is invertible,}\\
\alpha & \mbox{ if $A=1$ and $0<\alpha\leq 1$,}\\
\alpha^{-1} & \mbox{ if $A=-1$ and $\alpha\geq 1$, and}\\
|A\alpha| & \mbox{ if $0<A\leq 1$ and $-1\leq \alpha <0$}.
\end{array}\right.$$

The invertible case is trivial, while the cases where $A=1$ follow because the map reduces to an invertible map on a strip height $\alpha$. The appendix gives a constructive proof for the case $0<A\leq 1$, $-1\leq \alpha<0$. Because $f_{-A,-1}$ is conjugate to $f_{A,-1}$ by interchanging $x$ and $y$ we have $$\ell(X^+)=\left\{ 
\begin{array}{cc}
|A|~, &  ~~ 0<|A|\le 1~, \alpha=-1\\
 1~,  &  ~~ A=0 ~, \alpha=-1.
\end{array}\right.$$ Note in particular that there is a discontinuity in $\ell(X^+)$ at $(\alpha,A)=(1,0)$.

The following examples (especially that in Section \[secegii\]) show that a general expression, if it can be obtained, is likely to be non-trivial.

$$\begin{array}{l|llll}
A & \alpha=2/3 & 1/2 & 1/3 & 1/5 \\
\hline
1/4 &  0.261  & 0.337  & 0.251  & 0.196 \\
1/3 &  0.235  & 0.326  & 0.265  & 0.199 \\
2/3 &  0.215  & 0.342  & 0.257  & 0.149 \\
5/4 &  0.628  & 0.626  & 0.419  & 0.252 \\
3/2 &  0.338  & 0.562  & 0.505  & 0.303 
\end{array}$$

Example I
---------

The simplest nontrivial rational map 