1 Introduction

Let X be a smooth projective variety over an algebraically closed field k of characteristic $0$ . For a birational map $f:X\dashrightarrow X$ and $0\leq p\leq \mathrm {dim}(X)$ , the pth dynamical degree of f is defined by

$$ \begin{align*} \unicode{x3bb}_p(f)=\lim_{n\to\infty}((f^n)^{*}(H^p)\cdot H^{\mathrm{dim}X-p})^{{1}/{n}}, \end{align*} $$

where H is an ample divisor (cf. [Reference Truong16]).

The pullback map $f^{*}$ and the pushforward map $f_{*}$ are defined as follows. For a birational map $f:X\dashrightarrow X$ , let $\Gamma _f$ be the closure of the graph of f (cf. Figure 1). The pullback $f^{*}:A_{l}(X)\rightarrow A_{l}(X)$ and the pushforward $f_{*}:A_{l}(X)\rightarrow A_{l}(X)$ are defined as ${f^{*}(\gamma ):={\pi }_{*}([\Gamma _f]\cdot {\pi }^{\prime *}\gamma )}$ and $f_{*}(\gamma ):={\pi }^{\prime }_{*}([\Gamma _f]\cdot \pi ^{*}\gamma )$ (cf. [Reference Fulton12, Definition 16.1.2], [Reference Truong16, §3.1]), where $\gamma $ is an algebraic cycle on X.

In addition, the dynamical degrees are calculated as

$$ \begin{align*} \unicode{x3bb}_p(f)=\lim_{n\to\infty}\|(f^n)^{*}:N^p(X)\otimes_{\mathbb{Z}}\mathbb{R}\rightarrow N^p(X)\otimes_{\mathbb{Z}}\mathbb{R}\|^{{1}/{n}}, \end{align*} $$

where $N^p(X)$ is the group of algebraic cycles of codimension p on X modulo numerical equivalence (cf. [Reference Truong16, Reference Truong17]).

When $k=\mathbb {C}$ , the projective variety X can be considered as a compact Kähler manifold, the birational map f can be considered as a meromorphic map, and the dynamical degrees can be calculated as

$$ \begin{align*} \unicode{x3bb}_p(f)&=\lim_{n\to+\infty}\bigg(\int_{X} (f^n)^{*}(\omega_X^p)\wedge\omega_X^{\mathrm{dim}X-p}\bigg)^{{1}/{n}}\\ &=\lim_{n\to+\infty}\|(f^n)^{*}:\mathrm{H}^{p,p}(X)\rightarrow\mathrm{H}^{p,p}(X)\|^{{1}/{n}}, \end{align*} $$

where $\omega _X$ is a normalized Kähler form, $\mathrm {H}^{p,p}(X)$ ( $0\leq p\leq \mathrm {dim}X$ ) is the $(p,p)$ -Dolbeault cohomology, and $\|\cdot \|$ is a norm for linear transformations (cf. [Reference Dinh and Sibony9–Reference Dinh and Sibony11]).

The values of the dynamical degrees are not less than $1$ and $\unicode{x3bb} _0(f)=1$ . When f is a morphism, the dynamical degrees of f are algebraic, since $(f^n)^{*}=(f^{*})^n$ . In the case of $X=\mathbb {P}^d$ , the rational map $f:\mathbb {P}^d\dashrightarrow \mathbb {P}^d$ can be written as $f=[f_0:f_1:\cdots :f_d]$ by the homogeneous polynomials (over k) of the same degree. Define $\mathrm {deg}(f):=\mathrm {deg}(f_0)$ and then the first dynamical degree of f is calculated by

$$ \begin{align*} \unicode{x3bb}_1(f)&=\lim_{n\to\infty}((f^n)^{*}(H)\cdot H^{d-1})^{{1}/{n}}\\ &=\lim_{n\to\infty}(\mathrm{deg}(f^n))^{{1}/{n}}. \end{align*} $$

Figure 1

The graph of a birational map $f:X\dashrightarrow X$ .

When $d=2$ , it is known that the first dynamical degree of f is always algebraic by the following theorems.

These theorems imply that $\unicode{x3bb} _1(f)$ is either $1$ or a Pisot number, or a Salem number (see definition in [Reference Bertin, Decomps-Guilloux, Grandet-Hugot, Pathiaux-Delefosse and Schreiber5]) for the case where $d=2$ . Salem numbers, which can be realized as $\unicode{x3bb} _1(f)$ for some birational map $f:\mathbb {P}^2\dashrightarrow \mathbb {P}^2$ , are all determined in [Reference Uehara18, Theorem 1.1]. Pisot numbers, which can be realized as $\unicode{x3bb} _1(f)$ , are partially determined (cf. [Reference Kim13, Theorem C], [Reference Bedford and Kim2, §3], [Reference Blanc and Cantat6, §7] ).

In [Reference Bell, Diller, Jonsson and Krieger4], a birational map of $\mathbb {P}^d(d\geq 3$ ) with a transcendental first dynamical degree is constructed.

In [Reference Bell, Diller, Jonsson and Krieger4, §7.2], the specific birational map $\phi :\mathbb {P}^3\dashrightarrow \mathbb {P}^3$ with a transcendental (first) dynamical degree is presented.

Remark 1.4. The birational map f constructed in Theorem 1.3 is of the form $f=L_{B^{-1}}\circ h_{-I}\circ L_{B}\circ h_A$ , where:

• • $h_{M}:\mathbb {P}^d\dashrightarrow \mathbb {P}^d$ is the monomial map corresponding to $M\in \mathrm {GL}_{d}(\mathbb {Z})$ ;

• • $L_{M'}:\mathbb {P}^d\dashrightarrow \mathbb {P}^d$ is the linear transformation corresponding to $M'\in \mathrm {GL}_{d+1}(k)$

(see the details in [Reference Sugimoto15, §2.3]); and

$$ \begin{align*} I=\begin{pmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end{pmatrix}\!,\quad B=\begin{pmatrix} 1 & -1 & 1 & \cdots & (-1)^{d} \\ 1 & 1 & -1 & \cdots & (-1)^{d-1} \\ -1 & 1 & 1 & \cdots & (-1)^{d-2} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ (-1)^{d-1} & (-1)^{d-2} & (-1)^{d-3} & \cdots & 1 \end{pmatrix}\!. \end{align*} $$

In addition, the specific example $\phi :\mathbb {P}^3\dashrightarrow \mathbb {P}^3$ in [Reference Bell, Diller, Jonsson and Krieger4, §7.2] is the case where $d=3$ and

$$ \begin{align*} A=\begin{pmatrix} -3 & -14 & -12 \\ 4 & 11 & 6 \\ -2 & -4 & -1 \end{pmatrix}\!. \end{align*} $$

In [Reference Sugimoto15, §3], we calculate the second dynamical degree of the map $\phi $ and confirm that $\unicode{x3bb} _1(\phi )<\unicode{x3bb} _2(\phi )$ . In addition, we construct another birational map $\psi :\mathbb {P}^3\dashrightarrow \mathbb {P}^3$ with a transcendental first dynamical degree such that $\unicode{x3bb} _1(\psi )>\unicode{x3bb} _2(\psi )$ (cf. [Reference Sugimoto15, §4]) by taking

$$ \begin{align*} A=\begin{pmatrix} 56 & -19 & -17 \\ -16 & 6 & 5 \\ 207 & -71 & -63 \end{pmatrix}\!. \end{align*} $$

It is confirmed that the second dynamical degree $\unicode{x3bb} _2(\psi )$ is algebraic, as follows.

Theorem 1.5. (Cf. [Reference Sugimoto15, §4])

There exists a birational map $\psi :\mathbb {P}^3\dashrightarrow \mathbb {P}^3$ whose dynamical degrees are

$$ \begin{align*} \unicode{x3bb}_0(\psi)=1,\quad \unicode{x3bb}_1(\psi)=\unicode{x3bb},\quad \unicode{x3bb}_2(\psi)=a,\quad \unicode{x3bb}_3(\psi)=1, \end{align*} $$

with $291\leq \unicode{x3bb} \leq 669$ being a transcendental number and $a=174.6660\cdots $ being a root of

$$ \begin{align*} a^9-173a^8-291a^7-2a^6+332a^5+334a^4+238a^3+75a+75=0. \end{align*} $$

We use this result to obtain a transcendental arithmetic degree in [Reference Sugimoto15, Corollary 4.5] by combining with the results of [Reference Matsuzawa and Wang14, Theorem B].

In this paper, we use Theorem 1.5 to resolve the next question by using the algebraicity of a.

It is known that the intermediate dynamical degrees of an automorphism f of a compact Kähler manifold (or a projective variety over $\mathbb {C}$ ) are either irrational or equal to $1$ [Reference Bedford1, Theorem 6.1].

This theorem is proved in §3. In the counterexample that we found, all intermediate dynamical degrees are transcendental, but they are algebraically dependent.

Another question about dynamical degrees is presented in [Reference Bell, Diller, Jonsson and Krieger4, Question 1.6].

This question is resolved for the case $d\geq 3$ in [Reference Sugimoto15, Remark 4.6]. In addition, for the case where $d=3$ , we construct an example where the first dynamical degree $\unicode{x3bb} _1(f)$ is strictly maximal among the dynamical degrees in Theorem 1.5. However, for the case where $d>3$ , the first dynamical degree is not strictly maximal among the dynamical degrees in our example (see [Reference Sugimoto15, Remark 4.6]).