Re: Fractional iteration again (sorry)

Thank you very much for your reply David, and sorry it took me so long
to say so. Comments follow.

On 23 Jun, 08:55, ah...@xxxxxxxxxxxxxxxxxxx (David Libert) wrote:
Rotwang (sg...@xxxxxxxxxxxxx) writes:
On 19 Jun, 04:07, Rotwang <sg...@xxxxxxxxxxxxx> wrote:

[deletions...]

Based on the recent discussions it seems that there is no known way to
define non-integer iteration in general, but I was wondering whether
anybody here knows whether it can always be defined in the restricted
setting given here, or alternatively whether there are any no-go
theorems which show that it can't.

Of particular interest is the case where f (z) = exp (z), for z in {z
in C | 0 <= im z < 2 pi} since this is an important transformation in
the book I'm reading. In this case f^t (z) would be a generalisation
of tetration to non-integer height...

Sorry, this is wrong. I mean a generalisation of iterated
exponentiation to non-integer height. Does anybody know if such a
thing exists?

This question came up here and in other recent threads. I have not read
all of the large volume of writing about this in all threads, so I may have
overlooked something, but from what I have so far seen this has not yet been
answered.

I don't know the answer either, nor have I found one on the Internet.
I will note some things related to the question, some I think I have proven
myself, others found.

[...]

By smooth I mean all finite iterated derivatives exist everywhere and are continous.

For f : R -> R non-decreasing and smooth there are #R many g non-decreasing and smooth
with f = g o g.

Sounds plausible. I got as far as showing how to construct an infinite
number of continuous and piecewise smooth functions f: |R -> |R such
that f o f = exp. This didn't really get me any closer to finding an
analytic function defined on the complex plane though.

[...]

For f non-decreasing, you can keep repeating the .5 construction of new g's above. You
can define dyadic rational exponentiation on these, by using the contruction to take .5 's
and ordinary integer powers to take multiples. The right things commute for this to be
well-defined.

The ordering relations among these are as expected and monotonic, so by taking appropriate
sups or infs of dyadic rational exponents you can extend this to real exponents of iteration.

Unfortunately, if f started as continuous or smooth, you have each of the slices is
respectivelty continous or smooth,

Is this easy to prove?

but overall operation considered over variable exponent
need not be continuous or smooth in that exponent variable. Because each time you halved
the exponent you made a non-canonical choice of how to do so, and could have taken wildly
different functions.

Indeed none of the functions produced in any of these claims are canonical. The internal
proofs made arbitrary choices along the way, that is why they produced many functions
instead of one.

On a different note, it has occured to me that the relationship
between continuous iterations of functions and holomorphic flows
generating those functions is not as simple as I had assumed. In
particular one can have a function defined on a region of the complex
plane which is not 1-to-1, but which nonetheless appears to have a
continuous iteration and can therefore be seen as being generated by a
vector field. For example with f(z) = z^2 we get (in the notation of
the OP)

f^t(z) = z^(2^t) = exp(2^t*ln z)

v(z) = df^t(z)/dt_{t = 0} = ln 2*z*ln z

and the apparent contradiction is resolved by noting that v(z) and
f^t(z) are not holomorphic on C but on the Riemann surface of the
complete analytic function of ln(z), and the flows they generate on
that surface are 1-to-1 (sorry if this sounds handwavy, I'm not sure
how to make it precise). Assuming that this is a general feature of
continuous holomorphic iterations, it would seem that any set of
functions interpolating between the identity function on the strip {z
in C | 0 < im z < 2*pi} and the exponential function would need to be
generated by a holomorphic vector field on some Riemann surface.
.

Relevant Pages

  • Re: Fractional iteration again (sorry)
    ... In this case f^t would be a generalisation ... of tetration to non-integer height... ... can define dyadic rational exponentiation on these, by using the contruction to take .5 's ... sups or infs of dyadic rational exponents you can extend this to real exponents of iteration. ...
    (sci.math)
  • Re: Fractional iteration again (sorry)
    ... of tetration to non-integer height... ... I mean a generalisation of iterated ... I Googled on fractional iteration and it does seem to be a big topic. ... previously heard of tetration in making higer iterations of exponentiation possibibly ...
    (sci.math)
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    ... turned out to have a simple closed form from trigonometric functions. ... a closed form, however, could there be some sort of power series or ... other non-closed-form formula for its general iteration, ... That way one could see a smooth, ...
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  • tetration and logaritms
    ... and from looking at the first terms of this taylor series we see that we can get series reversion which of course gives exp- 1. ... tetration is by this construction smooth ... ... expis the continu iteration of e*x. ...
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