# Salem/Pisot Numbers in the Weyl Spectrum

Kyounghee Kim  Department of Mathematics
Florida State University
Tallahassee, FL 32308<EMAIL_ADDRESS>
###### Abstract.

In this article, we define the orbit data of birational maps of
$\mathbf{P}^{2}(\mathbb{C})$ and show that the orbit data determine the
dynamical degree by providing the minimal polynomials of the dynamical degree
in terms of orbit data. Using this, we determine all Salem numbers and Pisot
numbers that appear in the Weyl Spectrum of Coxeter group $W_{n}$ associated
to $E_{n}$ and the Dynamical Spectrum of $\mathbf{P}^{2}(\mathbb{C})$.

###### Key words and phrases:

Weyl Spectrum, Dynamical Spectrum, Salem number, Pisot number, Orbit Data

###### 2023 Mathematics Subject Classification:

20F55,37F99,14E07

##  1\. Introduction

A Pisot number $\tau$ is an algebraic integer greater than $1$ such that all
Galois conjugates, except $\tau$, have modulus $\lneq 1$. A Salem number
$\lambda$, defined by Salem [20], is an algebraic integer such that its
minimal polynomial is of degree at least $4$, and all Galois conjugates except
$\lambda$ and $\lambda^{-1}$ have modulus $1$. Salem numbers and Pisot numbers
are closely related in the sense that the closure of the Salem numbers is the
union of the set of Salem numbers and the set of Pisot numbers. Thus, every
Pisot number is a limit of a sequence of Salem numbers. However, it is not
known that every limit of a non-eventually-constant sequence of Salem numbers
is a Pisot number. Salem numbers and Pisot numbers have appeared in many
different areas of Mathematics. (See [9, 4, 22, 10, 1].) For a detailed
history of Salem and Pisot numbers, we refer to a comprehensive survey by
Smyth [21].

Salem numbers and Pisot numbers appear in Dynamical Systems. By Diller and
Favre [6] and Blanc and Cantat [3], it is known that the exponential growth
rate of the degree of a birational map on $\mathbf{P}^{2}(\mathbb{C})$ under
iterations are given by either a Salem number or a Pisot number. This
exponential growth rate of the degree is called the dynamical degree. However,
not every Salem/Pisot number can be realized. In fact, Blanc and Cantat [3]
show that infinitely many Pisot numbers bigger than the golden ratio can not
be realized as the exponential growth rate of the degree of a birational map
on $\mathbf{P}^{2}(\mathbb{C})$.

By Diller and Favre [6], McMullen [17], Uehara [23], Kim [13], and Blanc and
Cantat [3], we know that the set of all dynamical degrees of birational maps
on $\mathbf{P}^{2}(\mathbb{C})$, is identical to the Weyl Spectrum, the
closure of the set of spectral radii of elements in Coxeter group associated
to $E_{n},n\geq 10$. The main goal of this article is to determine Salem and
Pisot numbers in the Weyl Spectrum and study their properties as dynamical
degrees of birational maps on $\mathbf{P}^{2}(\mathbb{C})$.

The Coxeter group $W_{n},n\geq 3$ associated to $E_{n}$ diagram in Figure 1 is
a group generated by $s_{i}$’s with relations

$(s_{i}s_{j})^{m_{i,j}}=1\qquad\text{where}\ \ m_{i,j}\ =\ \left\\{\
\begin{aligned} &1\ \ \text{ if }i=j\\\ &2\ \ \text{ if there is no edge
joining }s_{i}\text{ and }s_{j}\\\ &3\ \ \text{ if there is an edge joining
}s_{i}\text{ and }s_{j}\\\ \end{aligned}\right.$

$s_{1}$
---
$s_{2}$
---
$s_{3}$
---
$s_{4}$
---
$s_{5}$
---
$s_{n-1}$
---
$s_{0}$
---
Figure 1. Coxeter Graph $E_{n}$

It is known [14, 15] that the spectral radius of $\omega\in W_{n}$ is either
$1$ or a Salem number. Not every Salem number is realized as a spectral radius
of an element of $W_{n}$. However, McMullen [16] showed that the known
smallest Salem number $\lambda_{10}$ is, in fact, realized as a spectral
radius of the Coxeter element in $W_{10}$. The known smallest Salem number
$\lambda_{10}\approx 1.176$ is the number known as Lehmer’s number whose
minimal polynomial is given by the Lehmer’s polynomial

$L(t)=t^{10}+t^{9}-t^{7}-t^{6}-t^{5}-t^{4}-t^{3}+t+1.$

McMullen also showed that if $\omega\in W_{n},n\geq 10$ is not conjugate into
a proper subgroup isomorphic to a Coxeter group $W_{m},m<n$, then the spectral
radius of $\omega$ is always bigger than equal to $\lambda_{10}$, in other
words, $\lambda_{10}$ is the smallest Salem number in $\Lambda_{W}$ where
$\Lambda_{W}$ is the closure of the set of all spectral radii in $W_{n},n\geq
4$

$\Lambda_{W}:=\overline{\\{\lambda(\omega):\omega\in W_{n},n\geq 4\\}}$

and $\lambda(\omega)$ denote the spectral radius of $\omega$. We call this set
$\Lambda_{W}$ is the Weyl spectrum.

One interesting relation between Salem and Pisot numbers is that the limit
point of Salem numbers is either a Salem number or a Pisot number. It follows
that the Weyl spectrum is a union of $\\{1\\}$, the set of Salem number
$\Lambda_{S,W}\subset\Lambda_{W}$, and the set Pisot numbers
$\Lambda_{P,W}\subset\Lambda_{W}$ in the Weyl spectrum.

$\Lambda_{W}=\\{1\\}\sqcup\Lambda_{S,W}\sqcup\Lambda_{P,W}$

On the other hand, we can see the Weyl specturm $\Lambda_{W}$ as the dynamical
spectrum on $\mathbf{P}^{2}(\mathbb{C})$. Let us consider a birational
transformation $f=[f_{1}:f_{2}:f_{3}]$ on $\mathbf{P}^{2}(\mathbb{C})$ with
degree $d$ polynomials $f_{1},f_{2},f_{3}$ without a common factor. Then, the
degree of $f$ is given by the common degree $d$ of coordinate polynomials and
the dynamical degree $\lambda(f)$ is defined by

$\lambda(f)\ =\ \lim_{n\to\infty}(\deg f^{n})^{1/n}.$

Because of possible common factors in the $n$-fold composition of $f$, we have
$\deg f^{n}\leq(\deg f)^{n}$, and thus, computing the dynamical degree of a
given birational map is a challenging task. For a given birational map
$f\in\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))$, Diller and Favre [6] showed that
there is always a rational surface $X$ obtained by blowing up
$\mathbf{P}^{2}(\mathbb{C})$ along a finite set of points such that the
induced map $F:X\to X$ satisfies

$\lambda(F)=\lambda(f)=\rho(F_{*})$

where $\rho$ denote the spectral radius and $F_{*}$ is the induced linear map
on $\text{Pic}(X)$. Also, by Diller and Favre [6] and Blanc and Cantat [3], we
have the following dichotomy:

######  Theorem 1.1.

[6, 3] Suppose $f\in\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))$ is a birational
transformation with the dynamical degree $\lambda(f)>1$. Then we have one of
the following:

  * •

The dynamical degree $\lambda(f)$ is a Salem number. In this case, $f$ is
birationally equivalent to an automorphism.

  * •

The dynamical degree $\lambda(f)$ is a Pisot number. In this case, $f$ is NOT
birationally equivalent to an automorphism.

Let us denote $\Lambda_{\mathbb{C}}$ the set of all dynamical degrees of
birational maps on $\mathbf{P}^{2}(\mathbb{C})$. Then by above Theorem 1.1, we
have $\Lambda_{\mathbb{C}}$ is the disjoint union of the set
$\Lambda_{S,\mathbb{C}}$ of Salem numbers in $\Lambda_{\mathbb{C}}$ and the
set $\Lambda_{P,\mathbb{C}}$ of Pisot numbers in $\Lambda_{\mathbb{C}}$. By
Nagata, [18, 19] if $F$ is an automorphism on a rational surface $X$ over
$\mathbb{C}$ with infinite order, then the induced cation $F_{*}$ on
$\text{Pic}(X)$ is an element of $W_{n}$ where $n+1$ is the Picard rank of
$X$. Thus we have $\Lambda_{S,\mathbb{C}}\subset\Lambda_{S,W}$. McMullen [17],
Uehara [23], and Kim [13] show that, in fact,

$\Lambda_{S,\mathbb{C}}=\Lambda_{S,W}.$

To simplify our notations, let us use $\Lambda_{S/P}$ for both
$\Lambda_{S/P,W}$ and $\Lambda_{S/P,\mathbb{C}}$. Also, Blanc and Cantat [3]
show that the dynamical spectrum $\Lambda_{\mathbb{C}}$ is well-ordered and
closed. It follows that

$\Lambda_{W}=\Lambda_{\mathbb{C}}.$

In this article, using the theories in the Coxeter group and birational maps
on $\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))$, we show the following:

######  Theorem A.

The Weyl spectrum $\Lambda_{W}$ is given by

$\Lambda_{W}=\\{\rho(\chi_{M,\bar{n}}:M\in\bigcup_{d}(\mathcal{M}_{d}\cup\mathcal{M}^{\prime}_{d}),\bar{n}\in\mathbb{N}^{k},k=|M|,d\geq
2\\}$

where $\rho$ denote the largest real root, $|M|$ denote the size of $M$,
$\chi$ is defined in (3.7), $\mathcal{M}_{d},d\geq 2$ is given recursively by
the Theorem 5.6, and $\mathcal{M}^{\prime}_{d},d\geq 2$ is defined in (5.1).

For a birational map $f\in\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))$, there is
the set $\mathcal{E}(f)$ of exceptional curves mapped to a point and the set
$\mathcal{I}(f)$ of points of indeterminacy. Let us set $\mathcal{E}^{*}(f)$
be the set of exceptional curves mapped to the points of indeterminacy under
finite iterations.

$\mathcal{E}^{*}(f)=\\{C\in\mathcal{E}(f):f^{n}(C)\in\mathcal{I}(f),\dim
f^{n+1}(C)=1,n\geq 1\\}$

For each $C_{i}\in\mathcal{E}^{*}(f)$, let $n_{i}$ be the smallest number $n$
of iterations such that $\dim f^{n+1}(C)=1$. We call the ordered tuple
$\mathcal{O}_{\ell}(f)=(n_{1},\dots,n_{k})$, $k=|\mathcal{E}^{*}(f)|$ of these
numbers the orbit lengths of $f$. We have the following monotonicity result.

######  Theorem B.

Suppose $f\in\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))$ with $\lambda(f)>1$.
Then, the dynamical degree $\lambda(f)$ increases in orbit lengths. In other
words, suppose $f,g\in\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))$ are birational
maps on $\mathbf{P}^{2}(\mathbb{C})$ satisfying
$\mathcal{E}^{*}(f)=\mathcal{E}^{*}(g)$ and $\lambda(f),\lambda(g)>1$. If
$\mathcal{O}_{\ell}(f)=(n_{1},\dots,n_{k})\lneq\mathcal{O}_{\ell}(g)=(m_{1},\dots,m_{k})$
in the sense that $n_{i}\leq m_{i}$ for all $i$, then $\lambda(f)<\lambda(g)$.

The smallest accumulation point in Pisot numbers is the golden ratio. With the
construction given by Diller [5], we show that

######  Theorem C.

All Pisot numbers smaller than or equal to the golden ratio are realized as
dynamical degrees of birational maps on $\mathbf{P}^{2}(\mathbb{C})$.

This article is organized as follows: In Section 2, we discuss basic facts for
birational maps on $\mathbf{P}^{2}(\mathbb{C})$ and define orbit data for
birational maps. In Section 3, we discuss the properties of the Coxeter group
and give a formula for the characteristic polynomial of any element in the
Coxeter group. The proof of Theorem A is in Section 5, and the proof of
Theorem B is in Section 4. Section 6 lists orbit data corresponding to Pisot
numbers smaller than equal to the golden ratio and the proof of Theorem C.

##  2\. Brational maps on $\mathbf{P}^{2}(\mathbb{C})$

Suppose $X$ is a rational surface over $\mathbb{C}$. Let $f:X\dasharrow X$ be
a birational transformation on $X$. Then, there is a well-defined linear map
$f_{*}:\text{Pic}(X)\to\text{Pic}(X)$ and the dynamical degree $\lambda(f)$ of
$f$ is defined by

$\lambda(f)=\lim_{n\to\infty}\rho((f^{n})_{*})^{1/n}$

where $\rho$ denotes the spectral radius.

We say a birational transformation $f$ on a rational surface $X$ is
algebraically stable if

$(f^{n})_{*}=(f_{*})^{n}\qquad\text{for all }n\in\mathbb{Z}.$

Thus, we see that if a birational transformation $f$ is algebraically stable,
then the dynamical degree $\lambda(f)$ of $f$ is given by the spectral radius
of the induced linear action $f_{*}$ on $\text{Pic}(X)$.

###  2.1. $\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))$

Let $f$ be a birational map on $\mathbf{P}^{2}(\mathbb{C})$. There are three
homogeneous polynomials $f_{1},f_{2}$ and $f_{3}$ of the same degree $d$ and
without a non-constant common factor such that

$f[z_{1}:z_{2}:z_{3}]\ =\
[f_{1}(x_{1},x_{2},x_{3}):f_{2}(x_{1},x_{2},x_{3}):f_{3}(x_{1},x_{2},x_{3})].$

In this case, the degree of $f$ is given by the common degree of $f_{i}$’s

$\deg(f)=d.$

Because of a possible common factor in the iterations of $f$, the degree of
$f^{2}=f\circ f$ is not necessarily equal to $\deg(f)^{2}$. For example, for
the cremona involution

$\sigma[z_{1}:z_{2}:z_{3}]=[z_{2}z_{3}:z_{1}z_{3}:z_{1}z_{2}]$

we have $\deg(\sigma)=2$ and $\deg(\sigma^{2})=1$. In fact, we have

$\deg(f^{n})\leq(\deg f)^{n}\ \ \ \text{ for all }n\geq 1$

Since $\text{Pic}(\mathbf{P}^{2}(\mathbb{C}))$ is generated by a generic line,
the dynamical degree $\lambda(f)$ is also given by

$\lambda(f)=\lim_{n\to\infty}(\deg f^{n})^{1/n}.$

Determining $\deg f^{n}$ for all $n$ is very challenging because of possible
cancellations in coordinate functions. However, Diller and Favre [6] showed
that one can always construct a rational surface $X$ by blowing up a finitely
many (possibly infinitely near) points on $\mathbf{P}^{2}(\mathbb{C})$ such
that the induced map on $X$ is algebraically stable. Since the dynamical
degree is invariant under a birational conjugacy, we have
$\lambda(f)=\rho(F_{*})$.

###  2.2. Orbit Data

Suppose $f\in\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))$ is given by homogeneous
polynomials $f_{1},f_{2}$ and $f_{3}$ of the same degree $d$ without a non-
constant common factor. Then we have exceptional locus $\mathcal{E}(f)$ is
given by the zero set of $\det Df$. Let $C_{1},\dots,C_{m}$ be the degree
$c_{i}$ irreducible components of $E(f)$ counted with multiplicity

$\mathcal{E}(f)=\\{C_{1},\dots,C_{m}\\},\quad\text{with}\ \ \deg
C_{i}=c_{i}\\}.$

By counting the degree, we see that $\sum d_{i}=3(d-1)$. Also, there is an
indeterminacy locus

$\mathcal{I}(f)=\\{z\in\mathbf{P}^{2}(\mathbb{C}):f_{1}(z)=f_{2}(z)=f_{3}(z)=0\\}.$

For any curve $V\in\mathbf{P}^{2}(\mathbb{C})$, let us denote
$f(V)=\overline{f(V-\mathcal{I}(f))}$, the strict transform of $V$ under $f$.
Let us define a subset $\mathcal{E}^{*}(f)\subset\mathcal{E}(f)$ by

$\mathcal{E}^{*}(f)=\\{C_{i}\in\mathcal{E}(f):\text{dim}f^{n+1}(C_{i})=1\ \
\text{ for some }n\geq 1\\}.$

It follows that if $C_{i}\in\mathcal{E}^{*}(f)$ then
$f^{n}(C_{i})\in\mathcal{I}(f)$.

With renaming $C_{i}$’s, if necessary, we may assume that
$\mathcal{E}^{*}(f)=\\{C_{1},\dots,C_{k}\\}$ for some $k\leq m$. For
$i=1,\dots,k$, let $n_{i}$ be the smallest positive integer $n$ such that
$\text{dim}f^{n+1}(C_{i})=1$

$n_{i}=\min\\{n\geq 1:\text{dim}f^{n+1}(C_{i})=1\\}.$

Also let $\mathcal{O}_{i}$ be the orbit of $C_{i}$ under $f$

$\mathcal{O}_{i}=\\{p_{i,j}=f^{j}(C_{i}),1\leq j\leq n_{i}\\}.$

It follows that

$\\{p_{i,1},1\leq i\leq
k\\}\subset\mathcal{I}(f^{-1})\quad\text{and}\quad\\{p_{i,n_{i}},1\leq i\leq
k\\}\subset\mathcal{I}(f).$

Without loss of generality, we may assume that
$|\mathcal{O}_{i}|\leq|\mathcal{O}_{i+1}|$ for all $i=1,\dots,k-1$. Let us
consider a sequence of blowups

$\pi:X=X_{k}\to X_{k-1}\to\cdots\to X_{1}\to X_{0}=\mathbf{P}^{2}(\mathbb{C})$

where $\pi_{i}:X_{i}\to X_{i-1}$ is a blowup along the finite set of points in
the strict transform of $\mathcal{O}_{i}=\\{p_{i,j},1\leq j\leq
n_{i}\\}\subset X_{i-1}$. By Diller and Favre [6], the lift $F:X\dasharrow X$
of $f$ is algebraically stable.

To simplify the notation, let us also use $\mathcal{O}_{i}$ for its strict
transform in $X_{i-1}$. Then we have

$\text{Pic}(X)=\langle e_{0},e_{i,j},1\leq j\leq n_{i},1\leq i\leq k\rangle$

where $e_{0}$ is the class of a line and $e_{i,j}$ is the class of the total
transform of $p_{i,j}$. It follows that the class $D_{i}$ of the strict
transform of $f^{n_{i}+1}C_{i}$ is given by a linear combination of $e_{0}$
and $e_{i,1}$’s

(2.1)
$D_{i}=F_{*}(e_{i,n_{i}})=d_{i}e_{0}-\sum_{j}m_{j,i}\,e_{j,1},\qquad\text{where
}\ \ d_{i}>0\text{, and }m_{j,i}\geq 0.$

Since a generic line in $\mathbf{P}^{2}(\mathbb{C})$ intersects $C_{i}$ at
$c_{i}$ points (counted with multiplicity), we have

(2.2) $F_{*}(e_{0})=de_{0}-\sum_{i}c_{i}\,e_{j,1}$

and

(2.3) $F_{*}(e_{i,j})=e_{i,j+1}\qquad\text{for }1\leq i\leq k,1\leq j\leq
n_{i}-1.$

Using above equations (2.1) - (2.3), we have $F_{*}$ on $\text{Pic}(X)$.

Let us use the ordered basis $\langle
e_{0},e_{1,1},e_{1,2},\dots,e_{1,n_{1}},e_{2,1},\dots,e_{k,1},\dots,e_{k,n_{k}}\rangle$.
Except the columns corresponding $e_{0}$,$e_{i,n_{i}},1\leq i\leq k$, all
other columns of the matrix representation of $F_{*}$ are given by all zeros
except $1$ in the sub-diagonal entry. Let a $(k+1)\times(k+1)$ matrix $M$ be a
principal sub-matrix of $F_{*}$ with the columns and rows corresponding
$e_{0},e_{i,n_{i}},1\leq i\leq k$ and let $\bar{n}=(n_{1},n_{2},\dots,n_{k})$
be a $k$ tuple of strictly positive integers such that
$n_{i}=|\mathcal{O}_{i}|$. It is clear that we can reconstruct $F_{*}$ using
$M$ and $\bar{n}$.

######  Definition 2.1.

With the notations above, we call a $(k+1)\times(k+1)$ matrix $M$ the orbit
permutation and a $k$-tuple $\bar{n}$ of positive integers the orbit lengths.
Also, we call the pair $(M,\bar{n})$ Orbit data $\mathcal{O}$ of $f$

$\mathcal{O}(f)\ =\ (M,\bar{n}).$

A matrix $M$ in the orbit data of a birational map
$f\in\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))$ is called the orbit permutation
because it describes the correspondence between the exceptional curves and the
points of indeterminacy. Notice that the orbit data is uniquely determined if
the order of exceptional curves is given.

######  Theorem 2.2.

The dynamical degree of a birational map
$f\in\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))$ is determined by its orbit data
$\mathcal{O}(f)$.

###### Proof.

The orbit data $\mathcal{O}(f)$ gives $F_{*}:\text{Pic}(X)\to\text{Pic}(X)$
for an algebraically stable modification $F$. By Diller and Favre[6], we have
$\lambda(f)=$ the spectral radius of $F_{*}$. ∎

######  Example 2.3.

Suppose $\sigma$ is the Cremona involution

$\sigma[z_{1}:z_{2}:z_{3}]=[z_{2}z_{3}:z_{1}z_{3}:z_{1}z_{2}].$

Then we have

$\mathcal{O}(\sigma)=\left(M=\begin{bmatrix}2&1&1&1\\\ -1&0&-1&-1\\\
-1&-1&0&-1\\\ -1&-1&-1&0\end{bmatrix},\bar{n}=(1,1,1)\right)$

######  Example 2.4.

Let $L\in\text{Aut}(\mathbf{P}^{2}(\mathbb{C}))$ and let
$f=L\circ\sigma\in\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))$. Then we have

$\mathcal{E}(f)=\\{E_{1}=\\{z_{1}=0\\},E_{2}=\\{z_{2}=0\\},E_{3}=\\{z_{3}=0\\}\\}$

and

$\mathcal{I}(f)=\\{p_{1}=[1:0:0],p_{2}=[0:1:0],p_{3}=[0:0:1]\\}$

Now suppose that there is a permutation $\rho\in S_{3}$,
$n_{1},n_{2},n_{3}\geq 1$ such that

$f^{n_{i}+1}(E_{i})=p_{\rho(i)},\qquad f^{j}(E_{i})\not\in\mathcal{I}(f)\text{
for all }1\leq j\leq n_{i}.$

Then we have $\mathcal{O}(f)=(M_{\rho}=M\cdot m(\rho),(n_{1},n_{2},n_{3})$
where $M$ is the matrix given in the previous example and $m(\rho)$ is the
matrix representation corresponding to $\rho\in S_{3}$. For example, if
$\rho=(1,2)$ is a transposition of $1$ and $2$, then

$m((1,2))=\begin{bmatrix}1&0&0&0\\\ 0&0&1&0\\\ 0&1&0&0\\\
0&0&0&1\end{bmatrix}.$

###  2.3. Dynamical Spectrum

Let us denote $\Lambda_{\mathbb{C}}$ the Dynamical spectrum, the set of all
dynamical degrees of birational maps on $\mathbf{P}^{2}(\mathbb{C})$

$\Lambda_{\mathbb{C}}=\\{\lambda(f):f\in\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))\\}.$

There are only two possibilities for the dynamical degrees in
$\Lambda_{\mathbb{C}}$. By Diller-Favre [6] and Blanc-Cantat [3], we know

######  Theorem 2.5.

[6, 3] Suppose $f$ is a birational map on a surface
$\mathbf{P}^{2}(\mathbb{C})$. If $\lambda(f)\neq 1$, then

  * •

either the dynamical degree is a Salem number and $f$ is birationally
equivalent to an automorphism

$\Lambda_{S}:=\\{\lambda(f):f\in\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))\text{
and }f\ \text{is birationally equivalent to an automorphism}\\}$

  * •

or the dynamical degree is a Pisot number, and $f$ is not birationally
equivalent to an automorphism.

$\Lambda_{P}:=\\{\lambda(f):f\in\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))\text{
and }f\ \text{is NOT birationally equivalent to an automorphism}\\}$

The Salem number $\lambda_{s}$ is an algebraic integer $>1$ with a monic self-
reciprocal minimal polynomial such that all Galois conjugates except
$1/\lambda_{s}$ are on the unit circle. The Pisot number $\lambda_{p}$ is an
algebraic integer $>1$ such that all Galois conjugates are of modulus strictly
less than $1$.

Suppose $f\in\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))$ is birationally
equivalent to an automorphism with $\lambda(f)>1$ and let $F:X\to X$ be its
algebraically stable modification on a rational surface $X$. Then $F$ is an
automorphism on a rational surface $X$, and by Nagata, [18, 19], we have
$F_{*}\in W_{n}$ where $W_{n}$ is the Coxeter group corresponding to $E_{n}$
diagram and $n+1$ is the Picard number of $X$. We will discuss the Coxeter
group in the next Section 3. The Converse is not true. Kim [12] shows that
there is an element $\omega\in E_{14}$ such that there is no rational surface
automorphism $F$ such that $F_{*}=\omega$. However, by McMullen [17], Uehara
[23], and Kim [13], for any spectral radius of $\omega\in E_{n},n\geq 10$,
there is a birational map $f\in\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))$ such
that $\lambda(f)=\rho(\omega)$.

For a complex number $\lambda\geq 1$, we say $\lambda$ is realized if there is
a birational map $f\in\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))$ with
$\lambda(f)=\lambda$.

##  3\. Coxeter Group

Let $\mathbb{Z}^{1,n},n\geq 3$ be a lattice with the ordered basis
$(e_{0},e_{1},\dots,e_{n})$ and the symmetric bilinear form given by

$e_{0}\cdot e_{0}=1,\quad\ e_{i}\cdot e_{i}=-1,\ \text{for
}i=1,\dots,n,\quad\text{ and }\quad e_{i}\cdot e_{j}=0\ \text{for }i\neq j.$

Thus for any $x=(x_{0},x_{1},\dots,x_{n})=\sum x_{i}e_{i}\in\mathbb{Z}^{1,n}$,

$x\cdot x\ =x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-\cdots-x_{n}^{2}.$

The canonical vector $\kappa_{n}=(-3,1,\dots,1)\in\mathbb{Z}^{1,n}$ and let
$L_{n}$ denote the orthogonal complement of $\kappa_{n}$. Also, let

$\alpha_{0}=e_{0}-e_{1}-e_{2}-e_{3},\qquad\text{and}\qquad\alpha_{i}=e_{i}-e_{i+1},\
i=1,\dots,n-1.$

Simple computations show

$\alpha_{i}\cdot\kappa_{n}=0,\ \ \alpha_{i}\cdot\alpha_{i}=-2,\ \ \ \text{for
all }i=0,1,\dots,n-1.$

Thus we see that $L_{n}$ is generated by these vectors
$\alpha_{0},\alpha_{1},\dots,\alpha_{n-1}$

$L_{n}=\kappa^{\perp}=\langle\alpha_{0},\alpha_{1},\dots,\alpha_{n-1}\rangle.$

For each $i=0,\dots,n-1$, define a reflection $s_{i}$ through $\alpha_{i}$ on
$\mathbb{Z}^{1,n}$

$s_{i}(x)\ :=\ x+(x\cdot\alpha_{i})\,\alpha_{i}.$

Then we have

$s_{i}(\kappa_{n})=\kappa_{n},\quad s_{i}(\alpha_{i})=-\alpha_{i},\quad
s_{i}(\alpha_{j})=\alpha_{j},\ \text{for }i\neq j,$

and for $x,y\in L_{n}$

$s_{i}(x)\in L_{n},\qquad s_{i}(x)\cdot s_{i}(y)\ =x\cdot y.$

Thus $s_{i}\in O(L_{n})$ for $i=0,\dots,n-1$ and satisfy the following
relations $(s_{i}s_{j})^{m}_{ij}=1$ where

(3.1) $m_{ij}\ =\ \left\\{\begin{aligned} &1\qquad\text{if }i=j\\\
&3\qquad\text{if }\\{i,j\\}\in\\{\\{0,3\\},\ \\{i,i+1\\},\
i=1,2,\dots,n-2\\}\\\ &2\qquad\text{otherwise}\\\ \end{aligned}\right.$

The subgroup $W_{n}\subset O(L_{n})$ generated by reflections
$s_{0},\dots,s_{n-1}$ is called the Coxeter group. For $m\leq n$, we may
consider $W_{m}$ as a subgroup of $W_{n}$ by fixing $e_{i}$ for $i\geq m+1$,

$W_{m}\cong\langle s_{0},s_{1},\dots,s_{m-1}\rangle\subset W_{n}.$

With this identification, we define the infinite Coxeter group $W_{\infty}$ as
a union of $W_{n}$ for $n\geq 3$.

$W_{\infty}:=W_{\infty}.$

###  3.1. Root system

Let $V_{n}$ be an inner product $(\cdot,\cdot)$ space over $\mathbb{R}$ with
the basis $(\alpha_{0},\dots,\alpha_{n-1})$ where

$(\alpha_{i},\alpha_{j})=-2\cos(\pi/m_{ij}).$

Thus $(\alpha_{i},\alpha_{j})=2,0,-1$ if $m_{ij}=1,2,3$. For any $\alpha\in
V_{n}$ with $(\alpha,\alpha)=\pm 2$, we can define a reflection $s_{\alpha}$
by

$s_{\alpha}(\lambda)\ =\
\lambda-\frac{2(\lambda,\alpha)}{(\alpha,\alpha)}\alpha.$

Thus we may assume that $L_{n}\subset V_{n}$ and each $s_{i}\in W_{n}$ gives
the reflection in $O(V_{n})$ preserving $L_{n}$.

The generating vectors $\alpha_{i}$ are called simple roots and the root
system $\Phi_{n}=\cup_{i}W_{n}\alpha_{i}$ is the $W_{n}$ orbits in $V_{n}$ of
simple roots. (General theories and precise definitions are available in many
literatures, including [11].) It is useful to remember that any root is either
positive or negative; in other words, any root can be written as a positive
sum of simple roots or a negative sum of simple roots.

$\Phi_{n}=\Phi_{n}^{+}\cup\Phi_{n}^{-}$

where $\Phi_{n}^{\pm}=\\{\sum c_{i}\alpha_{i}:\pm c_{i}\geq 0\,\text{for all
}i\\}$.

###  3.2. Weyl degree

For each element $\omega\in W_{\infty}$, let us define Weyl degree(or simply
degree), $\deg(\omega)$, of $\omega$ by

$\deg(\omega):=\omega e_{0}\cdot e_{0}.$

If $\deg(\omega)=1$, then $\omega e_{0}=e_{0}$, and $\omega$ permutes
$e_{i}$’s. Thus the spectral radius $\lambda(\omega)=1$. For a positive
integer $d\geq 1$, let $\Omega_{d}$ denote the set of elements in $W$ with the
Weyl degree $=d$

$\Omega_{d}\ :=\ \\{\omega\in W_{\infty}\ :\ \deg(\omega)=d\\}.$

Also, we let $\Lambda_{d}$ denote the set of spectral radii of elements in
$\Omega_{d}$

$\Lambda_{d}:=\\{\lambda(\omega):\deg(\omega)=d\\}.$

Clearly, the set of all spectral radii of $W_{\infty}$ is equal to the union
$\cup_{d}\Lambda_{d}$.

$\\{\lambda(\omega):\omega\in W_{\infty}\\}\ =\
\\{\lambda(\omega):\omega\in\cup_{d\geq 1}\Omega_{d}\\}\ =\
\cup_{d}\Lambda_{d}$

McMullen [16] showed that the characteristic polynomial of any element in
$W_{n}$ is either a product of cyclotomic polynomials or a product of a Salem
polynomial and cyclotomic polynomials. Thus, the spectral radius is either $1$
or a Salem number. By Theorem 2.5 of Diller-Favre and Blanc-Cantat we have

$\cup_{d}\Lambda_{d}=\\{1\\}\cup\Lambda_{S},$

where $\Lambda_{S}$ is all dynamical degrees $>1$ of birational maps
equivalent to an automorphism.

Let us denote $\Lambda_{W}$, the Weyl spectrum, the closure of the set of all
spectral radii of $W_{\infty}$.

$\Lambda_{W}\ :=\ \overline{\cup_{d}\Lambda_{d}}.$

###  3.3. Matrix Representations

Let $\omega\in W_{n}$ such that

(3.2) $\displaystyle\omega(e_{0})=de_{0}-\sum_{i=1}^{n}c_{i}e_{i}$
$\displaystyle\omega(e_{j})=d_{i}e_{0}-\sum_{i=1}^{n}m_{i,j}e_{i}\qquad
j=1,\dots,n$

where $d=\deg(\omega)$ and $d_{i},c_{i},m_{i,j}$ are non-negative integers.
Let $\bar{d}=(d_{1},\dots,d_{n})$, $\bar{c}=(c_{1},\dots,c_{n})$ be $n$-tuples
of non-negative integers. Also let $m=\\{m_{i,j}\\}$ be an $n\times n$ matrix
with non-negative integers. Then with an ordered basis
$\\{e_{0},e_{1},\dots,e_{n}\\}$, we can write $\omega$ as a $(n+1)\times(n+1)$
matrix

$\omega=\begin{bmatrix}d&\bar{d}\\\ -\bar{c}&-m\end{bmatrix}.$

Because $\omega\in O(L_{n})$, we have the following basic facts (known as
Noether’s equalities and inequalities).

######  Lemma 3.1.

Suppose $\omega\in W_{n}$ is given by (3.2).Then we have

  1. (1)

$3(d-1)=\sum_{i}d_{i}=\sum_{i}c_{i}$,

  2. (2)

$d^{2}-1=\sum_{i}d_{i}^{2}=\sum_{i}c_{i}^{2}$,

  3. (3)

$m_{ij}<c_{i}$ and $m_{ij}<d_{j}$ for all $i,j$,

  4. (4)

the sum of three largest $d_{i}$’s is bigger than or equal to $d+1$ and the
same is true for $\bar{c}$,

$d_{1}+d_{2}+d_{3}\geq d+1\ \ \text{after rearranging indices so that
}d_{1}\geq d_{2}\geq\cdots\geq d_{n}$

  5. (5)

the sum of two $d_{i}$’s is smaller than or equal to $d$ and the same is true
for $\bar{c}$

$d_{i}+d_{j}\leq d\ \ \ \text{for }i\neq j$

  6. (6)

$\omega^{-1}$ is given by the matrix representation

$\begin{bmatrix}d&\bar{c}\\\ -\bar{d}&-m^{t}\end{bmatrix},$

where $m^{t}$ is the transpose of $m$.

###### Proof.

Except for the last statement, the proof is written in several articles,
including [3] and [7]. The last statement can be checked by multiplying two
matrices using other statements in this Lemma. ∎

###  3.4. Decompositions

For any $\omega\in W_{n}$, we have $\omega e_{j}\cdot\omega e_{j}=-1$ for
$j=1,\dots,n$. It follows that for each $j=1,\dots,n$ we have

$\text{either }\ \ \omega e_{j}=e_{k}\ \ \text{for some }k\geq 1\
\quad\text{or }\quad\omega e_{j}\cdot e_{0}\neq 0.$

For each $\omega\in W_{n}$, let us define

(3.3) $\displaystyle I_{e}(\omega)\ :=\ \\{0\leq i\leq n:\omega^{k}e_{i}\cdot
e_{0}\neq 0\quad\text{for some }k\geq 0\\},\quad\text{and}$ $\displaystyle
I_{p}(\omega)\ :=\ \\{0\leq i\leq n:\omega^{k}e_{i}\cdot e_{0}=0\quad\text{for
all }k\geq 0\\}.$

Clearly, if $\omega e_{i}=e_{j}$ with $i\in I_{e}$ then $j\not\in I_{p}$. Also
if $i\in I_{p}$ then $\omega e_{i}=e_{j}$ for some $j\in I_{p}$. Let us denote
$n_{e}:=|I_{e}(\omega)|-1$. Then by reordering $e_{i}^{\prime}s$ we may assume
that

$I_{e}(\omega)=\\{0,1,\dots,n_{e}\\},\quad\text{and}\quad
I_{p}(\omega)=\\{n_{e}+1,\dots,n\\}.$

Let us define two elements $\omega_{e},\omega_{p}\in W_{n}$ be

(3.4) $\displaystyle\omega_{e}e_{i}\ =\ \omega e_{i},\text{ for }i\in
I_{e},\qquad\text{and}\qquad\omega_{e}e_{i}\ =\ e_{i},\text{ for }i\in I_{p}$
$\displaystyle\omega_{p}e_{i}\ =\ e_{i},\text{ for }i\in
I_{e},\qquad\text{and}\qquad\omega_{p}e_{i}\ =\ \omega e_{i},\text{ for }i\in
I_{p}$

######  Lemma 3.2.

Suppose $\omega\in W_{n}$ such that $I_{e}(\omega)=\\{0,1,\dots,m\\}$ for some
$m\leq n$. Then with $\omega_{e},\omega_{p}$ given in (3.4) we have

$\omega=\omega_{e}\cdot\omega_{p},$

and

  1. (1)

$\omega_{e}$ is generated by reflections $s_{0}\dots,s_{m-1}$ and
$\deg(\omega)=\deg(\omega_{e})$ ,

$\omega_{e}\in\langle s_{0},s_{1},\dots,s_{m-1}\rangle,\qquad\omega e_{0}\cdot
e_{0}=\omega_{e}e_{0}\cdot e_{0}$

  2. (2)

$\omega_{p}$ is generated by reflections $s_{m+1},\dots,s_{n-1}$, and

  3. (3)

the spectral radius $\lambda(\omega)$ of $\omega$ is given by the spectral
radius of $\omega_{e}$

$\lambda(\omega)\ =\ \lambda(\omega_{e}).$

###### Proof.

Form (3.4), we see that $\omega=\omega_{e}\cdot\omega_{p}$ and the first two
assertions. Also, by (3.4), it is clear that the matrix representation of
$\omega$ can be written as a block diagonal matrix. Thus we have
$\lambda(\omega)\ =\ \lambda(\omega_{e})$. ∎

Since $\omega_{p}$ does not contribute to the spectral radius of $\omega$, let
us focus on elements with $\omega_{p}=Id$. Then by reordering
$e_{1},\dots,e_{n}$, there are $k\leq n$ distinct positive integers, $1\leq
n_{1}<n_{2}<\cdots<n_{k}\leq n$, such that

  * •

$\omega e_{i}\ =\ e_{i}+1$ if $i\not\in\\{n_{j},j=1,\dots,k\\}$, and

  * •

$\omega e_{n_{j}}\cdot e_{0}=d_{j}>0$ for all $j=1,\dots,k$.

Let us set $n_{0}=0$ and define $\omega_{c}\in W_{n}$ by

(3.5) $\begin{aligned} \omega_{c}\ :\ &e_{0}\mapsto\omega e_{0},\\\
&e_{n_{i-1}+1}\mapsto\omega e_{n_{i}},i=1,\dots,k,\quad\text{and}\\\
&e_{j}\mapsto e_{j},\text{ for }j\neq n_{i}+1,i=0,1,\dots,k-1\\\
\end{aligned}.$

Also define $\omega_{s}\in W_{n}$ by

(3.6) $\displaystyle\omega_{s}\ :$ $\displaystyle e_{0}\mapsto e_{0}$
$\displaystyle e_{1}\mapsto e_{2}\mapsto\cdots\mapsto e_{n_{1}}\mapsto e_{1}$
$\displaystyle e_{n_{i-1}+1}\mapsto e_{n_{i-1}+2}\mapsto\cdots\mapsto
e_{n_{i}}\mapsto e_{n_{i-1}+1},\ \ \text{for }i=2,\dots,k$

With Lemma 3.2, we see that $\omega\in W_{n}$ is conjugate to a product of
three elements defined in (3.4 \- 3.6).

######  Proposition 3.3.

Suppose $\omega\in W_{n}$ such that $\deg(\omega)=d\geq 2$. Then, there is a
permutation $\rho$ in $e_{i},i=1,\dots,n$ we have

$\omega=\rho^{-1}\cdot\omega_{c}\cdot\omega_{s}\cdot\omega_{p}\cdot\rho.$

Furthermore, we have $\deg(\omega_{c})=\deg(\omega)$ and
$\lambda(\omega_{c}\cdot\omega_{s})=\lambda(\omega)$.

###### Proof.

The reordering elements of $e_{i}$’s, $i=1,\dots,n$ is given by conjugating by
a permutation in $e_{i},i=1,\dots,n$. Thus, this Proposition comes from Lemma
3.2. ∎

###  3.5. Formulas for Spectral radii

In this section, we obtain a formula for the characteristic polynomial of a
given $\omega\in\Omega_{d}$ up to cyclotomic factors. Since $\omega_{p}$ in
Proposition 3.3 is only related to the cyclotomic factor, we may assume both
$\omega_{p}$ and $\rho$ in Proposition 3.3 are identity.

For $\omega=\omega_{c}\cdot\omega_{s}\in\Omega_{d}$, let $\tilde{\omega}_{c}$
be a $(k+1)\times(k+1)$ submatrix of $\omega_{c}$ given by columns and rows
corresponding $e_{0},e_{n_{0}+1},\dots,e_{n_{k-1}+1}$. In other words,
$\tilde{\omega}_{c}$ is a matrix obtained by deleting columns and rows
corresponding to $e_{i}$’s fixed by $\omega_{c}$. Also, let $n(\omega)$ be a
$k$ tuple of positive integers given by lengths of cycles in $\omega_{s}$

$n(\omega)\ :=\ (n_{1},n_{2}-n_{1},n_{3}-n_{2},\cdots,n_{k}-n_{k-1}).$

For each $\omega\in W$, numerical data $k,\tilde{\omega}_{c},$ and $n(\omega)$
are uniquely determined up to reordering indices. We define orbit data
$\mathcal{O}(\omega)$ of $\omega$ by numerical data
$(\tilde{\omega}_{c},n(\omega))$.

$\mathcal{O}(\omega):=(\tilde{\omega}_{c},n(\omega)).$

Also, we call $n(\omega)\in\mathcal{O}(\omega)$ orbit lengths. The element
$\tilde{\omega}_{c}$ is determined by the images of $e_{i}$, which maps to a
vector $\neq e_{j}$ for some $j\geq 1$ under $\omega$. We call
$\tilde{\omega}_{c}$ the orbit permutation.

######  Remark 3.4.

This definition of orbit data for an element of a Coxeter group is comparable
with the orbit data of a birational map defined in Section 2. If
$f\in\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))$ is birationally equivalent to an
automorphism $F:X\to X$ on a rational surface $X$. Then, the orbit data of $f$
is equal to the orbit data of the linear action $F_{*}\in W_{\infty}$ up to
relabelling $e_{i}$’s.

######  Lemma 3.5.

For $\omega\in\Omega_{d},d\geq 2$, the spectral radius $\lambda(\omega)$ of
$\omega$ is determined by orbit data, a $(k+1)\times(k+1)$ matrix
$\tilde{\omega}_{c}$, and a $k$ tuple $n(\omega)$ of positive integers.

###### Proof.

From (3.5) and (3.6), it is clear that we can reconstruct $\omega_{c}$ and
$\omega_{s}$ from orbit data. ∎

######  Lemma 3.6.

Let $d\geq 2$. The set $\mathcal{M}_{d}$ of $\tilde{\omega}_{c}$ for all
$\omega\in\Omega_{d}$ is finite.

$\mathcal{M}_{d}:=\\{\tilde{\omega}_{c}:\omega\in\Omega_{d}\\},\qquad|\mathcal{M}_{d}|<\infty$

###### Proof.

Suppose $\begin{bmatrix}d&\bar{d}\\\
-\bar{c}&-m\end{bmatrix}\in\mathcal{M}_{d}$. Then $d_{i},c_{i}\neq 0$ for all
$i$. By Lemma 3.1, we have

$3(d-1)=\sum_{i}d_{i},\quad\text{and}\quad d^{2}-1=\sum_{i}d_{i}^{2}$

Since $d_{i}$ are all positive integers, there are only finitely many possible
solutions. Also, by Lemma 3.1, we see that $0\leq m_{ij}\leq d_{j}$ for all
$i,j$. It follows that $\mathcal{M}_{d}$ is a finite set. ∎

Suppose $M$ is a $(k+1)\times(k+1)$ matrix and $\bar{n}$ be a $k$ tuple of
positive integers. Let $M_{t}$ be a matrix obtained from $M$ by subtracting
$t$ from the $1,1$ entry. Also, for $I\subset\\{1,\dots,k\\}$ let
$n_{I}=\sum_{i\in I}n_{i}$ and $M_{I}$ be a principal sub-matrix of $M_{t}$
obtained by deleting $i+1^{\text{th}}$ column and row for all $i\not\in I$.
Let us define a polynomial

(3.7)
$\chi_{M,\bar{n}}(t):=\sum_{I\subset\\{1,\dots,k\\}}(-1)^{|I|-1}\,t^{N-n_{I}}\,\det
M_{I}.$

Let $I(\bar{n})=\\{i+j:1\leq j\leq n_{i},i=1,\dots,k\\}$ and let $M(\bar{n})$
be a $(N+k+1)\times(N+k+1)$ matrix such that

  * •

a sub-matrix of $M(\bar{n})$ obtained by removing $i^{\text{th}}$ rows and
$i^{\text{th}}$ columns for $i\in I(\bar{n})$ is identical to $M$, and

  * •

for $i\in I(\bar{n})$, the $i^{\text{th}}$ column of $M(\bar{n})$ has $1$ in
$i+1^{\text{th}}$ entry and $0$ otherwise.

Notice that this matrix $M(\bar{n})$ corresponds to the matrix representation
of $\omega$ with the orbit data $\mathcal{O}(\omega)=(M,\bar{n})$.

######  Lemma 3.7.

With the above settings, the characteristic polynomial of $M(\bar{n})$ is
given by $\chi_{M,\bar{n}}(t)$ given in (3.7).

###### Proof.

Suppose $k=1$ and $n_{1}=1$. Then we have

$\omega-tId=\begin{bmatrix}d-t&d_{1}\\\ -c_{1}&-m_{1,1}-t\end{bmatrix}$

and thus

$\det(\omega-tId)=-t(d-t)+\det\begin{bmatrix}d-t&d_{1}\\\
-c_{1}&-m_{1,1}\end{bmatrix}.$

If $n_{1}=n>1$, then we have

$\omega-tId=\begin{bmatrix}d-t&0&0&&d_{1}\\\ -c_{1}&-t&0&&-m_{1,1}\\\
0&1&-t&&0\\\ 0&0&1&&\\\ 0&&&\ddots&0\\\ 0&&&0&-t\end{bmatrix}$

Then, the principal sub-matrix obtained by deleting the $2^{\text{nd}}$ column
and the $2^{\text{nd}}$ row has a row with a single element $-t$ in the
diagonal, and thus using minors corresponding $-t$ in such a row, we see that
the cofactor corresponding to $(2,2)$-entry is $(-t)^{n-1}(d-t)$. Similarly,
we use minors corresponding to a single $1$ in a row for the minor of
$(2,3)$-entry. Then we see that the cofactor corresponding $(2,3)$-entry is
$(-1)^{n-1}\det\begin{bmatrix}d-t&d_{1}\\\ -c_{1}&-m_{1,1}\end{bmatrix}$. It
follows that

$\det(\omega-tId)=(-t)^{n}(d-t)+(-1)^{n-1}\det\begin{bmatrix}d-t&d_{1}\\\
-c_{1}&-m_{1,1}\end{bmatrix}.$

Suppose $k=2$, $n_{1},n_{2}$ are two strictly positive integers. Then

$\omega-tId=\begin{bmatrix}d-t&|&0&&d_{1}&|&&&d_{2}\\\ -&-&-&-&-&-&-&-&-\\\
-c_{1}&|&-t&&-m_{1,1}&|&&&-m_{1,2}\\\ 0&|&1&\ddots&0&|&&&\\\ 0&|&&1&-t&|&&&\\\
-&-&-&-&-&-&-&-&-\\\ -c2&|&&&-m_{2,1}&|&-t&&-m_{2,2}\\\ &|&&&&|&1&\ddots&0\\\
&|&&&&|&&1&-t\end{bmatrix}$

Using $(-t)$’s and $1$’s in the $(3,3)$ block, we have

(3.8) $\displaystyle\det(\omega-tId)$
$\displaystyle=(-t)^{n_{2}}\det\begin{bmatrix}d-t&0&&d_{1}\\\
-c_{1}&-t&&-m_{1,1}\\\ 0&1&\ddots&0\\\ 0&&1&-t\end{bmatrix}$
$\displaystyle+(-1)^{n_{2}-1}\det\begin{bmatrix}d-t&0&&d_{1}&d_{2}\\\
-c_{1}&-t&&-m_{1,1}&-m_{1,2}\\\ 0&1&\ddots&0&0\\\ 0&&1&-t&0\\\
-c_{2}&&&-m_{2,1}&-m_{2,2}\end{bmatrix}$

For the second term in the right-hand side of the above equation, we use,
again, $(-t)$’s and $1$’s for the cofactor expansion. Then we see the second
term is equal to

(3.9) $(-1)^{n_{2}-1}\left((-t)^{n_{1}}\det\begin{bmatrix}d-t&d_{2}\\\
-c_{2}&-m_{2,2}\end{bmatrix}+(-1)^{n_{1}-1}\det\begin{bmatrix}d-t&d_{1}&d_{2}\\\
-c_{1}&-m_{1,1}&-m_{1,2}\\\ -c_{2}&-m_{2,1}&-m_{2,2}\end{bmatrix}.\right)$

Combining equations (3.8) and (3.9), we see that the characteristic polynomial
of $\omega$ is given by the formula given in (3.7). For $k\geq 3$, we work
with cofactor expansions using $(-t)$’s and $1$’s for each block. The
computation is essentially identical. ∎

######  Theorem 3.8.

Suppose orbit data of $\omega$ is given by
$\mathcal{O}(\omega)=(M=\begin{bmatrix}d&\bar{d}\\\
-\bar{c}&-m\end{bmatrix},\bar{n}=(n_{1},\dots,n_{k}))$. Then the
characteristic polynomial of $\omega_{c}\cdot\omega_{s}$ is a monic polynomial
$\chi_{\mathcal{O}(\omega)}(t)$ defined in (3.7). Thus, the spectral radius of
$\omega$ is the largest real root of $\chi(t)$.

###### Proof.

Since the matrix representation of $\omega_{c}\cdot\omega_{s}$ is given by
$M(\bar{n})$, this Theorem is due to the previous Lemma 3.7. ∎

Let $f\in\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))$ be a birational map with the
orbit data $\mathcal{O}(f)=(M,\bar{n})$. Then there exists a birational map
$F$ on a rational surface $X$ such that $F$ is an algebraically stable
modification of $f$ such that the induced action of $F$ on $\text{Pic}(X)$ is
given by $M(\bar{n})$. Thus we have

######  Corollary 3.9.

Let $f\in\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))$ be a birational map with the
orbit data $\mathcal{O}(f)=(M,\bar{n})$. Then the dynamical degree of $f$ is
the largest real root of $\chi_{M,\bar{n}}$ defined in (3.7).

######  Example 3.10.

Let $\omega\in\Omega_{3}$ given by

(3.10) $\omega\ :\ \left\\{\begin{aligned} &e_{0}\ \mapsto
3e_{0}-2e_{1}-e_{4}-e_{6}-e_{10}-e_{11}\\\ &e_{1}\mapsto e_{2}\mapsto
e_{3}\mapsto e_{0}-e_{1}-e_{6}\\\ &e_{4}\mapsto e_{5}\mapsto
2e_{0}-e_{1}-e_{4}-e_{6}-e_{10}-e_{11}\\\ &e_{6}\mapsto e_{7}\mapsto
e_{8}\mapsto e_{9}\mapsto e_{0}-e_{1}-e_{10}\\\ &e_{10}\mapsto
e_{0}-e_{1}-e_{4}\\\ &e_{11}\mapsto e_{12}\mapsto e_{0}-e_{1}-e_{11}.\\\
\end{aligned}\right.$

Then we have $k=5$, $n(\omega)=(3,2,4,1,2)$,

$\begin{bmatrix}d&\bar{d}\\\ -\bar{c}&-M\end{bmatrix}\ =\ \begin{bmatrix}\ 3&\
1&\ 2&\ 1&\ 1&\ 1\\\ -2&-1&-1&-1&-1&-1\\\ -1&\ 0&-1&\ 0&-1&\ 0\\\ -1&-1&-1&\
0&\ 0&\ 0\\\ -1&\ 0&\ -1&\ -1&\ 0&\ 0\\\ -1&\ 0&-1&\ 0&\ 0&-1\\\
\end{bmatrix}.$

And for $I\subset\\{1,\dots,5\\}$,

$M_{\emptyset}\ =\ \begin{bmatrix}3-t\end{bmatrix},\ \ \ M_{\\{1\\}}\ =\
\begin{bmatrix}3-t&1\\\ -2&-1\end{bmatrix},\ \ M_{\\{2\\}}\ =\
\begin{bmatrix}3-t&2\\\ -1&-1\end{bmatrix},\ \ M_{\\{3\\}}\ =\
\begin{bmatrix}3-t&1\\\ -1&0\end{bmatrix},\cdots$ $M_{\\{1,2\\}}\ =\
\begin{bmatrix}3-t&1&2\\\ -2&-1&-1\\\ -1&0&-1\\\ \end{bmatrix},\ \
M_{\\{1,3\\}}\ =\ \begin{bmatrix}3-t&1&1\\\ -2&-1&-1\\\ -1&-1&0\\\
\end{bmatrix},\ \ \dots\ ,\ \ M_{\\{1,2,\dots,k\\}}=M_{t}$

Using (3.7), we see that the characteristic polynomial of $\omega$ is given by

$\chi(t)=(t^{12}-2t^{11}+t^{10}-2t^{9}+t^{7}-t^{6}+t^{5}-2t^{3}+t^{2}-2t+1)(t-1)$

and the spectral radius is $1.96683..$, the largest real root of $\chi(t)$.

###  3.6. Special Case.

For three distinct positive integers $i,j,k\geq 1$, let $\kappa_{i,j,k}\in
W_{\infty}$ be the reflection through the vector $e_{0}-e_{i}-e_{j}-e_{k}$.
For any $d\geq 2$, let

$\omega_{d}=\kappa_{1,3,2}\cdot\kappa_{1,5,4}\cdot\kappa_{1,7,6}\cdots\kappa_{1,2d-1,2d-2}.$

Then we see that

$\omega_{d}=\begin{bmatrix}d&d-1&1&\cdots&1\\\ -(d-1)&-(d-2)&-1&\cdots&-1\\\
-1&-1&&&\\\ \vdots&\vdots&&-Id&\\\ -1&-1&&\end{bmatrix},$

and

$\mathcal{O}(\omega_{d})=(\omega_{d},\bar{n}=(1,1,\dots,1)).$

Any permutation $\sigma$ in $\\{1,2,2d-1\\}$ is generated by reflections
$s_{i},i\geq 1$ through the vectors $e_{i}-e_{i+1}$. Thus we have

$\mathcal{O}(\omega_{d}\cdot\sigma)=(\omega_{d}\cdot\sigma,\bar{n}=(1,1,\dots,1)).$

With the above notations, let us consider a subset $\Omega\in W_{\infty}$
given by

$\Omega=\\{\omega\in
W_{\infty}:\mathcal{O}(\omega)=(\omega_{d}\cdot\sigma,\bar{n}),\sigma\in\langle
s_{1},\dots,s_{2d-2}\rangle,\bar{n}\in\mathbb{N}^{2d-1},d\geq 2\\}.$

For any $\omega\in\Omega$, the corresponding $M_{I}$’s in (3.7) and their
determinants are relatively simple. Due to these simplicities, the Salem
polynomials for the spectra radii can be obtained using Linear Algebra. Not to
board the readers with basic computations, let us describe the Salem
polynomials appearing in $\Omega$.

Suppose $d\geq 2$ is a positive integer and $\sigma$ is a permutation in
$\\{1,2,\dots,2d-1\\}$. For each subset $I\subset\\{1,2,\dots,2d-1\\}$, let us
define $\sigma^{-1}(I)=\\{i:\sigma(i)\in I\\}$ and

$\rho(I)\ =\ \left\\{\begin{aligned} \ &5\qquad\text{if
}I=\sigma^{-1}(I),\text{ and }1\in I\\\ &4\qquad\text{if
}|I\setminus\sigma^{-1}(I)|=1,\text{ and }1\in I\cap\sigma^{-1}(I)\\\
&3\qquad\text{if }|I\setminus\sigma^{-1}(I)|=1,\text{ and }1\in
I\cup\sigma^{-1}(I),1\in I\cap\sigma^{-1}(I)\\\ &2\qquad\text{if
}I=\sigma^{-1}(I),\text{ and }1\not\in I\cup\sigma^{-1}(I)\\\ &1\qquad\text{if
}|I\setminus\sigma^{-1}(I)|=1,\text{ and }1\not\in I\cup\sigma^{-1}(I)\\\
&0\qquad\qquad\text{otherwise}\end{aligned}\right.$

And also let

$\text{Sgn}(I)=\left\\{\begin{aligned} \ \ &+\qquad\text{if }\sigma|_{I}\text{
is given by a product of even transposition }\\\ &-\qquad\text{if
}\sigma|_{I}\text{ is given by a product of odd transposition, }\\\
\end{aligned}\right.$

where $\sigma|_{I}$ is a permutation such that $\sigma|_{I}(i)=\sigma(i)$ for
$i\in I$ and $\sigma|_{I}(i)=i$ otherwise. Finally, let $n=|I|$ and let us
define

$\psi(I)\ =\ \left\\{\begin{aligned} \ \ &Sgn(I)((n+1-d)t-1)\qquad\ \ \text{
if }\rho(I)=5\\\ &Sgn(I)t\qquad\qquad\qquad\qquad\qquad\text{ if }\rho(I)=4\\\
&-Sgn(I)(t-1)\qquad\qquad\qquad\ \text{ if }\rho(I)=3\\\
&-Sgn(I)(t-d+n)\qquad\ \ \qquad\text{ if }\rho(I)=2\\\
&-Sgn(I)\qquad\qquad\qquad\qquad\quad\text{ if }\rho(I)=1\\\
&0\qquad\qquad\qquad\qquad\qquad\qquad\quad\text{ if }\rho(I)=0.\\\
\end{aligned}\right.$

######  Lemma 3.11.

Suppose $\omega\in\Omega$ with orbit data
$\mathcal{O}(\omega)=(\omega_{d}\cdot\sigma,\bar{n}=(n_{1},n_{2},\dots,n_{2d-1}))$.
Then, the spectral radius is the largest real root of $\chi(t)$ where

$\chi(t)=t^{N}\left[(d-t)+\sum_{I\neq\emptyset}\frac{1}{t^{I}}\psi(I)\right]$

where $N=n_{1}+n_{2}+\cdots+n_{2d-1}$ and $t^{I}=t^{\sum_{i\in I}n_{i}}$.
Furthermore, if $\sigma(1)=1$ and $n_{1}=1$, then the spectral radius is equal
to $1$ for all choices of $n_{i}$, $i\geq 2$.

##  4\. Weyl Spectrum vs. Dynamic Spectrum

The Dynamical spectrum $\Lambda_{\mathbb{C}}$ does not contain all Salem
numbers, and it is difficult to know all birational maps and their dynamical
degrees. However, it is known that the dynamical spectrum is, in fact, the
same as the Weyl spectrum. By McMullen [17], Uehara [23], and Kim [13]. Thus,
we have

######  Theorem 4.1.

Supose $\lambda\in\cup_{d}\Lambda_{d}$. If $\lambda>1$, then there is
$f\in\text{Bir}(\mathbf{P}^{2})$ such that $f$ is birationally equivalent to
an automorphism and $\lambda(f)=\lambda$. Thus we have

$\cup_{d}\Lambda_{c}\ =\ \\{1\\}\cup\Lambda_{S}.$

######  Theorem 4.2.

[3, Theoem D] The dynamical spectrum
$\Lambda(\mathbf{P}^{2}(\mathbb{C}))\subset\mathbb{R}$ is well ordered and is
closed.

Since any Pisot number is approximated by Salem numbers, the following
Corollary is the immediate consequence of the above Theorem 4.2

######  Corollary 4.3.

Any Pisot number $\lambda\in\Lambda_{P}$ is a limit from below of sequences of
Salem numbers in $\Lambda_{S}$.

###  4.1. Limits of Salem numbers in $\lambda_{S}$

Suppose $I\subset\\{1,\dots,k\\}$ and $n_{\star}$ is a positive integer. Let
$\bar{n}=(n_{1},\dots,n_{k})$ be a $k$ tuple with $n_{i}=n_{\star}$ for $i\in
I$. Suppose $\chi(t)$ is the characteristic polynomial (up to a product of
cyclotomic polynomials) of an element with orbit data
$(M=\begin{bmatrix}d&\bar{d}\\\
-\bar{c}&-m\end{bmatrix},\bar{n}=(n_{1},n_{2},\dots,n_{k}))$ defined in (3.7)
and let $\lambda_{M,\bar{n}}$ be the largest real root of $\chi(t)$. Now, let
$\tilde{m}$ be the principal sub-matrix of $m$ obtained by deleting
$i^{\text{th}}$ columns and rows for $i\in I$. And let $\tilde{d}$ and
$\tilde{c}$ be $k-|I|$ sub-tuples of $\bar{d}$ and $\bar{c}$ obtained by
deleting $i^{\text{th}}$ entry for $i\in I$, and $\tilde{n}$ be the $k-|I|$
tuple consists of $n_{j},j\not\in I$. By inspection, we see that

$\chi(t)=\sum_{j=0}^{|I|}t^{j\cdot n_{\star}}\psi_{j}(t),$

where

(4.1) $\psi_{j}(t)\ =\ \sum_{|I\cap J|=j}t^{N-(n_{J}-j\cdot n_{\star})}\det
M_{J},\quad N=n_{1}+n_{2}+\cdots+n_{k},\ \ n_{J}=\sum_{\ell\in J}n_{\ell}.$

Also, let us set $\chi_{I}(t)$ be the polynomial corresponding to
$(\tilde{M}=\begin{bmatrix}d&\tilde{d}\\\
-\tilde{c}&-\tilde{m}\end{bmatrix},\tilde{n})$ defined in (3.7). Then, we see
that $\psi_{|I|}(t)=\chi_{I}(t)$. Furthermore, since $\chi(t)$ is a self-
reciprocal polynomial for all $n_{\star}\in\mathbb{N}$ and all polynomials
$\psi_{j}(t)$, $j=0,\dots,|I|$ are independent of $n_{\star}$, we see that

$\psi_{|I|-j}(t)=\pm t^{\deg\psi_{j}(t)}\psi_{j}(1/t),\quad j=0,\dots,|I|.$

By letting $n_{\star}\to\infty$, we get

$\lim_{n_{\star}\to\infty}\lambda_{M,\bar{n}}=\lambda_{P}$

where $\lambda_{P}$ is the largest root of $\psi_{|I|}(t)$. By Theorem 4.2 of
Blanc and Cantat [3], we see that $\lambda_{P}$ is the unique largest real
root $>1$ of $\psi_{|I|}(t)$.

######  Proposition 4.4.

Suppose $\chi(t)$ is the characteristic polynomial (up to a product of
cyclotomic polynomials) of an element with orbit data
$(M\in\cup\mathcal{M}_{d},(n_{1},n_{2},\dots,n_{k}))$ defined in (3.7). Also
let $\lambda_{(n_{1},n_{2},\dots,n_{k})}$ be the largest real root of
$\chi(t)$. Fix a subset $I\in\\{1,\dots,k\\}$ and let the polynomial
$\psi_{|I|}(t)$ defined in (4.1). Then we have

$\lim_{n_{i}\to\infty,i\in I}\lambda_{(n_{1},n_{2},\dots,n_{k})}=\lambda_{P}$

where $\lambda_{P}$ is the largest real root of $\psi_{|I|}(t)$. In fact,
$\lambda_{P}$ is a Pisot number and $\lambda_{(n_{1},n_{2},\dots,n_{k})}$
approaches to $\lambda_{P}$ from below.

###### Proof.

We may assume that $n_{i}$ are all equal for $i\in I$. From the discussion
above, it is clear that the limit is a root $\lambda_{P}$ of $\psi_{|I|}(t)$.
By Theorem 4.2, $\lambda_{P}$ is either a Salem number or a Pisot number. It
follows that $\psi_{|I|}(t)$ has unique real root $>1$. If $\lambda_{P}$ is a
Salem number, by Theorem 2.5 and Theorem 4.1, $\lambda_{P}$ is a spectral
radius of element in $\cup_{n}W_{n}$. The discussion above shows that
$\tilde{M}$ does not satisfy Noether’s inequalities in Lemma 3.1. The last
assertion follows from the fact that $\Lambda_{\mathbb{C}}$ is well ordered. ∎

###  4.2. Monotonicity

In this subsection, we show that for a given $M\in\cup_{d}\mathcal{M}_{d}$,
the spectral radius of $\omega\in\Omega_{d}$ is increasing in orbit lengths.

######  Theorem B.

Suppose $\omega$ and $\omega^{\prime}$ have distinct orbit data
$\mathcal{O}(\omega)=(M,(n_{1},\dots,n_{k}))$ and
$\mathcal{O}(\omega^{\prime})=(M,(n^{\prime}_{1},\dots,n^{\prime}_{k}))$
respectively. If $n_{i}\leq n^{\prime}_{i}$ for all $i=1,\dots,k$, and
$\lambda(\omega)>1$ then $\lambda(\omega)\lneq\lambda(\omega^{\prime})$.

###### Proof.

It is sufficient to consider $n_{1}<n_{1}^{\prime}$ and $n_{i}=n_{i}^{\prime}$
for all $i\neq 1$. For $\omega$ with
$\mathcal{O}(\omega)=(M,(n_{1},\dots,n_{k}))$, by Theorem 3.8 we have that the
spectral radius of $\omega$ is the largest real root of

$\chi(t)=t^{n_{1}}\psi(t)+\phi(t),\quad\text{where}$ $\psi(t)\ =\ \sum_{1\in
I}t^{(N-n_{1})-n_{I}}\det M_{I},\quad\phi(t)\ =\ \sum_{1\not\in
I}t^{N-n_{I}}\det M_{I}.$

Consider the system of equations

(4.2) $y=-\phi(t)/\psi(t),\qquad\text{and}\qquad y=t^{n_{1}}.$

Then, the spectral radius of $\omega$ is given by the $t$-coordinates of the
unique intersection $>1$ of the above system of equations. Notice that both
$\phi(t)$ and $\psi(t)$ are independent of $n_{1}$. Also, using (4.1) we see
that $\psi(t)=0$ has the unique real root $t_{\star}>1$ and $\phi(t)=0$ has no
root $>1$. Since the system (4.2) has the unique solution $>1$, we have

$\lim_{t\nearrow t_{\star}}-\frac{\phi(t)}{\psi(t)}=+\infty.$

Therefore, the $t$-coordinates of the unique solution $>1$ of in (4.2)
increases in $n_{1}$. ∎

##  5\. Salem and Pisot numbers in the Weyl Spectrum

In this section, we determine all possible Salem and Pisot numbers in the Weyl
spectrum inductively. By Theorem 3.8, we need to find all possible orbit data
$\mathcal{O}(\omega),\omega\in\cup_{d}\Omega_{d}$ to determine all possible
Salem numbers in $\Lambda_{W}=\Lambda_{\mathbb{C}}$.

######  Lemma 5.1.

Let $M\in\mathcal{M}_{d},d\geq 2$ be a $(k+1)\times(k+1)$ matrix. Then for all
$k$ tuple $\bar{n}$ of positive integers, there is $\omega\in\Omega_{d}$ such
that $\mathcal{O}(\omega)=(M,\bar{n})$.

###### Proof.

For any positive integer $m$, let $s=s_{1}s_{2}\dots s_{m-1}$ be a production
of $m-1$ reflections. It follows that $s\in\cup_{n}W_{n}$ such that

$s:e_{1}\to e_{2}\to\cdots\to e_{m}\to e_{1}.$

Thus, using the formula in (3.6), we can construct any orbit data with a given
$M$. ∎

######  Lemma 5.2.

Let $M\in\mathcal{M}_{d},d\geq 2$ be a $(k+1)\times(k+1)$ matrix. If
$M^{\prime}$ is a matrix obtained by permuting columns of $M$ except the first
column, then $M^{\prime}\in\mathcal{M}_{d}$. Similarly, if $M^{\prime}$ is a
matrix obtained by permuting rows of $M$ except the first row, then
$M^{\prime}\in\mathcal{M}_{d}$.

###### Proof.

If $M^{\prime}$ is obtained by permuting the second and the third columns,
then $M^{\prime}=M\cdot s_{1}$ where $s_{1}$ is the reflection through the
vector $e_{1}-e_{2}$. Also, $s_{1}\cdot M$ gives a matrix obtained by
permuting the second and the third rows of $M$. This Lemma follows by applying
this argument repeatedly. ∎

Also, we have

######  Lemma 5.3.

Let $M\in\mathcal{M}_{d}$ be a $(k+1)\times(k+1)$ matrix. Suppose $M^{\prime}$
is a matrix obtained by permuting $2^{\text{nd}}$ – $(k+1)^{\text{th}}$ rows
of $M$. Then by renaming $e_{1},\dots,e_{k}$ if necessary, $M^{\prime}$ is
equal to a matrix obtained by permuting $2^{\text{nd}}$ – $(k+1)^{\text{th}}$
columns of $M$.

###### Proof.

Suppose $M^{\prime}\in\mathcal{M}_{d}$ is obtained by permuting the second and
the third columns of $M\in\mathcal{M}_{d}$. Then we have $M^{\prime}=M\cdot
s_{1}$ and therefore we have

$s_{1}\cdot M^{\prime}\cdot s_{1}=s_{1}\cdot M\in\mathcal{M}_{d}.$

In all other cases, we can get the desired result by applying the above
argument repeatedly. ∎

Suppose $M\in\mathcal{M}_{d}$ is a $(k+1)\times(k+1)$ matrix. Let
$S_{M}\subset W_{k}$ be the set of all permutations in $e_{1},\dots e_{k}$,

$S_{M}:=\\{w\in W_{k}:w(e_{0})=e_{0}\\}.$

Then by above Lemmas 5.2\- 5.3, we see that the orbit of $M$ is in
$\mathcal{M}_{d}$;

$\mathcal{O}_{M}:=\\{M\cdot s:s\in S_{M}\\}\subset\mathcal{M}_{d}.$

Since $\mathcal{M}_{d}$ is finite, we can write $\mathcal{M}_{d}$ as a finite
disjoint union by choosing one representative for each orbit.

$\mathcal{M}_{d}=\sqcup\mathcal{M}_{d,i}\qquad\text{where}\qquad\mathcal{M}_{d,i}=\mathcal{O}_{M_{i}}.$

For instance, we have

$\mathcal{M}_{2}=\mathcal{O}_{M},\qquad\text{where}\qquad
M=\begin{bmatrix}2&1&1&1\\\ -1&0&-1&-1\\\ -1&-1&0&-1\\\
-1&-1&-1&0\end{bmatrix},$
$\mathcal{M}_{3}=\mathcal{O}_{M},\qquad\text{where}\qquad
M=\begin{bmatrix}3&2&1&1&1&1\\\ -2&-1&-1&-1&-1&-1\\\ -1&-1&-1&0&0&0\\\
-1&-1&0&-1&0&0\\\ -1&-1&0&0&-1&0\\\ -1&-1&0&0&0&-1\end{bmatrix},$

and

$\mathcal{M}_{4}=\mathcal{O}_{M_{1}}\sqcup\mathcal{O}_{M_{2}}$

where

$M_{1}=\begin{bmatrix}4&3&1&1&1&1&1&1\\\ -3&-2&-1&-1&-1&-1&-1&-1\\\
-1&-1&-1&0&0&0&0&0\\\ -1&-1&0&-1&0&0&0&0\\\ -1&-1&0&0&-1&0&0&0\\\
-1&-1&0&0&0&-1&0&0\\\ -1&-1&0&0&0&0&-1&0\\\ -1&-1&0&0&0&0&0&-1\end{bmatrix},\
\ M_{2}=\begin{bmatrix}4&2&2&2&1&1&1\\\ -2&-1&-1&-1&-1&-1&0\\\
-2&-1&-1&-1&-1&0&-1\\\ -2&-1&-1&-1&0&-1&-1\\\ -1&-1&-1&0&0&0&0\\\
-1&-1&0&-1&0&0&0\\\ -1&0&-1&-1&0&0&0\end{bmatrix}.$

######  Lemma 5.4.

Suppose $M=\begin{bmatrix}d&\bar{d}\\\
-\bar{c}&-m\end{bmatrix}\in\mathcal{M}_{d}$. Then there is
$\omega\in\Omega_{d}$ such that

$\deg(M\cdot\omega)\lneq d.$

In this case, we have

  * •

$M\cdot\omega(e_{p})\cdot e_{0}=M(e_{p})\cdot e_{0}$ for $p\neq 0,i,j,k$, and

  * •

$M\cdot\omega(e_{p})\cdot e_{0}=d-d_{q}-d_{r}$ for $\\{p,q,r\\}=\\{i,j,k\\}$.

###### Proof.

Suppose $M\in W_{n}$ for some $n\geq 3$. By Lemma 3.1, there are three indices
$i,j,k$ such that $d_{i}+d_{j}+d_{k}\geq d+1$. Let $\omega\in W_{n}$ be a
reflection through the vector $e_{0}-e_{i}-e_{j}-e_{k}$, that is,

$\begin{aligned} \omega\ :\ &e_{0}\mapsto 2e_{0}-e_{i}-e_{j}-e_{k}\\\
&e_{p}\mapsto e_{0}-e_{i}-e_{j}-e_{k}+e_{p}\quad\text{for }\ p=i,j,k\\\
&e_{p}\mapsto e_{p}\quad\text{otherwise}\end{aligned}.$

Thus we have

$\deg(M\cdot\omega)\ =\ 2d-d_{i}-d_{j}-d_{k}\leq d-1.$

The second part of this Lemma follows from a simple matrix multiplication. ∎

Let $\kappa_{i,j,k}$ is the reflection through the vector
$e_{0}-e_{i}-e_{j}-e_{k}$. Since $\kappa_{i,j,k}$ is an involution, we see
that any $M\in\mathcal{M}_{d}$ can be obtained by
$M^{\prime}\cdot\kappa_{i,j,k}$ for some $M^{\prime}\in M_{d}^{\prime}$ with
$d^{\prime}<d$. In fact, by the previous Lemma 5.4, we see that

$d^{\prime}_{i}+d^{\prime}_{j}+d^{\prime}_{k}=3d-2(d_{i}+d_{j}+d_{k})\leq d-2$

where $d_{i}^{\prime},d_{j}^{\prime},d_{k}^{\prime}$ are the corresponding
entries in the first row of the matrix representation of
$M^{\prime}=\begin{bmatrix}d^{\prime}&\bar{d}^{\prime}\\\
-\bar{c}^{\prime}&m^{\prime}\end{bmatrix}$. Notice that $d_{i}^{\prime}$ could
be zero if $d_{j}+d_{k}=d$.

Suppose $M=\begin{bmatrix}d&\bar{d}\\\
-\bar{c}&-m\end{bmatrix}\in\mathcal{M}_{d}$ is given by a $(k+1)\times(k+1)$
matrix. For each $s\leq d-1$, let

$\displaystyle
I_{s,3}(M)=\\{(i_{1},i_{2},i_{3}):d_{i_{1}}+d_{i_{2}}+d_{i_{3}}=s,1\leq
i_{1}\leq i_{2}\leq i_{3}\leq k\\}$ $\displaystyle
I_{s,2}(M)=\\{(i_{1},i_{2},k+1):d_{i_{1}}+d_{i_{2}}=s,1\leq i_{1}\leq
i_{2}\leq k\\}$ $\displaystyle I_{s,1}(M)=\\{(i_{1},k+1,k+2):d_{i_{1}}=s,1\leq
i_{1}\leq k\\}$ $\displaystyle I_{s,0}(M)=\\{(k+1,k+2,k+3)\\}.$

And let

$I_{s}(M)=I_{s,3}(M)\cup I_{s,2}(M)\cup I_{s,1}(M)\cup I_{s,0}(M).$

Notice that this set $I_{s}(M)$ is not necessarily non-empty. Also, let us
define

$h_{s}(\mathcal{M}_{d})=\\{M\cdot\kappa_{i,j,k}:M=\begin{bmatrix}d&\bar{d}\\\
-\bar{c}&-m\end{bmatrix}\in\mathcal{M}_{d},(i,j,k)\in I_{s}(M)\\}$

######  Lemma 5.5.

Suppose $0\leq s\leq d-1$. Then

$h_{s}(\mathcal{M}_{d})\subset\mathcal{M}_{2d-s}.$

###### Proof.

Suppose $M=\begin{bmatrix}d&\bar{d}\\\
-\bar{c}&-m\end{bmatrix}\in\mathcal{M}_{d}$. Also suppose
$d_{i_{1}}+d_{i_{2}}+d_{i_{3}}=s,1\leq i_{1}\leq i_{2}\leq i_{3}\leq k+3$.
Then we have

$\begin{aligned} M\cdot\kappa_{i,j,k}\ :\ &e_{0}\mapsto
M(2e_{0}-e_{i}-e_{j}-e_{k})\\\ &e_{s}\mapsto
M(e_{0}-e_{i}-e_{j}-e_{k}+e_{s})\qquad\text{ for }s\in\\{i,j,k\\}\\\
&e_{s}\mapsto M(e_{s})\qquad\text{ for }s\not\in\\{i,j,k\\}\end{aligned}.$

Thus, we see that the first row of $M\cdot\kappa_{i,j,k}$ is given by strictly
positive numbers and therefore $M\cdot\kappa_{i,j,k}\in\mathcal{M}_{2d-s}$. ∎

######  Theorem 5.6.

For each $d\geq 2$, the set $\mathcal{M}_{d}$ is determined recursively:

$\mathcal{M}_{d}=\bigcup_{i=1}^{d/2}h_{d-2i}(\mathcal{M}_{d-i}),$

with

$\mathcal{M}_{2}=\left\\{\begin{bmatrix}2&1&1&1\\\ -1&0&-1&-1\\\ -1&-1&0&-1\\\
-1&-1&-1&0\end{bmatrix},\ \begin{bmatrix}2&1&1&1\\\ -1&-1&0&-1\\\
-1&0&-1&-1\\\ -1&-1&-1&0\end{bmatrix},\ \begin{bmatrix}2&1&1&1\\\
-1&-1&-1&0\\\ -1&0&-1&-1\\\ -1&-1&0&-1\end{bmatrix}\right\\}$

###### Proof.

This follows from Lemmas 5.4 and 5.5. ∎

######  Theorem 5.7.

The set $\Lambda_{S,d}$ of spectral radii of $\omega\in\Omega_{d}$ is given by

$\Lambda_{S,d}=\\{\rho(\chi_{M,\bar{n}}):\bar{n}=(n_{1},\dots,n_{k}),n_{i}\geq
1,k+1=|M|,M\in\mathcal{M}_{d}\\}$

where $\rho$ denote the largest real root, $|M|$ is the number of rows of $M$,
and $\chi_{M,\bar{n}}$ is the polynomial defined in (3.7). Thus, the set
$\Lambda_{S}$ of all Salem numbers in the Weyl spectrum is given by

$\Lambda_{S}=\cup_{d}\Lambda_{S,d}.$

###### Proof.

The first part of this theorem is an immediate consequence of Theorem 3.8 and
Lemma 5.1. Also, since we have

$\mathcal{M}_{2}=\mathcal{O}_{M},\qquad\text{where}\qquad
M=\begin{bmatrix}2&1&1&1\\\ -1&-1&0&0\\\ -1&0&-1&0\\\ -1&0&0&-1\end{bmatrix},$

we can determine all matrices in $\mathcal{M}_{d}$ by Theorem 5.6. Thus, we
have the set of all Salem numbers in the Weyl Spectrum. ∎

Suppose $M\in\mathcal{M}_{d}$ is given by $(k+1)\times(k+1)$ matrix. For a
proper subset $I\subset\\{2,3,\dots,k+1\\}$, let $M(I)$ be the principal sub-
matrix of $M$ obtained by deleting $i^{\text{th}}$ column and row for
$i\not\in I\cup\\{1\\}$. And let $\mathcal{M}^{\prime}_{d}$ be the set of all
such sub-matrices of $M(I)$, that is,

(5.1)
$\mathcal{M}^{\prime}_{d}=\\{M(I):I\subsetneq\\{2,\dots,k+1\\},k=|M|,M\in\mathcal{M}_{d}\\}.$

######  Theorem 5.8.

The set $\Lambda_{P}$ of all Pisot numbers in the Weyl spectrum is given by

$\Lambda_{P}=\\{\rho(\chi_{M^{\prime},\bar{n}^{\prime}}):\bar{n}^{\prime}=(n_{1},\dots,n_{k}),n_{i}\geq
1,k+1=|M^{\prime}|,M^{\prime}\in\mathcal{M}^{\prime}_{d},d\geq 2\\},$

where $\rho(p)$ is the largest real root of $p$, $|M|$ is the number of rows
of $M$, and $\chi_{M,\bar{n}}$ is the polynomial defined in (3.7).

###### Proof.

In the discussion above Proposition 4.4, we see that all Pisot numbers can be
obtained using certain sub-matrices of $M$ for some
$M\in\cup_{d}\mathcal{M}_{d}$. Since we have $\mathcal{M}_{d}$ for all $d\geq
2$, we can determine all Pisot numbers in the Weyl spectrum. ∎

###### Proof of Theorem A.

This is an immediate consequence of Theorems 5.7 and 5.8. ∎

##  6\. Realization of small Pisot numbers in the Dynamical Spectrum

A Pisot number is an algebraic integer $>1$ whose all Galois conjugates have
modulus $<1$. The smallest Pisot number $p_{1}\approx 1.3247$ is the largest
real root of $t^{3}-t-1$, and the smallest accumulation point in Pisot numbers
is the golden ratio $\phi=(1+\sqrt{5})/2\approx 1.6180$ which is the largest
real root of $t^{2}-t-1$. Thus, the golden ratio $\phi$ is also a Pisot
number.

All Pisot numbers smaller than the golden ratio are listed in [8]. Let us
define

$\displaystyle\theta_{1}(n)\ :=\ t^{n}(t^{2}-t-1)+t^{2}-1$
$\displaystyle\theta_{2}(n)\ :=\ \frac{t^{2n}(t^{2}-t-1)+1}{t-1}$
$\displaystyle\theta_{3}(n)\ :=\ \frac{t^{2n+1}(t^{2}-t-1)+1}{t^{2}-1}$
$\displaystyle\theta_{s}\ :=\ t^{6}-2t^{5}+t^{4}-t^{2}+t-1$

Then the $n^{\text{th}}$ smallest Pisot number $p_{n}$ is a largest real root
of $P_{n}(t)$ where

(6.1) $\displaystyle P_{1}=\theta_{1}(1),$ $\displaystyle
P_{2}=\theta_{1}(2),\ P_{3}=\theta_{1}(3),\ P_{4}=\theta_{3}(1),$
$\displaystyle P_{5}=\theta_{1}(4),\ P_{6}=\theta_{2}(2),\
P_{7}=\theta_{1}(5),\ P_{8}=\theta_{s},\ P_{9}=\theta_{3}(2)$

and for $P_{m},m>9$,

(6.2) $P_{m}\ =\ \left\\{\begin{aligned} &\theta_{1}(m/2+1)\qquad\qquad\ \
\text{if }m>9,m\equiv 0(\text{mod }2)\\\ &\theta_{2}((m-3)/4+1)\qquad\
\text{if }m>9,m\equiv 3(\text{mod }4)\\\
&\theta_{3}((m-1)/4)\qquad\qquad\text{if }m>9,m\equiv 1(\text{mod
}4)\end{aligned}\right.$

###  6.1. Small Pisot numbers in the Weyl Spectrum

There are three possible orbit permutations for $\omega$ with
$\deg(\omega)=2$.

$\mathcal{M}_{2}=\left\\{M_{1}=\begin{bmatrix}2&1&1&1\\\ -1&0&-1&-1\\\
-1&-1&0&-1\\\ -1&-1&-1&0\end{bmatrix},\ M_{2}=\begin{bmatrix}2&1&1&1\\\
-1&-1&0&-1\\\ -1&0&-1&-1\\\ -1&-1&-1&0\end{bmatrix},\
M_{3}=\begin{bmatrix}2&1&1&1\\\ -1&-1&-1&0\\\ -1&0&-1&-1\\\
-1&-1&0&-1\end{bmatrix}\right\\}$

Let us set

$M_{2}^{\prime}=\begin{bmatrix}2&1&1\\\ -1&-1&0\\\ -1&0&-1\end{bmatrix},\quad
M^{\prime}_{3}=\begin{bmatrix}2&1&1\\\ -1&-1&-1\\\
-1&0&-1\end{bmatrix},\quad\text{ and }\quad M^{\prime}=\begin{bmatrix}2&1\\\
-1&-1\end{bmatrix}$

Also, let

$f_{i}(n_{1},n_{2})\ =\ \chi_{M^{\prime}_{i},(n_{1},n_{2})}\quad i=2,3$
$f(n_{1})\ =\ \chi_{M^{\prime},(n_{1})}$

where $\chi_{M,\bar{n}}$ is the polynomial defined in (3.7). Since
$M_{2}^{\prime}$ and $M_{3}^{\prime}$ are sub-matrices of $M_{2}$ and $M_{3}$
obtained by deleting $4^{\text{th}}$ column and row, by Theorem 5.8 we know
that the largest real root of $f_{i}(n_{1},n_{2})$ is a Pisot number for all
$i=1,2$ and $n_{1},n_{2}\geq 1$. Similarly, since $M^{\prime}$ is a principal
sub-matrix of both $M_{2}$ and $M_{3}$ obtained by deleting $3^{\text{rd}}$
and $4^{\text{th}}$ columns and rows, the largest real root of $f(n_{1})$ is a
Pisot number for all $n_{1}\geq 1$. With the formula (3.7), we see that

$f_{3}(n_{1},n_{2})=t^{n_{1}+n_{2}+1}-2t^{n_{1}+n_{2}}+t^{n_{1}+1}+t^{n_{2}+1}-t^{n_{1}}-t^{n_{2}}+t-1,$
$f_{2}(n_{1},n_{2})=t^{n_{1}+n_{2}+1}-2t^{n_{1}+n_{2}}+t^{n_{1}+1}+t^{n_{2}+1}-t^{n_{1}}-t^{n_{2}}+t,$

and

$f(n_{1})=t^{n_{1}+1}-2t^{n_{1}}+t-1$

Thus we have

######  Lemma 6.1.

For each $n\geq 1$, we have

$f_{3}(1,n)=\theta_{1}(n)$

and

$f_{3}(2,3)=\theta_{s}.$

######  Lemma 6.2.

For each $n\geq 1$, we have

$f_{2}(1,2n+2)=t^{2}(t-1)\theta_{2}(n),\qquad
f_{2}(1,2n+3)=t^{2}(t^{2}-1)\theta_{3}(n).$

######  Lemma 6.3.

For $n_{1}=1$, we have

$f(1)=t^{2}-t-1.$

Therefore, we have

######  Proposition 6.4.

Every Pisot number smaller than or equal to the golden ratio is realized as a
dynamical degree of a birational map.

###### Proof.

Since the dynamical spectrum is identical to the Weyl spectrum, this
Proposition is followed by the previous two Lemmas 6.1 \- 6.3. ∎

######  Remark 6.5.

For a given $d\geq 2$, the set of orbit permutations $\mathcal{M}_{d}$ is
finite. For a given orbit permutation $M\in\mathcal{M}_{d}$, the Proposition
Theorem B shows that the spectral radius is increasing in orbit lengths. Thus,
it is possible to check small Pisot numbers in $\Lambda_{d}$ for a given
$d\geq 2$. Since the conjugacy class of an element $\omega\in W_{\infty}$ is
infinite, we would have $\Lambda_{d}\subset\Lambda_{d}^{\prime}$ for $d\leq
d^{\prime}$ where $\Lambda_{d}$ is the set of spectral radii of
$\omega\in\Omega_{d}$ of degree $d$. With computer experiment, we made
interesting observations:

  * •

In $\Lambda_{2}$, the smallest Pisot number $p_{\star}$ strictly bigger than
the golden ratio is a root of

$t^{7}-2t^{6}+t^{5}-t^{4}+t^{3}-t^{2}+t-1=0$

and it is approximately $\sim 1.64073$. This Pisot number is the limit of
spectral radii of $\omega_{n}\in\Omega_{2}$ as $n\to\infty$ where

$\mathcal{O}(\omega_{n})=(M_{3},\bar{n}=(2,4,n)).$

Also, this Pisot number can be realized as a dynamical degree of a birational
map of degree 2.

  * •

In $\Lambda_{3}$, the smallest Pisot number strictly bigger than the golden
ratio is again $p_{\star}$. This Pisot number is the limit of spectral radii
of $\omega_{n}\in\Omega_{3}$ as $n\to\infty$ where

$\mathcal{O}(\omega_{n})=\left(\begin{bmatrix}3&1&2&1&1&1\\\
-2&-1&-1&-1&-1&-1\\\ -1&-1&-1&0&0&0\\\ -1&0&-1&0&0&-1\\\ -1&0&-1&-1&0&0\\\
-1&0&-1&0&-1&0\end{bmatrix},\bar{n}=(1,2,1,3,n)\right).$

  * •

In $\Lambda_{3}\setminus\Lambda_{2}$, the smallest Pisot number strictly
bigger than the golden ratio is the root of

$t^{7}-t^{6}-2t^{5}+2t^{3}+t^{2}-t-1=0$

and it is approximately $\sim 1.75827$.

  * •

Based on the computer experiment, it seems that $p_{\star}$ is the smallest
Pisot number strictly bigger than the golden mean in the Weyl spectrum.
However, we could not get rid of the possibilities of the existence of a Pisot
number between the golden mean and $p_{\star}$ in $\Lambda_{d}$ for
sufficiently large $d$.

###  6.2. Realization

Notice that for each $i=1,2$ and $n_{1},n_{2}\geq 1$, we have

$\rho(f_{i}(n_{1},n_{2}))=\lim_{n_{3}\to\infty}\rho(\chi_{M_{i},(n_{1},n_{2},n_{3})})$

where $\rho(p)$ is the largest real root of $p$. Diller [5] gave the method of
constructing a quadratic birational map properly fixing a cubic curve.

######  Theorem 6.6.

[5, Theorem 1] Let $C\subset\mathbf{P}^{2}(\mathbb{C})$ be an irreducible
cubic curve. Suppose $p_{1},p_{2},p_{3}\in C_{\text{reg}}$ are points in the
smooth locus of $C$, $b\neq 0\in C_{\text{reg}}$, and $a\in\mathbb{C}^{*}$.
Then there exists the unique quadratic birational map $f$ properly fixing $C$
with $\mathcal{I}(f)=\\{p_{1},p_{2},p_{3}\\}$ and the restriction
$f|_{C_{\text{reg}}}:z\to az+b$ if and only if $a(p_{1}+p_{2}+p_{3})=3b$. In
this case the points of indeterminacy for $f^{-1}$ are given by
$\\{ap_{i}-2b,i=1,2,3\\}$.

Using the group law of a cubic curve in $\mathbf{P}^{2}(\mathbb{C})$, Diller
[5] also gives a simple method to construct a birational map with given orbit
data. Let us set
$C=\\{\gamma(t)=[1:t:t^{3}]:t\in\mathbb{C}\cup\\{\infty\\}\\}\subset\mathbf{P}^{2}(\mathbb{C})$.
Also let
$p_{1}=\gamma(t_{1}),p_{2}=\gamma(t_{2}),p_{3}=\gamma(t_{3}),a\in\mathbb{C}^{*}$
and set $b=1-a$. With the same notation as in Example 2.4, Diller [5] shows
that for sufficiently large $n_{3}$, there exist
$f\in\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))$ with
$\mathcal{O}(f)=(M_{3},(n_{1},n_{2},n_{3}))$ determined by $t_{1},t_{2},t_{3}$
and $a$ satisfying

(6.3) $\left\\{\begin{aligned} &(t_{2}-1)/a^{n_{1}}+1+t_{2}+t_{3}=0\\\
&(t_{3}-1)/a^{n_{2}}+1+t_{1}+t_{3}=0\\\
&(t_{1}-1)/a^{n_{3}}+1+t_{2}+t_{1}=0\\\
&t_{1}+t_{2}+t_{3}=3(1-a)/a\end{aligned}\right.$

The first three conditions in (6.3) come from the fact that the total of
parameters of all intersection points of a line with a cubic curve $C$ is
equal to zero. The first equation is given by the total of parameters of
$E_{i}\cap C$, and the last equation is due to Theorem 6.6.

To obtain a birational map $f\in\text{Bir}(\mathbf{P}^{2}(\mathbb{C}))$ with
orbit data $(M_{3}^{\prime},(n_{1},n_{2}))$, $t_{1},t_{2},t_{3}$ and $a$ need
to satisfy the first, second and the last equations in (6.3) and to fail the
third equation for all $n_{3}\in\mathbb{Z}_{>0}$. By counting the number of
equations, we see that there are infinitely many such $t_{1},t_{2},t_{3}$ and
$a$. In fact, if $f$ is birationally equivalent to an automorphism, then $a$
has to be equal to $\lambda(f)$ or a Galois conjugate of the dynamical degree
$\lambda(f)$. Thus, by choosing $a\in\mathbb{Z}$, we can construct a
birational map with orbit data $(M_{3}^{\prime},(n_{1},n_{2}))$ properly
fixing a cubic curve $C$. A detailed discussion of constructing a birational
map with given parameter values $t_{1},t_{2},t_{3}$, and $a$ is available in
[5, 2, 12]. As a summary, we have

###### Proof of Theorem C.

Theorem C follows from Proposition 6.4. Let us describe concrete construction
using the method in [5]. For $(n_{1},n_{2})\in\\{(2,3),(1,n),n\geq 1\\}$, let
$t_{1},t_{2},t_{3}$ and $a$ be complex numbers satisfying

(6.4) $\displaystyle\frac{1}{a^{n_{1}}}(t_{2}-1)+1+t_{2}+t_{3}=0$
$\displaystyle\frac{1}{a^{n_{2}}}(t_{3}-1)+1+t_{1}+t_{3}=0$ $\displaystyle
t_{1}+t_{2}+t_{3}=3\frac{1-a}{a}$
$\displaystyle\frac{1}{a^{k}}(t_{1}-1)+1+t_{2}+t_{1}\neq 0\quad\text{ for all
}k\geq 1.$

The first three equations in (6.4) can be written as a matrix form

$\begin{bmatrix}0&1/a^{n_{1}}+1&1\\\ 1&0&1/a^{n_{2}}+1\\\
1&1&1\end{bmatrix}\cdot\begin{bmatrix}t_{1}\\\ t_{2}\\\ t_{3}\end{bmatrix}\ =\
\begin{bmatrix}1/a^{n_{1}}-1\\\ 1/a^{n_{2}}-1\\\ 3(1-a)/a\end{bmatrix}$

Since the determinant of the $3\times 3$ matrix on the left hand side is equal
to $1/a^{n_{1}+n_{2}}+1/a^{n_{2}}+1$, for any choice of $a\neq 0,1$, there is
a non-trivial solution. Since the last condition in (6.4) is an open
condition, we see that there is always such $t_{1},t_{2},t_{3}$ and $a$. By
Diller [5], there is a birational map with orbit data
$(M_{3}^{\prime},(n_{1},n_{2})$. Similarly, there are complex numbers
$t_{1},t_{2},t_{3}$ and $a$ satisfying

(6.5) $\displaystyle\frac{1}{a^{n_{1}}}(t_{2}-1)+1+t_{2}+t_{3}=0$
$\displaystyle\frac{1}{a^{n_{2}}}(t_{1}-1)+1+t_{1}+t_{3}=0$ $\displaystyle
t_{1}+t_{2}+t_{3}=3\frac{1-a}{a}$
$\displaystyle\frac{1}{a^{k}}(t_{3}-1)+1+t_{2}+t_{1}\neq 0\quad\text{ for all
}k\geq 1.$

and thus there exists a birational map with orbit data
$(M_{2}^{\prime},(1,n))$. Finally, we see that there exists a birational map
with orbit data $(M^{\prime},(1))$ using the conditions

(6.6) $\displaystyle\frac{1}{a^{n_{1}}}(t_{2}-1)+1+t_{2}+t_{3}=0$
$\displaystyle t_{1}+t_{2}+t_{3}=3\frac{1-a}{a}$
$\displaystyle\frac{1}{a^{\ell}}(t_{1}-1)+1+t_{1}+t_{3}\neq 0\quad\text{ for
all }\ell\geq 1$ $\displaystyle\frac{1}{a^{k}}(t_{3}-1)+1+t_{2}+t_{1}\neq
0\quad\text{ for all }k\geq 1.$

By Lemmas 6.1 \- 6.3, we see that the set of dynamical degrees of the above
birational maps is the set of all Pisot numbers smaller than or equal to the
golden ratio. ∎

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