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

[Lasse Rempe]

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

-----Original Message-----
From: Epstein, Adam
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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