Tetration Forum

In order to understand it myself and as a service for the non-German community, in the following posts I will provide the construction of Kneser (for an analytic super logarithm) which he gives in his paper:

Hellmuth Kneser, Reelle analytische Lösungen der Gleichung $\phi(\phi(x))=e^x$ und verwandter Funktionalgleichungen. J.Reine Angew. Math. 187, (1949), 56-67

I will proceed in several posts because my time is currently quite limited, and so I will give account of succeeding portions of his construction.

As mentioned in the title of his paper Kneser originally seeks for an analytic solution $\phi$ of the functional equation $\phi(\phi(x))=e^x$ (This functional equation was considerably discussed at the conference of the German mathematician's association, October 1941 in Jena. So there the idea was born to construct an analytic solution.)

As is very well-known the construction of a half-iterate can be reduced to the solution of the Abel equation, which Kneser gives here in a bit more generality:
$\psi(f(x))=\psi(x)+\beta$ .

If one has such a solution $\psi$ then it is easy to construct the half iterate by:
$\phi(x)=\psi^{-1}(\frac{\beta}{2}+\psi(x))$
because then
$\begin{align*} \phi(\phi(x)) &=\psi^{-1}(\frac{\beta}{2}+\psi(\psi^{-1}(\frac{\beta}{2}+\psi(x)))\\ &=\psi^{-1}(\frac{\beta}{2}+\frac{\beta}{2} +\psi(x))\\ &=\psi^{-1}(\beta+\psi(x))\\ &= \psi^{-1}(\psi(f(x))\\ &=f(x) \end{align*}$

Closely related to the Abel equation is the Schroeder equation:
$\chi(f(x))=\gamma \chi(x)$
If one has a solution of the Schroeder equation, then $\psi(x)=\log(\chi(x))$ is a solution of the Abel equation with $\beta=\log(\gamma)$ , because:
$\psi(f(x))=\log(\chi(f(x))=\log(\gamma)+\log(\chi(x))=\log(\gamma)+\psi(x)$

And one can also derive the fractional iterates directly from the Schroeder function by:
$f^{t}(x)=\chi^{-1}(\gamma^t\chi(x))$ .

The good thing is now that for holomorphic functions with fixed point $c$ and $a:=f'( c)\neq 0,1$ there is always the so called principal Schroeder function (holomorphic in a vicinity of $c$ ) which satisfies the Schroeder equation for $\gamma=a$ and which satisfies $\chi'( c)=1$ .

There are two possibilities to construct this principal Schroeder function. First by development of the powerseries in $c$ and chosing the coefficients of $\chi$ such that they satisfy the Schroeder equation.
And second by the limit
$\chi(x)=\lim_{n\to\infty} \frac{f^n(x)-c}{a^n}$

As you can easily verify if $\chi$ is a solution of the Schroeder equation then $b\chi(z)$ is also a solution for any constant $b$ . They are called "regular" (by Szekeres) if $\chi$ is the principal Schroeder function.
Those regular solutions are characterized by that $\chi^{-1}(\gamma^t \chi(x))$ is holomorphic in $c$ .

As a first step Kneser computes this principal Schroeder function $\chi$ of $\exp(z)$ at $\exp$ 's first fixed point $c\approx0.318+1.337i$ in the upper half plane, the fixed point nearest to the real axis. $\chi(z)$ is however not real on the real axis, so he determines some mapping properties of $\chi$ and later manipulates $\chi$ to become real and analytic on the real axis.

So up to now we have a Schroeder function $\chi$ in a vicinity $H_0$ of the fixed point $c$ . This function satisfies:
$\chi(\log(z))=\chi(z)/c$
$\chi'( c)=1$

The next thing Kneser does is to analytically continue $\chi$ from this small vicinity $H_0$ to the whole upper halfplane without the points $\exp^n(0)$ , that is:
$H=\{z\in\mathbb{C}: \Im(z)\ge 0, z\neq \exp^n(0), n= 0,1,\dots\}$ .
He first verifies that $\lim_{n\to\infty} \log^n(z) = c$ for each $z\in H$ (where the cut of the logarithm is the usual one, i.e. $-\pi<\Im(\log(z))\le \pi$ .)

And then he continues the function along the increasing sets $H_n$ , which contain all the points $z$ such that $\log^n(z)$ is contained in that initial vicinity $H_0$ of $c$ .

The inverse function (in a vicinity of $c$ ) $\chi^{-1}$ satisifies:
$\chi^{-1}(cz)=\exp(\chi^{-1}(z))$ and hence can be continued to the whole complex plane, i.e. is an entire function.

To see the properties of $\chi$ on $H$ he considers the following lines and areas. Each increasing index number indicates the application of exponentiation, for example $A_1=\exp(A_0)$ , $H_{-1}=\log(H_0)$ . The lines are without end points, the areas are without boundaries.

These areas are mapped by $\chi$ to (the letters indicate the source area):

Now $\chi(H_{-1})\cup \chi(H_0)\cup \chi(H_1)$ is simply connected and does not contain 0 (only in the boundary). Hence it is possible to define a holomorphic logarithm $\underline{\log}$ on that domain, he defines
$\psi(z)=\underline{\log}(\chi(z))$ on $H_{-1}\cup H_0\cup H_1$ . Which then satisfies
$\psi(e^z)=\psi(z)+\underline{\log}( c)=\psi(z)+c$ .

And this is the image under $\psi$ , considering that $\underline{\log}$ is biholomorphic:

Define $L_0=\psi(H_0)$ and define $L_n=L_0+nc$ , then we have $L_{-1}=\psi(H_{-1})$ and $L_1=\psi(H_1)$ . Define $L=\bigcup_{n=-\infty}^\infty L_n$ . So one can see that the boundary of $L$ consists only of the cyan and violet arcs, hence the points of the real axis in $H$ are mapped to the boundary of $L$ . This property is used in the final step of Kneser's construction, which follows in my next post.

As the last step we use the Riemann mapping theorem, to map the area $L$ biholomorphically to the upper halfplane, which maps the boundary to the real axis. By some reason (to be explained) the corresponding function $\Psi$ indeed satisfies the Abel equation.

But we do that in two steps first we map $L$ to the unit disk $\mathbb{E}$ via $\alpha: L\leftrightarrow\mathbb{E}$ and then we map $\mathbb{E}$ to the upper (open) halfplane $\mathbb{H}$ via $\beta:\mathbb{E}\leftrightarrow\mathbb{H}$ .

The existence of $\alpha$ is guarantied by the Riemann mapping theorem. The translation $z\mapsto z+c$ maps $L$ to $L$ without having a fixed point, hence the corresponding mapping in $\mathbb{E}$ , $\tau:\alpha(z)\mapsto \alpha(z+c)$ maps the unit disk into the unit disk without a fixed point. It is well known that such a function needs to be linear fractional, i.e. of the form $\tau(z)=\frac{d_1 z+d_2}{d_3 z +d_4}$ . Kneser shows that must be parabolic, that means it has only one fixed point $p$ at the boundary of $\mathbb{E}$ .

To map the unit disk to the upper half plane we can again use a linear fractional transformation $\beta(z)$ . There are enough parameters to chose it such that $p$ is mapped to infinity and such that $\beta(\tau(z))=z+1$ .

Now we can define
$\Psi(z)=\beta(\alpha(\psi(z))$ which has the property
$\Psi(e^z)=\beta(\alpha(\psi(z)+c))=\beta(\tau(\alpha(\psi(z)))=1+\Psi(z)$ .
and is biholomorphic on the interior of $H_{-1}\cup H_0\cup H_1$ .

The interior of $H_{-1}\cup H_0\cup H_1$ is mapped to some area bordering on the real line. The boundary on the real axis is hence mapped to the real axis. More precisely the interval $(\log(\Re( c)),e^{|c|})$ is mapped to the real axis. By the Schwarz reflection principle it can be continued to the complex conjugate of $H_{-1}\cup H_0 \cup H_1$ especially it is analytic on $(\log(\Re( c)),e^{|c|})$ and can from there be continued to a vicinity of the whole real axis by $\Psi(e^x)=\Psi(x)+1$ and $\Psi(\ln(x))=\Psi(x)-1$ .

bo198214

bo198214

bo198214

bo198214