[FOM] Analyticity of half-exponentials
Lasse Rempe
L.Rempe at liverpool.ac.uk
Thu Apr 19 14:40:22 EDT 2007
> If Kneser created his increasing real analytic solution to f(f(x))= e^x
> by a process involving arbitrary choices, that's not very satisfactory,
> and the question remains how to define the "best" solution (by which
> *I* mean a solution constructed by a process which generalizes to find
> g such that g(g(x))= f(x), h such that h(h(x))=g(x), etc., with all
> functions increasing and preferably analytic).
>
I will make some comments related to this question, but I have to be
brief as I am short of time.
Let us begin with a function such as f(x) = exp(x) - 2. This function
has two real fixed points a and r, a<0<r. Hence the first fixed point is
attracting, i.e. f'(a)=exp(a)<1, while r is repelling, i.e. f'(r)>1.
(If you prefer, you can replace f by the map exp(x)-1 which you have
already mentioned, which has a single fixed point at 0, and replace the
Schroeder equation below by the Abel equation. This may in a way be more
natural.)
By Koenigs's theorem, f is conformally conjugate to the map z -> mu*z in
a complex neighborhood of r, where mu=f'(r). That is, there is a
conformal map phi defined near r with phi(r)=0 and
phi(f(z)) = mu phi(z).
This function phi is unique up to multiplication by a complex constant
(e.g. we can make phi unique by requiring phi'(r)=1). As mentioned
before, this gives rise to a half-iterate of the exponential which is
analytic. A priori, this function will only be defined on a neighborhood
of r, but clearly we can extend it to the entire interval [r,infinity).
Moreover, since \phi is unique up to multiplication by a constant, there
are no arbitrary choices in the construction, and the resulting
half-iterate is monotonic, as required.
Now consider the case of the actual exponential map exp. As has been
mentioned, the fact that exp has no real fixed points means that we
cannot apply this method directly. However, here is a potential way to
construct a half-iterate.
The map exp on the real line is conjugate to the map f above, restricted
to some open interval (x_0,infty), with x_0 > r. Moreover, if we require
the conjugacy to be reasonable nice (e.g. to have orbits under f and
under exp be related in a way that is Holder), then this conjugacy is
unique, and hence deserves to be called natural. We can now use this
conjugacy to transfer the half-iterate of f to a half-iterate of the
exponential.
There are two questions, however, to which I do not know the answer:
a) Is the conjugacy described real-analytic? If not, then we have no
reason to expect our half-iterate, however natural it may be, to be
real-analytic. This is, in fact, related to a more general question
about whether 'dynamic rays' (or 'hairs') of exponential maps are
real-analytic curves. This question is still open (however, Viana proved
a long time ago that these curves, and in particular the conjugacy we
consider, is C^infty).
b) Is the half-exponential obtained independent of the initial choice of
the map f? I.e., if we instead considered exp(z)-3 etc., would we get
the same result?
This turned out to be rather less brief than intended, so I think I need
to leave off here.
Lasse
--
--------------------------------------------------------------
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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