly be
turned into a flow somehow. And I was wondering if some Riemann
surface could be used for this domain D, maybe that of f^-1(z).
This investigation was inspired by this web page:
http://www.xs4all.nl/~westy31/ContFract/Continuous_iteration_of_fractals.html
and I was wondering if there was any way to do it without the
discontinuity, or at least to minimize arbitrary choices as much as
possible (the location of said discontinuity was chosen arbitrarily
by the arctan2 branch used in computing the "sum" of points on the
surface when represented as polars (r_1, theta_1) and (r_2, theta_2)
with unbounded theta.).
Addendum: Now that I think about it, I don't think the Riemann
surface of f^-1 is suitable. Consider two applications of f: f(f(z)).
The inverse now needs a Riemann surface that is different, having 4
"leaves". It would seem that for the general problem, we'd need one
with infinitely many "leaves", like the one mentioned on the web page.
Although then comes the problem of suitably defining addition which
is needed to extend the map...