# [FOM] [Fwd: FW: Analyticity of half-exponentials]

Lasse Rempe L.Rempe at liverpool.ac.uk
Thu Apr 19 19:54:13 EDT 2007

I am forwarding the following message on behalf of my colleague Adam
Epstein, who seems to have trouble posting it.

-----Original Message-----
Sent: Wed 18/04/2007 22:23
To: fom at cs.nyu.edu
Subject: Analyticity of half-exponentials

>> Is there a monotonic real analytic function defined  on the
>> non-negative real numbers such that f(f(x)) = 2^x, or f(f(x))=e^x?

It is well-known (though I have no ready reference) that the exponential
function is not a composite in the semigroup of entire functions C->C.
That is, if g and h are entire with g o h = exp then either g or h is
affine (of the form az+b).

From this it follows that e^z, 2^z, e^z-1 etc. have no entire iterative
square-roots.

>If I look at a modification of the question and ask for a function f
>such that f(f(x)) = (e^x - 1) instead of e^x, then I can actually build
>a formal power series that uniquely solves the functional equation, but
>which has some negative coefficients (and is therefore probably
>non-monotonic) and whose radius of convergence I cannot calculate

Consider f_a(z) = a^z -1  where a>1.

For a \new e there are two fixed points: 0 and some real number zeta_a
\neq 0. Here, zeta_a is positive < = > a<e
For a = e there is a unique, 'double' fixed point at 0.

Consider first a>e. The local theory of fixed points of analytic maps
gives a unique real-analytic vectorfield defined near 0, for which f_a
is the time 1 map. The functional equation which characterizes this
vectorfield also guarantees its analytic continuation to (zeta_a,
infinity), and its nonvanishing there. Thus, the time 1/2 map of the
flow is the desired 'best solution' on a neigborhood of the nonnegative
real-axis.

This analysis rests on two facts: that 0 is a fixed point, and
repelling: the derivative has absolute value strictly greater than 1.

For a<e one can do something similar, but now starting with the
'linearization' (embedding in a local flow) near the repelling point
zeta_a > 0, and then extending as previously, to a real-analytic map on
(0,infinity).

One can play similar tricks to get flows, half-iterates, etc  on
(a>e)  (-infinity, 0)
(a<e)  (-infinity, zeta_a)

now using the fact that the other fixed point (0 for a<e, zeta_a for
a>e) is attracting: |derivative|<1.

Now, for various subtle reasons, there is no guarantee that the two
solutions, defined on the positive/negative real axes, fit together. In
the general situation (a monotonic smooth map with two fixed points, one
attracting, the other repelling) they do not. This phenomenon is not
something which can be read off from Taylor expansions.

The case a=e is hardest. Here the local theory gives real-anaytic
vectorfields, whence fractional iterates, to the left and to the right
of 0. But again, for related subtle reasons, one cannot expect them to
fit together (or even be, separately, real analytic AT 0). In this
particular case I would expect not, but checking this could involve
consideration of even more global properties of the map - that it is an
entire function, having a Julia set, etc.

--
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Dr. Lasse Rempe
Dept. of Math. Sciences, Univ. of Liverpool, Liverpool L69 7ZL
Office 505; tel. +44 (0)151 794 4058, fax +44 (0)151 794 4061
http://pcwww.liv.ac.uk/~lrempe
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