def_Reverse.tex

On $\Lambda=\mathbb{R}^3_+$, the map
\[
F (x_1,x_2,x_3) = (x'_1,x'_2,x'_3)
\]
is defined by
\[
\left(\begin{array}{r}
	x'_{\pi 1} \\
	x'_{\pi 2} \\
	x'_{\pi 3}
\end{array}\right) =
\begin{cases}
\left(\begin{array}{l}
    x_{\pi 1}\\
    x_{\pi 2}\\
    x_{\pi 3}-x_{\pi 1}-x_{\pi 2}
\end{array}\right)
    &\mbox{if } x_{\pi 3}>x_{\pi 1}+x_{\pi 2}\\
\frac{1}{2}
\left(\begin{array}{r}
     -x_{\pi 1}+x_{\pi 2}+x_{\pi 3}\\
      x_{\pi 1}-x_{\pi 2}+x_{\pi 3}\\
      x_{\pi 1}+x_{\pi 2}-x_{\pi 3}
\end{array}\right)
     &\mbox{otherwise.}
\end{cases}
\]
where $\pi\in\mathcal{S}_3$ is the permutation of $\{1,2,3\}$ such that
$x_{\pi 1}<x_{\pi 2}<x_{\pi 3}$
\cite{arnoux_symmetric_2015}.
\subsection{Matrix Definition}
The subcones are
\begin{align*}
	\Lambda_i &= \{(x_1,x_2,x_3)\in\Lambda\mid 
	2x_i > x_1+x_2+x_3\},
	&i\in\{1,2,3\},\\
	\Lambda_4 &= \Lambda\setminus(\Lambda_1\cup\Lambda_2\cup\Lambda_3)
\end{align*}
The matrices are given by the rule
\[
    M(\mathbf{x}) = M_i
    \qquad\text{ if and only if }\qquad
    \mathbf{x}\in\Lambda_i.
\]
The map $F$ on $\Lambda$ and
the projective map $f$ on
$\Delta=\{\mathbf{x}\in\Lambda\mid\Vert\mathbf{x}\Vert_1=1\}$ are:
\[
    F(\mathbf{x}) = M(\mathbf{x})^{-1}\mathbf{x}
    \qquad\text{and}\qquad
    f(\mathbf{x}) = \frac{F(\mathbf{x})}{\Vert F(\mathbf{x})\Vert_1}.
\]