Riemann surface for iteration of complex map?

Hi.

I once had a thread here where I mentioned the idea of using a Riemann
surface to "continuously iterate" a complex map:

http://groups.google.com/group/sci.math/msg/3a51089884ec50d2?hl=en&dmode=source

and got hung up on how to do the "addition". So I'm wondering: what if
maybe one doesn't need that -- but instead needs a different Riemann
surface? What if the "best" choice of Riemann surface depends on the
map we want to iterate continuously, i.e. even for the mapping z^2 +
c, it would depend on c?

First of all, some introduction: consider a holomorphic function f on
the complex plane, or perhaps the plane minus some isolated points
(not sure exactly what types of domain should be allowed). Then
consider a Riemann surface S that covers f's domain C-plane with
covering map mu, and another map g: S->S. These are all interrelated
by

mu(g(s)) = f(mu(s))

and we say "g casts f under mu".

Here I call f the "shadow" of g, or g a "bulk" or "expansion" of f,
and S an "iterative Riemann surface" or a "dynamical Riemann surface"
of f, for lack of better terms.

Note that in this space, iteration of g corresponds to that of f,
which is the property we exploit.

Now the question is, how can we find the other items for a given f-
map?

For some simple elementary functions, this is not too hard. Consider
the power map: f(z) = z^n, n a positive integer. When iterated, this
map gives f^k(z) = z^(n^k). However, multivaluedness problems pop up
for real k, and worse yet, for a single-valued "branch" we do not even
have a true dynamical system!

Yet we can define a bulk for f by removing z = 0 and using the Riemann
surface of the complex logarithm for the dynamical surface of f. On
this surface, points are pairs s = (r, theta), where r is a positive
real and theta any real. Then mu(s) = re^(itheta) is the covering map,
and the bulk is g(s) = (r^n, n theta). It is easy to see that this
will cast f. We can now produce iteration: g^k(s) = (r^(n^k), n^k
theta), and this _is_ a dynamical system.

For the linear family, i.e. f(z) = az + b, the DRS is just the complex
plane, C.

However, what about for other maps, like f(z) = z^2 - 2? This one also
has a closed-form iterate f^k(z) = cos(2^k arcsin(z/2)), that of
course is not dynamical for real k. But can we extend this to some
sort of Riemann surface in the same manner we did for the other map,
even if it is _not the same one_ (which may therefore obviate the need
to "define the subtraction of 2" that I was hung up with before.)? If
so, what's this Riemann surface, what is the g-map, and how do we
represent points on the surface? What about the map f(z) = z^2 - 1? It
has no closed-form iterate (at least not in terms of elementary
functions, or any widely-used nonelementary special functions, as far
as I know), but can we construct a dynamical Riemann surface for it?
How about for maps z^2 + c with c-parameters from several other areas
of the Mandelbrot set -- do these DRSes get more intricate and
interesting forms? What about transcendental mappings, say the
exponential family exp(cz), which is the basis of tetration? Does exp
(z) (natural exponential), for example, have an interesting-looking
DRS (DRSes?)? If so, what would the graph or graphs look like?
.

Relevant Pages

  • great and typical tommy1729 idea :)
    ... Note that sqrtsends points on the Riemann ... the z-plane to points on the Riemann surface. ... of z^2 + c in such a way that the inverse map is ... indeed mike. ...
    (sci.math)
  • Re: Symmetrized complex space
    ... I doubt it's a Riemann domain at all, that is no global map to C^3 that's ... Why would there exist such a map? ... C^3 is simply connected, it has no coverings and, in ... there exists a map Z --> C^3 which is locally a homeomorphism. ...
    (sci.math)
  • Re: use remove_if in map
    ... RemoveInvalid(std::setvalidset) ... valid map elements into another map (possibly via a vector, ... log m) where s is the size of the valid set, and m that of the map. ... Option 2 is to iterate over the map and check each element for validity. ...
    (microsoft.public.vc.stl)
  • Why is not "sub" necessary? What is the difference: block and expression?
    ... I'm trying to understand the map function in perl and have been studying ... When David calls his iterate function, why does he not need to use the ... keyword "sub"? ...
    (perl.beginners)
  • Re: Collection
    ... The purpose of a Map is to hold associations; using a Map as a List is overkill. ... If, OTOH, you do need the associative lookup and only occasionally need to loop through the entrySet, then just iterate over the entrySet. ... It is unlikely that you actually need both Mapness and sortedness. ...
    (comp.lang.java.programmer)