\magnification=\magstep1

\def\CC{{\bf C}}
\def\RR{{\bf R}}
\def\DD{{\bf D}}
\def\TT{{\bf T}}
\def\oper{\diamond}


\noindent Given a complex number $a\in\CC$, we shall write
$$M_a(z) := {z+a \over 1+\overline{a} z} \ , \qquad
L_a z := a z$$ for the standard M\"obius transformation taking the
origin into the point $a$ and the multiplication operator with the
factor $a$, respectively. Recall that the family ${\rm Aut}(\DD)$ of
all surjective biholomorphic maps of the unit disc $\DD$ has the
form
$${\rm Aut}(\DD) =
\{ L_k \circ M_a \vert\DD : \ \vert a\vert <1 , \ \vert k \vert=1\}
= \{ M_a \circ L_k \vert\DD : \ \vert a\vert <1 , \ \vert k
\vert=1\}$$ and the decompositions $L_k \circ M_a $ respectively
$M_a \circ L_k$ are unique. Notice that
$$M_a^{-1} = M_{-a} \ ,\quad L_k \circ M_a \circ L_{\overline{k}} = M_{ka} .$$
Let ${\rm Fix}(f) := \{ x:\ f(x)=x\}$ denote the set of all fixed
points of any mapping $f:X\to X$.


\bigskip
\noindent Concerning the fixed points of the automorphisms in
$${\rm Aut}(\DD\cup\TT) :=
\{ L_k \circ M_a \vert\DD\cup\TT : \ \vert a\vert <1 , \ \vert k
\vert=1\} = \{ M_a \circ L_k \vert\DD\cup\TT : \ \vert a\vert <1 , \
\vert k \vert=1\}$$ of the closed unit disc $\DD\cup\TT$ where
$\TT:=\{ z: \vert z\vert=1\}$ denotes the unit circle, we can assert
the following.


\bigskip
\noindent \bf 1. Lemma \rm [Stach\'o op.9]. \it The M\"obius
transformation \ $M:=L_{\exp(it)} \circ M_a \vert \DD\cup\TT$ \ has
\medskip
\item{{\rm a)}} a unique fixed point if \ \
$\vert a \vert < \vert \sin(t/2)\vert = \vert( e^{it}-1)/2 \vert$,
\medskip
\item{{\rm b)}} two distinct fixed points lying in \ $\TT$ \ iff \ \
$\vert a \vert > \vert \sin(t/2)\vert$,
\medskip
\item{{\rm c)}} a unique fixed point lying in \ $\TT$ \ iff \ \
$\vert a \vert = \vert \sin(t/2)\vert$.

\bigskip
\noindent \rm The following Lemma is folklore.


\bigskip
\noindent \bf 2. Lemma. \it The group \ ${\rm Aut}(\DD\cup\TT)$ \ is
transitive on \ $\DD$ \ and 2-transitive on \ $\TT$. \hfill\break In
particular $M_a:0\mapsto a$ for any $a\in\DD$ and for any pair
$b,c\in\TT$ with $b\ne c$ there is a M\"obius transformation
$N_{b,c}$ such that $N_{b,c}:-1\mapsto b ,\, 1\mapsto c$.


\bigskip
\noindent \rm Direct calculation shows the following


\bigskip
\noindent \bf 3. Lemma. \it For a M\"obius transformation $M\in{\rm
A}(\DD\cup\TT)$ we have
\medskip
\item{{\rm a)}} ${\rm Fix}(M) =\{ 0\}$ \ iff \ $M=L_k$ for some $k\in\TT$,
\medskip
\item{{\rm a)}} ${\rm Fix}(M) =\{-1,1\}$ \ iff \ $M=M_a$ for some $a\in(-1,1)$,
\medskip
\item{{\rm a)}} ${\rm Fix}(M) =\{ 1\}$ \ iff \ $M=L_{\exp(it)} \circ M_{\sin(t/2)}$ for some $t\in(-\pi,\pi)$.


\bigskip
\noindent \rm In terms of flows
$\exp\big(tV(z){\partial\over\partial z}\big): \DD\cup\TT \ni
z\mapsto \big[ x_t:\ d x_\tau/d\tau = V(x_\tau, \ x_0=z \big]$ of
vector fields we can reformulate Lemma 3 as follows.

\bigskip
\noindent \bf 3$^\prime$. Lemma. \it The subfamilies of ${\rm
Aut}(\DD\cup\TT)$ with common fixed points $\{ 0\},\,\{-1,1\}$ resp.
$\{ 1\}$ are one-parameter Lie subgroups. Namely
\medskip
\item{{\rm a)}} $\big\{ M\in{\rm Aut}(\DD\cup\TT) : {\rm Fix}(M) =\! \{0\} \big\} \!=
\{ L_{e^{it}} : \ t\in\RR\} = \big\{ \exp\big[t (iz)
{\partial\over\partial z} \big] : \ t\in\RR \big\}$,
\medskip
\item{{\rm b)}}
$\big\{ M\in{\rm Aut}(\DD\cup\TT) :  {\rm Fix}(M) \!\!=\! \{\pm 1\}
\big\} \!= \big\{ M_{\tanh(t)} : \ t\in\RR \big\} = \big\{ \exp\big[
t (iz) {\partial\over\partial z} \big] : \ t\in\RR \big\}$,
\medskip
\item{{\rm c)}} $\big\{ M\in{\rm Aut}(\DD\cup\TT) :  {\rm Fix}(M) =\! \{1\} \big\} \!=
\big\{ \exp\big[ t \big( i(1-z)^2 \big) {\partial\over\partial z}
\big]: \ t\in\RR \big\} = \hfill\break \phantom{x} \hskip52mm =
\Big\{ L_{{1-it \over 1+1t}} \circ M_{{ it \over 1-it }} : \ t\in\RR
\Big\}$.


\bigskip
\noindent \rm The following is trivial.

\bigskip
\noindent \bf 4. Lemma. \it If the maps $f,g:X\to X$ commute then
$g({\rm Fix}(f)) \subset {\rm Fix}(f)$. In particular if
$f,g:X\leftrightarrow X$
\medskip
\item{{\rm a)}}
then \ ${\rm Fix}(f) = \{ x_0\}$ \ implies \ ${\rm Fix}(g) = \{
x_0\}$,
\medskip
\item{{\rm b)}}
${\rm Fix}(f) = \{ x_1,x_2\}$ \ implies \ ${\rm Fix}(g) = \{
x_1,x_2\}$ \ or \ $g:x_1\leftrightarrow x_2$.


\bigskip
\noindent \rm Direct calculation shows the following

\bigskip
\noindent \bf 5. Lemma. \it Given \ $a\in\RR$, \ the M\"obius
transformation \ $M\in{\rm Aut}(\DD\cup\TT)$ \ commutes with \ $M_a$
\ iff we have \ $M=M_b$ \ for some $b\in\RR$.



\bigskip
\noindent \bf 6. Theorem. \it The maximal commutative subfamilies of
${\rm Aut}(\DD\cup\TT)$ are one-parameter Lie-subgroups. Namely
given a maximal commutative subfamily \ ${\cal A} \subset {\rm
Aut}(\DD\cup\TT)$, one of the following alternatives holds:
\medskip
\item{{\rm a)}} ${\cal A} \!=
\{ L_a \circ L_{e^{it}} \circ L_{-a} : \ t\in\RR\} = L_a \circ
\big\{ \exp\big[t (iz) {\partial\over\partial z} \big] \circ L_{-a}
: \ t\in\RR \big\}$ \hfill\break for some $a\in\RR$,
\medskip
\item{{\rm b)}}
${\cal A} \!= \big\{ N_{b,c} \circ M_{\tanh(t)} \circ N_{b,c}^{-1} :
\ t\in\RR \big\} = \big\{ N_{b,c} \circ \exp\big[ (1 - z^2)
{\partial\over\partial z} \big] \circ N_{b,c}^{-1} : \ t\in\RR
\big\}$ \hfill\break for some $c\ne b,c\in \TT$,
\medskip
\item{{\rm c)}} ${\cal A} \!=
\big\{ L_k \circ \exp\big[ t \big( i(1-z)^2 \big)
{\partial\over\partial z} \big] \circ L_{\overline{k}}: \ t\in\RR
\big\} = \Big\{ L_k \circ L_{{1-it \over 1+1t}} \circ M_{{ it \over
1-it }} \circ L_{\overline{k}} : \ t\in\RR \Big\}$ \hfill\break for
some $k\in\TT$.




\bigskip
\noindent \bf Proof. \rm According to Lemmas 1-2-3, the
transformations $F\in {\rm Aut}(\DD\cup\TT)$ admit at most two fixed
points and for $F\in {\rm Aut}(\DD\cup\TT)$ we have the alternatives

\medskip
\item{{\rm (i)}} ${\rm Fix}(F) = \{ a\}$ where $a\in\DD$ with $F$ necessarily in the form
$F=M_a \circ L_k \circ M_{-a}$ for some $k\in\TT$;

\medskip
\item{{\rm (ii)}} ${\rm Fix}(F) = \{ a,b\}$ where $b\ne a,b\in\TT$ with $F$ necessarily in the form
$F=N_{a,b} \circ M_r \circ N_{a,b}^{-1}$ for some $r\in(-1,1)$ and
$N_{a,b}\in{\rm Aut}(\DD\cup\TT)$ such that $N_{a,b}(1)=a$,
$N_{a,b}(-1)=b$;

\medskip
\item{{\rm (iii)}} ${\rm Fix}(F) = \{ c\}$ where $c\in\TT$ with $F$ necessarily in the form
$F=L_{\exp(it)} \circ M_{\exp(iv)\sin(t/2)}$ for some
$t\in(-\pi,\pi)$ and $v\in\RR$ such that
$c=\displaystyle{{\exp(iv)\over 2} \, {\exp(it)-1 \over
\sin(t/2)}}$.

\medskip
\noindent Taking into account Lemmas 4-5, we see that the fixed
point are the some for the transformations within a commutative
subfamily of ${\rm Aut}(\DD\cup\TT)$. Hence the statement is
immediate by Lemma $3^\prime$.


\bye
