Posted by: matheuscmss | May 31, 2013

## Dynamical degrees (after Blanc and Cantat)

About one week ago (May 23, 2013), I saw a very nice talk of Serge Cantat on Dynamical degrees at the seminar of the Topology and Dynamics team of Orsay.

Before entering on the details of Serge’s talk, let me mention that, generally speaking, this Topology and Dynamics seminar has a very interesting format: indeed, prior to the “main” talk, some local member makes an informal talk (where we are served with coffee and tea during the exposition…) to make the audience more comfortable with the topic of the main talk. An important feature of this format is that the main speaker is usually not allowed to see the informal talk, so that the main talk usually has some overlaps with the informal talk. Of course, this is a really nice feature since the audience is able to pose more questions after seeing the same idea for the second time (with different perspectives as they were presented by distinct persons).

In the case of Serge’s talk, the informal talk was delivered by Yves de Cornulier who introduced us to several aspects of the so-called Cremona group (the central object in Serge’s talk).

Below the fold, I will reproduce my notes from Serge’s talk around his joint work with Jérémy Blanc, while using from time to time my notes from Yves de Cornulier’s talk. As usual, any mistakes/typos in this post are my entire responsibility.

1. Preliminaries

Let ${k}$ be an algebraically closed field, e.g., ${k=\mathbb{C}}$. Denote by ${\mathbb{A}_k^2}$, resp. ${\mathbb{P}_k^2}$, the affine, resp. projective plane.

The so-called Cremona group ${Cr_2(k)}$ is the group of birational transformations of the plane, i.e., ${Cr_2(k):=Bir(\mathbb{P}_k^2)=Bir(\mathbb{A}_k^2):=}$

$\displaystyle \{f=(f_1,f_2), f_1, f_2 \in k(x,y):\,\exists\, g=(g_1,g_2), g_1, g_2\in k(x,y), f\circ g=g\circ f\}.$

Here, one has to be careful when defining the compositions ${f\circ g}$ and ${g\circ f}$ as they only make sense outside certain indetermination sets. In other words, we want to consider our birational transformations to be defined outside certain proper, Zariski closed subsets of ${\mathbb{A}_k^2}$ and/or ${\mathbb{P}_k^2}$. In particular, this explains the equality ${Bir(\mathbb{P}_k^2)=Bir(\mathbb{A}_k^2)}$: indeed, ${\mathbb{A}_k^2}$ can be viewed as a Zariski open and dense subset of ${\mathbb{P}_k^2}$ via ${(x,y)\mapsto [x:y:1]}$, where ${[x:y:z]}$ are the homogenous coordinates on ${\mathbb{P}_k^2}$.

In what follows, we’ll denote a birational transformation ${f\in Cr_2(k)}$ by ${f:\mathbb{A}_k^2\dashrightarrow \mathbb{A}_k^2}$ or ${f:\mathbb{P}_k^2\dashrightarrow \mathbb{P}_k^2}$, where the dashed arrow serves to indicate that we think of ${f}$ as well-defined outside some indetermination set (that will be a small set; see the next paragraph).

Note that, by definition, a birational transformation ${f:\mathbb{A}_k^2\dashrightarrow \mathbb{A}_k^2}$ has the form ${(x,y)\mapsto (f_1(x,y),f_2(x,y))}$, ${f_1, f_2\in k(x,y)}$ in affine coordinates, while a birational transformation ${f:\mathbb{P}_k^2\dashrightarrow \mathbb{P}_k^2}$ has the form ${[x:y:z]\mapsto [P(x,y,z):Q(x,y,z):R(x,y,z)]}$, where ${P, Q, R}$ are homogeneous polynomials of the same degree ${d\geq 1}$ without common factors of degree ${>1}$. In this setting, we define the degree ${deg(f)}$ of ${f}$ is the common degree ${d}$ of the rational functions ${P, Q, R}$. Also, we note that the set of indetermination of ${f}$ is ${Z(P)\cap Z(Q)\cap Z(R)}$ (where ${Z(M)}$ is the set of zeros of ${M}$). In particular, this indetermination set is finite (because ${P}$, ${Q}$ and ${R}$ have no common factors of degree ${>1}$).

Let us now give some concrete examples of elements of the Cremona group ${Cr_2(k)}$.

Example 1 From the point of view of function fields, ${Cr_2(k)\simeq Aut_{k}(k(x,y))}$ and it contains the projective linear group ${PGL_3(k)}$ of automorphisms (i.e., birational transformations of degree ${1}$) of the projective plane ${\mathbb{P}_k^2}$. For sake of comparison, recall that the Cremona group (of birational transformations) of the projective line ${\mathbb{P}_k^1}$ is ${Aut_k(k(x))}$ and it coincides with the projective linear group ${PGL_2(k)}$ of automorphisms (degree ${1}$) of the projective line (by thinking of a matrix ${\left(\begin{array}{cc} a & b \\ c & d \end{array}\right)\in PGL_2(k)}$ as a Moebius transformation ${x\mapsto (ax+b)/(cx+d)}$). Note that the set of indetermination of any automorphism of a projective plane (of any dimension) is empty.

Example 2 The standard quadratic involution ${\sigma([x:y:z])=\left[\frac{1}{x}:\frac{1}{y}:\frac{1}{z}\right] = [yz:xz:xy]}$ is an element of ${Cr_2(k)}$ not belonging to ${PGL_3(k)}$. In particular, ${Cr_2(k)}$ does not coincide with ${PGL_3(k)}$ (contrary to the case of the projective line ${\mathbb{P}_k^1}$). Nevertheless, Max Noether showed that ${Cr_2(k)}$ is generated by ${PGL_3(k)}$ and ${\sigma}$. Note that the set of indetermination of ${\sigma}$ consists of the three points ${[1:0:0]}$, ${[0:1:0]}$ and ${[0:0:1]}$.

Example 3 Let ${f(x,y)=(x/y, y)}$ or, more generally, ${f(x,y)=f_A(x,y)=(x^a y^b, x^c y^d)}$ with ${A=\left(\begin{array}{cc} a & b \\ c& d \end{array}\right)\in GL(2,\mathbb{Z})}$. Note that the standard quadratic involution ${\sigma}$ from Example 2 is ${\sigma=f_{-Id}}$ where ${-Id=\left(\begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array}\right)\in GL(2,\mathbb{Z})}$.

Example 4 Let ${h(x,y)=(y, x+y^2+c)}$, ${c\in k}$ (or ${h([x:y:z])=[yz:xz+y^2+cz:z^2]}$ in homogenous coordinates). In the literature, ${h}$ is known as the complex Hénon map.

2. Dynamical degree

Geometrically, the degree ${deg(f)}$ of a birational transformation ${f:\mathbb{P}_k^2\dashrightarrow \mathbb{P}_k^2}$ is the intersection of a generic line ${L\subset \mathbb{P}_k^2}$ with ${f^{-1}(L)}$. Thus, the quantity ${deg(f)}$ gives some coarse information on how ${f}$ acts on ${\mathbb{P}_k^2}$.

Hence, from the point of view of Dynamical Systems, it is natural to try to capture more information on the action of ${f}$ on the projective plane by investigating the sequence ${deg(f^n)}$ of degrees of the iterates ${f^n}$, ${n\in\mathbb{N}}$, of ${f}$. Here, it is not hard to show that the degree is a submultiplicative quantity (i.e., ${deg(f\circ g)\leq deg(f)\cdot deg(g)}$), so that the limit

$\displaystyle \lambda(f)=\lim\limits_{n\rightarrow \infty} (deg(f^n))^{1/n}$

exists.

The quantity ${\lambda(f)\geq 1}$ is called the dynamical degree of ${f}$.

Example 5 By direct computation, one can check that for the complex Hénon maps ${h}$ in Example 4 it holds ${deg(h^n)=2^n}$ for all ${n\in\mathbb{N}}$. Therefore, the dynamical degree of complex Hénon maps is ${\lambda(h)=2}$.

Example 6 The degree is a submultiplicative quantity but it is not multiplicative in general. Indeed, the standard quadratic involution ${\sigma}$ from Example 2 has degree ${2}$, but the degree of ${\sigma\circ\sigma=Id}$ is ${1}$. In particular, ${\lambda(\sigma)=1}$.

Example 7 The dynamical degrees of the maps ${f_A}$, ${A\in GL(2,\mathbb{Z})}$, from Example 3 are easy to determine: because of the functorial relation ${f_A^n=f_{A^n}}$, the degree of ${f_A^n}$ is the maximum of ${a_n+b_n}$ and ${c_n+d_n}$ where ${A^n=:\left(\begin{array}{cc} a_n & b_n \\ c_n & d_n \end{array}\right)}$, so that ${deg(f_A^n)}$ is comparable to ${\|A^n\|}$ (with a multiplicative comparison constant depending only on the choice of ${\|.\|}$); in particular, ${\lambda(f_A)}$ is the spectral radius of ${A}$.

Example 8 By applying the discussion of the previous example to ${f_A}$ with ${A=\left(\begin{array}{cc} 2 & 1 \\ 1 & 1 \end{array}\right)}$, we find that ${\lambda(f_A)=(3+\sqrt{5})/2}$. More generally, by considering the matrices ${A=\left(\begin{array}{cc} n-1 & n-2 \\ 1 & 1 \end{array}\right)}$ and ${\left(\begin{array}{cc} n-1 & n-2 \\ 1 & 1 \end{array}\right)}$ to make ${\lambda(f_A)}$ be the roots of the polynomials ${x^2-nx+1}$ for ${n\geq 2}$ and ${x^2-nx-1}$ for ${n\geq 1}$. In particular, the gold number ${(1+\sqrt{5})/2}$ is the dynamical degree of ${f_A}$, ${A=\left(\begin{array}{cc} 0 & 1 \\ 1 & 1 \end{array}\right)}$.

The main question guiding Serge’s talk was:

What is the structure of the set ${\Lambda(\mathbb{P}_k^2):=\{\lambda(f): f\in Cr_2(k)\}}$?

Remark 1 As Serge pointed out, besides the intrinsic interest of ${\Lambda(\mathbb{P}_k^2)}$, the analog of ${\Lambda(\mathbb{P}_k^2)}$ for other algebraic varieties might be an useful birational invariant in general. In other words, this dynamical invariant could be useful for treating classification problems in Algebraic Geometry.

Before discussing this question, let us list some basic properties of the dynamical degree:

• the dynamical degree is not larger than the (algebraic) degree: ${\lambda(f)\leq deg(f)}$;
• the dynamical degree is invariant under conjugation: ${\lambda(h^{-1}fh)=\lambda(f)}$;
• the logarithm of the dynamical degree is an upper bound for the topological entropy: ${\log\lambda(f)\geq h_{top}(f)}$.

Remark 2 In some sense, an element ${f\in Cr_2(k)}$ with ${\lambda(f)>1}$ is comparable to a pseudo-Anosov element ${\phi}$ in the mapping class group ${Mod(S_g)}$ of a genus ${g\geq 2}$ surface (and ${\lambda(f)}$ plays the role of the dilation of a pseudo-Anosov ${\phi}$ or the volume of the hyperbolic 3-manifold ${(S_g\times[0,1])/''\phi''}$ obtained by suspension). In fact, this comparison is partly supported by the following result of Diller and Favre showing (among other things) that one of the following possibility occurs:

• the sequence ${deg(f^n)}$ is bounded (and, a fortiori, ${\lambda(f)=1}$) if and only if there is an iterate ${f^n}$ conjugated to an automorphism ${\phi}$ isotopic to the identity;
• the sequence ${deg(f^n)}$ is a linear function of ${n}$ (and, a fortiori, ${\lambda(f)=1}$) if and only if ${f}$ preserves a rational fibration;
• the sequence ${deg(f^n)}$ is a quadratic function of ${n}$ (and, a fortiori, ${\lambda(f)>1}$) if and only if ${f}$ preserves an elliptic fibration;
• the sequence ${deg(f^n)}$ is an exponential function of ${n}$ (and, a fortiori, ${\lambda(f)>1}$).

Coming back to the main question, we will discuss in next section some results of Diller and Favre saying that ${\Lambda(\mathbb{P}_k^2)}$ consists of Pisot and Salem numbers.

3. The structure of ${\Lambda(\mathbb{P}_k^2)}$ after Diller and Favre

In this section we will sketch the proof of the following result:

Theorem 1 (Diller and Favre (2001)) If ${f\in Cr_2(k)}$, then ${\lambda(f)\in\{1\}\cup Pisot \cup Salem}$.

Here, ‘Pisot’ and ‘Salem’ stand for the sets of Pisot and Salem numbers. For sake of convenience, let us recall the definitions and some facts about these sets:

Definition 2 We say that ${\lambda\in\mathbb{R}}$ is a Pisot number if ${\lambda}$ is an algebraic integer such that ${\lambda>1}$ and all its Galois conjugates have moduli ${<1}$. Similarly, we say that ${\lambda\in\mathbb{R}}$ is a Salem number if ${\lambda}$ is an algebraic integer such that ${\lambda>1}$, all its Galois conjugates have moduli ${\leq 1}$, and there is at least one Galois conjugate of ${\lambda}$ of modulus ${1}$.

Among the basic features of Pisot and Salem numbers, we have:

• every positive integer ${>1}$ is a Pisot number;
• the smallest Pisot number ${\lambda_P\simeq 1.32...}$ is the root of ${x^3-x-1}$;
• the smallest Pisot number that is accumulated by other Pisot numbers (from both sides) is the golden ratio ${\lambda_G=(1+\sqrt{5})/2\simeq 1.61...}$ (and, actually, the discrete set of Pisot numbers between ${\lambda_P}$ and ${\lambda_G}$ is “known”);
• the set of Pisot numbers is closed;
• the closure of the set of Salem numbers contains the set of Pisot numbers;
• it is conjectured that the infimum of the set of Salem numbers is ${>1}$, and, more precisely, this infimum is believed to be the Lehmer number ${\lambda_L}$ (that is, the largest root of ${x^{10}+x^9-(x^7+x^6+x^5+x^4+x^3)+x+1}$).

Let us give briefly sketch the proof of Theorem 1. First, we will try to convince ourselves that, for any ${f\in Cr_2(f)}$, the quantity ${\lambda(f)}$ is an algebraic integer. Very roughly speaking, the idea is that ${\lambda(f)}$ is the spectral radius of a matrix with integer coefficients corresponding to a certain action on homology/cohomology.

In principle, it is tempting to try to relate ${\lambda(f)}$ to the action of ${f}$ on homology, but this naive idea does not work because the action of ${f}$ on the (second) cohomology group occurs by multiplication by ${deg(f)}$ but, in general, we might have that ${\lambda(f)}$ is strictly smaller than the spectral radius ${deg(f)}$ of the action of ${f}$ in cohomology because it might happen that ${deg(f^n) (cf. Example 2).

In order to overcome this difficulty, it is clear that we need to understand “what goes wrong” when ${deg(f^n). Here, we invoke the general fact that the equality ${deg(f\circ g)=deg(f)\cdot deg(g)}$ holds except if ${g}$ contracts a curve into an indetermination point of ${f}$: for instance, the reader is invited to think about this in the particular case of the standard quadratic involution ${\sigma}$ from Example 2 (while keeping in mind the fact that ${deg(f)}$ is the intersection of a line ${L}$ with ${f^{-1}(L)}$). In particular, if ${deg(f^n), this means that some curve meets an indetermination point of ${f}$. Fortunately, there is a well-known method to get around the difficulty created by the indetermination set: each time we see a curve being contracted to an indetermination point, we blow-up it (i.e., we replace the point by a projective plane).

By doing this blowing up procedure in a systematic (and careful) way, Diller and Favre end up by showing that, after finitely many blow-ups, one obtains a surface ${\pi:X\rightarrow\mathbb{P}_k^2}$ such that

• the sequence of “degrees” of the iterates of ${f_X=\pi^{-1}\circ f\circ \pi}$ is multiplicative, i.e., the action ${(f_X^n)^*}$ of ${f_X^n}$ on the “cohomology” (${H^2(X,\mathbb{Z})}$ for ${k=\mathbb{C}}$ and the Neron-Severi group of ${X}$ for a general algebraically closed field ${k}$) is the ${n}$-power ${(f_X^*)^n}$ of the action ${f_X^*}$ of ${f_X}$ on cohomology; in other words, by passing from ${\mathbb{P}_k^2}$ to ${X}$ and replacing ${f}$ by ${f_X}$, we got rid of the annoying technical problem that ${deg(f^n) might happen.
• ${\lambda(f)}$ is the spectral radius of ${(f_X)^*}$.

Of course, the last item readily implies that ${\lambda(f)}$ is an algebraic integer: indeed, ${\lambda(f)}$ is the spectral radius of ${(f_X)^*\in End(H^2(X,\mathbb{Z}))}$ (when ${k=\mathbb{C}}$).

Actually, one can improve the information on ${\lambda(f)}$ by noticing that the surface ${X}$ was obtained from a finite number of blow-ups from ${\mathbb{P}_k^2}$. In particular, ${H^2(X,\mathbb{Z})\simeq \mathbb{Z}[line]\oplus\bigoplus\limits_{i=1}^m\mathbb{Z}[E_i]}$, where ${[line]}$ is the class associated to a generic line of ${\mathbb{P}_k^2}$ and ${[E_i]}$ are the classes of the exceptional divisors introduced during the blowing up procedure. Furthermore, ${H^2(X,\mathbb{Z})}$ carries an intersection form ${\cdot}$ such that ${e_0\cdot e_0=1}$, ${e_0\cdot e_i=0}$, ${e_i\cdot e_j=0}$ for ${i\neq j}$ and ${e_i\cdot e_i=-1}$, that is, ${H^2(X,\mathbb{Z})}$ carries an intersection form of signature ${(1,m)}$. Thus, if ${(f_X)^*}$ preserved the intersection form, then ${(f_X)^*}$ would have at most ${1}$ eigenvalue of modulus ${>1}$ by an elementary linear algebra argument (see, e.g., this previous post here), and hence ${\lambda(f)}$ would ${1}$, Pisot or Salem. As it turns out, in general ${(f_X)^*}$ does not preserve the intersection, but it is possible to show that it “increases” it (i.e., there is some monotonicity phenomenon) and this is sufficient to conclude that ${(f_X)^*}$ has at most ${1}$ eigenvalue of modulus ${>1}$.

In any case, this completes the sketch of proof of Theorem 1 of Diller and Favre. In the next (final) section of this post, we will state some of the main results obtained by Serge (Cantat) and Jérémy Blanc.

4. The structure of ${\Lambda(\mathbb{P}_k^2)}$ after Blanc-Cantat

Once we know that ${\Lambda(\mathbb{P}_k^2)\subset \{1\}\cup Pisot\cup Salem}$, there are several natural questions that one can pose: What are the birational transformations leading to Pisot and/or Salem numbers? What is the infimum of ${\Lambda(\mathbb{P}_k^2)}$? Do all Pisot and/or Salem numbers belong to ${\Lambda((\mathbb{P}_k^2)}$?

Concerning the first question, Blanc and Cantat proved the following theorem:

Theorem 3 If ${f\in Cr_2(k)}$ (with ${k}$ algebraically closed) and ${\lambda(f)}$ is Salem, then there exists ${\pi:X\dashrightarrow\mathbb{P}_k^2}$ a birational morphism such that ${\pi^{-1}\circ f\circ \pi\in Aut(X)}$.

By combining this result with some previous works by Nagata and McMullen, they obtain the following corollary answering the second question posed above:

Corollary 4 The minimum of ${\Lambda(\mathbb{P}_k^2)}$ is Lehmer’s number ${\lambda_L}$.

Another interesting corollary of Blanc-Cantat theorem is:

Corollary 5 Let ${f, g\in Cr_2(k)}$ such that ${\lambda(f), \lambda(g) >1}$. If ${f}$ and ${g}$ are conjugated, then it is possible to conjugate them via a birational transformation whose degree is “bounded”: more precisely, there exists ${h\in Cr_2(k)}$ such that ${hfh^{-1}=g}$ and ${deg(h)\leq 2^{57}(deg(f)\cdot deg(g))^{28}}$.

In a similar spirit, Blanc and Cantat showed that the minimal degree ${mindeg(f)}$ of ${f\in Cr_2(k)}$, that is, ${mindeg(f):=\min\{deg(hfh^{-1}):h\in Cr_2(k)\}}$, is “comparable” to the dynamical degree ${\lambda(f)}$ in most cases:

Theorem 6 For all ${f\in Cr_2(k)}$ such that ${\lambda(f)>1}$, one has that ${mindeg(f)\leq \cosh(1000+200\log\lambda(f))}$. In particular, there are universal constants ${C}$ and ${N}$ such that ${\lambda(f)\leq mindeg(f)\leq C\lambda(f)^N}$.

Using this theorem, Blanc and Cantat show that, given a sequence ${\lambda_n=\lambda(f_n)\in \Lambda(\mathbb{P}_k^2)}$ converging to some ${\lambda_{\infty}}$, then, up to replacing ${f_n}$ by some of element in its conjugacy class, we can assume that the degrees ${deg(f_n)}$ are uniformly bounded. From this, they prove the following result answering the third question posed above:

Theorem 7 ${\Lambda(\mathbb{P}_k^2)}$ is a well ordered set (i.e., every subset has a least element) and it is closed if the field ${k}$ is uncountable (e.g., ${k=\mathbb{C}}$). In particular, since the golden ratio ${\phi\in \Lambda(\mathbb{P}_k^2)}$ is accumulated from the right by Pisot and Salem numbers, it follows that there are Pisot and Salem numbers that do not belong to ${\Lambda(\mathbb{P}_k^2)}$.

## Responses

1. Parabens pelo blog. Ta bem legal!