Re: Fractional iteration again (sorry)

On 19 Jun, 04:07, Rotwang <sg...@xxxxxxxxxxxxx> wrote:
Hi all

In a thread I started the other day the question came up of when a
holomorphic bijection f between two regions of C could be given by the
integral curve of a holomorphic function on some region of C, that is
when there exists v(z) such that

f (z) = f_v (z,1)

where f_v (z,t) satisfies

f_v (z,0) = z

and

d/dt f_v (z,t) = v (f_v (z,t))

(here v is assumed to be defined on a large enough domain that the
above makes sense). It occurs to me now that the question amounts to
asking whether there exists a way to compose f with itself "t times"
for t in [0,1], in the following sense: if we can find a set of
holomorphic functions f^t (z), t in [0,1], such that

f^0 (z) = z

f^1 (z) = f (z)

f^s (f^t (z)) = f^(s + t) (z) (call this the "group requirement")

and f^t(z) is differentiable wrt t then we can define v by v (z) = d/
dt f^t (z) at t = 0; letting f_v (z,t) = f^t (z) satisfies the
defining equations for f_v (subject to some assumptions about the
possibility of analytically continuing v to a larger domain).
Conversely if f is given by f_v (z,1) for some v then defining f^t (z)
= f_v (z,t) satisfies the defining equations for f^t.

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?
.

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