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.
\( \chi(\log(z))=\chi(z)/c \)
\( \chi'( c)=1 \)
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.
