The most naive definition of a rational map $f$ from ${\Bbb C}^n$ to itself is $f=(f_1,\dots,f_n)$, where
each coordinate $f_j=p_j/q_j$ is a quotient of two polynomials, i.e., a rational function. A rational map
defines a holomorphic function outside the zeros of the denominators $q_j$. Thus the iterates
$f^n:=f\circ\cdots\circ f$ are well defined on a dense, open set. We say that $f$ is *birational* if it has
a rational inverse, which is to say that there is a rational map $g$ of ${\Bbb C}^n$ such that $f\circ g$ and
$g\circ f$ are the identity on a dense, open set.

Let us consider the imbedding of ${\Bbb C}^n\ni z=(z_1,\dots,z_n)\mapsto [1:z_1:\cdots:z_n]\in{\Bbb P}^n$.
We may then write $f=[f_0:\cdots:f_n]:{\Bbb P}^n\to{\Bbb P}^n$, where the coordinates $f_j$ are homogeneous
polynomials, all of the same degree $d$, and without common factor. We say, then, that $d$ is the *degree*
of $f$. If we wish to consider $f$ as a dynamical system, we wish to study the behavior of the iterates $f^n$ as
$n\to\infty$. If $\varphi$ is birational, then the
iterates $\{f^n\}$ are conjugate to the iterates of $\{g^n\}$, where $g:=\varphi^{-1}\circ f\circ\varphi$.

The most naive question is whether $f$ is nontrivial or not. A birational map
$L$ is linear if and only if it has degree 1. However, if $\varphi$ is birational, $\varphi^{-1}\circ L\circ\varphi$
is likely not to be linear, which reflects the fact that *degree is not a birational invariant.*
However, the *dynamical degree*, defined by
$$\text{ddeg}(f) := \lim_{n\to\infty}(\text{deg}(f^n))^{1/n}$$
is birationally invariant. The dynamical degree is a measure of "dynamical nontriviality" of $f$. The dynamical
degree has been discussed in many places and we refer to this
expository article for more information and references.

We consider the space ${\cal M}_q$ of complex $q\times q$ matrices. For $x=(x_{i,j})\in{\cal M}_q$, we let
$I(x) = (x_{i,j})$ denote matrix inversion. Thus $I$ is a birational map of ${\cal M}_q\cong {\Bbb C}^{q^2}$
and in fact is an involution, $I^2=\text{identity}$. Another involution is given by *Hadamard involution*
$J(x)=(X_{i,j}^{-1})$, which takes the inverse of the individual entries in the matrix. This is also known
as the *Cremona involution*. The involutions $I$ and $J$ arise as natural symmetries in Lattice
Statistical Mechanics, and a question that had arisen was the behavior of the composition $K:=I\circ J$ on the
projectivized space ${\Bbb P}({\cal M}_q)$. In particular, the projectivized subspaces ${\Bbb P}({\cal S}_q)$ (and resp.
${\Bbb P}({\cal C}_q)$) of symmetric (resp. cyclic) matrices are also of interest.
Based largely on numerical work, it was conjectured by
Angles d'Auriac, Maillard and Viallet
that
$$\text{ddeg}(K|_{{\Bbb P}({\cal M}_q)})= \text{ddeg}(K|_{{\Bbb P}({\cal S}_q)}) \ \ \text{ and }\ \
\text{ddeg}(K|_{{\Bbb P}({\cal M}_q)})= \text{ddeg}(K|_{{\Bbb P}({\cal C}_q)})$$
It is interesting for the mathematician to try to imagine how one could do numerical experiments
to estimate any of these quantities. And it is doubly surprising because their numerical results were
so accurate. For instance, if $q=10$ (not yet a large number), then the projectivized space of $q\times q$ matrices
has dimension $q^2-1=99$; the degree of $I$ is $q-1=9$, and the degree of $J$ is the same as the dimension
of the space, again $=99$. So a direct calculation of the degree of $K^n$ is already difficult if $n>1$.

The
approach that these authors took was to start with a matrix $m=(m_{i,j})$ with coprime integer entries. They
then computed the growth rate of the denominators in $K^n(m)$. In other words, they are using the fact
that $K$ is a rational map with integer coefficients, and they are computing the exponential rate of growth of
height. Subsequently, it has been shown by
Kawaguchi and Silverman
that the height is in fact bounded above by the dynamical degree. The reverse inequality (for a Zariski open
set of points) is still open.

The left hand equality conjectured above was proved by
Tuyen Trung Truong
and the right hand equality was proved by
Bedford and Truong

There are other subspaces of the space of matrices that are of interest. One of them is the space
${\cal CS}_q:={\Bbb P}({\cal S}_q\cap {\cal C}_q)$. It was observed by Angl\`es d'Auriac, Maillard and Viallet that
$\text{ddeg}(K|_{\cal CS})$ must be smaller than the general number. Then
Bedford and Kim found a formula for the dynamical degree
$\text{ddeg}(K|_{\cal CS})$ in terms of $q$. It turns out that the dependence on $q$ is rather complicated and
splits into 3 quite different cases: $(i)$ $q$ is odd, $(ii)$ $q$ is 2 times odd, and $(iii)$ $q$ is divisible by 4.

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