Embedding flow
Zdun (\cite[Example 1.2, page 69-71]{Zdun1979-Book}) proved that
there exists a map \(h\) with
\[h(b)=a
\]
such that the local linear map
\[\begin{equation}
f(x)=
\begin{cases}
sx,~~~ & x\in [0,a),\\
h(x),~~~ & x\in [a,b),\\
1+M(x-1),~~~ & x\in [b,1],
\end{cases}
\end{equation}
\]
is globally \(C^1\)-embeddable if and only if
\[\begin{equation}
a=\frac{\ln M}{\ln \left(\frac{M}{s}\right)},
\end{equation}
\]
where the constants \(a,b,s\) and \(M\) satisfy \(0<a<b<1\), \(0<s<1<M\).
\({\large Proof}\):
We recall the fact that \(f\) is linear on \([0,a]\) and \([b,1]\) if and only if so is \(\phi\) on \([0,a]\) and
\([f(b),1]\).
Due to \(f(b)=a\), we have
\[\begin{equation}
\phi(x)=
\begin{cases}
x,~~~ & x\in [0,a),\\
\frac{a}{a-1}(x-1),~~~ & x\in [a,1].
\end{cases}
\end{equation}
\]
Furthermore, by using the relation
\[\frac{\phi^\prime(1)}{\phi^\prime(0)}=\frac{\ln f^\prime(1)}{\ln f^\prime(0)}
\]
\[\Rightarrow
\frac{a}{a-1}=\frac{\ln M}{\ln s}
\]
\[\Rightarrow
\frac{a-1}{a}=\frac{\ln s}{\ln M}
\]
\[\Rightarrow
\frac{1}{a}=1-\frac{\ln s}{\ln M}=\frac{\ln (M/s)}{\ln M}
\]
\[\Rightarrow
a=\frac{\ln M}{\ln (M/s)}.
\]
