introduction - École polytechnique · 2009-11-04 · in the last few years, there have been some...

MULTIPLICITY OF HOLOMORPHIC FUNCTIONS

CHARLES FAVRE

Abstract. The main result of this paper concerns an analytic version ofBirkhoff ergodic Theorem. It allows us to define asymptotic multiplicitiesgiven any generic rational map f : P

2→ P

2. This is the key for our studyof the singularities of the Green current T associated to f . We characterizethe points where the Lelong number of T is strictly positive.

Introduction

In the last few years, there have been some progress in the study of thedynamics of rational maps of the projective space Pk. The article [FS95] andthe recent survey [Si98] set the foundations of this study in the same spirit ofwhat has been done in the holomorphic case, whereas few papers [D97], [D98],[F1-98] have been devoted to birational mappings of P2. The paper [RS97] dealswith the problem of distributions of preimages of linear subspaces for generalrational maps.

The main problem while considering non-holomorphic mappings comes fromthe presence of points of indeterminacy. In dimension 2, there are only finitelymany of these, which makes things simpler. We will thus restrict ourselves toP2.

Let us consider f : P299K P2 a rational map of (algebraic) degree d. We

assume it to be dominant i.e. that its range is not contained in any hypersurfaceof P2. Let I(f) be its set of indeterminacy. The rational map f is said to begeneric if f−n(I(f)) is finite for any n ≥ 0. Under this condition, it is possibleto define an invariant positive closed (1, 1) current T (f) = limn→∞ Tn(f) whereTn(f) := d−n(fn)∗ω and ω is the Fubini-Study form on P2. The main purposeof this paper is to prove two Theorems which give some coarse informations onthe structure of T (f).

The first one deals with the study of the singularities of T (f). Note ν(T (f), z)the Lelong number of T (f) at z, and I(f∞) :=

⋃

n≥0 f−nI(fn).

Theorem 1. ν(T (f), z) > 0 if and only if z ∈ I(f∞).

Our proof uses an analytic version of Birkhoff Theorem, which is the mainresult of the paper.

We first define the function e(., f) : P2 → N ∪ {∞} as follows. On I(f) weset, e(z, f) = 0. At a point where f is not finite-to-one, we put e(z, f) = ∞.Otherwise, we define e(z, f) to be the local multiplicity of f at z. We then have

Date: July 20, 1998.1991 Mathematics Subject Classification. 32H50.Key words and phrases. Dynamics, multiplicity, Lelong number.

1

2 CHARLES FAVRE

Theorem 2. For all z ∈ P2, the limit e∞(z, f) := limn→∞ e(z, fn)1/n exists.

We will show in a forthcoming paper that the function e∞(., f) can be usefulfor getting estimates related to multiplicities and especially volume estimates.

The second Theorem concerning the current T (f) is a generalization of a cor-responding Theorem in [F1-98]. It shows that in the rational (non-holomorphic)case, the construction of an interesting invariant measure can not be performedby simply taking the wedge product T (f) ∧ T (f), even if it exists.

Theorem 3. There exists a function µ∞ : I(f∞) → R∗+ to the positive reals of

total mass∑

z∈I(f∞) µ∞(z) = 1 such that

limn→∞

Tn(f) ∧ Tn(f) =∑

z∈I(f∞)

µ∞(z)δz

Remark: Theorems 1. and 3. would be a direct consequence of [RS97] if weknew that T (f) ∧ T (f) existed in the sense of [De93]. However, this is false ingeneral as shown in [F1-98] example 4.

The plan of the article is as follows. In Section 1, we recall some basic defi-nitions and results we will use in the rest of the paper. In Section 2, we discussTheorem 2 in a more general setting and give some results on the asymptoticfunction e∞(., f). The rest of the paper is then completely devoted to the ap-plication of Theorem 2 to the study of the dynamics of rational mappings of P2.In Section 3, we show some basic properties of these mappings. In Section 4,we study precisely the function e∞(., f) in that case. Theorem 1 is then provedin Section 5. We conclude the paper by proving Theorem 3 in Section 6.

Acknowledgement: I would like to thank M. Jonsson and B. Teissier formany valuable discussions I had with them, for M. Benedicks to indicate meSzemeredi’s result, H. Furstenberg for his remarks on it, and finally N. Sibonyfor the great number of comments and improvements he made on the very firstversions. I also thank the department of mathematics of KTH in Stockholm forhis support and for providing me a very pleasant atmosphere for working.

1. Basics on multiplicity and positive closed currents.

In all this Section, we recall some well-known facts we will use in the otherparts of the paper. We make ad-hoc definitions and state ad-hoc results for ourpurpose, giving references for a general treatment of the notions introduced.

1.1. Multiplicity. Our basic refererence for this paragraph is [S65]. See also[M76]. Let On

0 be the local ring of holomorphic germs at (Cn, 0).Definition 1.1.

Let I ⊂ On0 be a primary ideal. Then there exists a polynomial P ∈ Q[X]

such that for all integers k large enough, one has dimC(On0 /Ik) = P (k). The

leading term of P can be written as P (k) = e(I)kn/n!+O(kn−1), where e(I) ∈N∗. We define e(I) to be the multiplicity of I at 0.

We will use the following proposition:Proposition 1.2.

MULTIPLICITY OF HOLOMORPHIC FUNCTIONS 3

If I ⊂ On0 is a complete intersection, that is if I = (f1, · · · , fn) is generated

by exactly n germs, one has

e(I) = dimC(On0 /I).

We will also need the following concept. Let V ⊂ Cn be an analytic set ofpure dimension k, and take a point p ∈ V .

Definition 1.3. The multiplicity of V at p, we denote by e(V, p), is the degreeof the ramified covering induced by the projection of a neighborhood of p in Vonto a generic small linear polydisk of dimension k passing through p.

In the sequel, we will denote by I(V ) ⊂ Onp the ideal of germs at p vanishing

along V .

1.2. Positive closed currents in dimension 2. A general reference for thestudy of currents on complex manifolds, for Lelong numbers and intersectiontheory of currents is [De93].

In the rest of this Section, X will denote a connected complex manifold ofdimension 2.Definition 1.4.

For a fixed k = 0, 1, 2, a current T on X of bidegree (k, k) is a continuouslinear form on test forms of bidegree (2 − k, 2 − k).

• A positive closed current of bidegree (2, 2) is a positive Borel measureon X.

• A positive closed current T of bidegree (1, 1) is given locally by T = ddcuwhere u is a plurisubharmonic (psh) function. This function is called alocal potential for T .

• A positive closed current of bidegree (0, 0) is a positive constant.

A very important class of positive closed current is given by integration onanalytic sets. More precisely:Proposition 1.5.

Let V ⊂ X be an analytic set of pure dimension k(≤ 2). Let Vreg be itsregular part. Then the current [V ] given by

< [V ], ω >=

∫

Vreg

ω,

for any (2 − k, 2 − k) test form ω, defines a positive closed current on X.

Note we can extend this definition to any (locally finite) positive analyticcycle by linearity.

Definition 1.6. A positive closed current T charges an analytic subset V ifand only if there exists a constant c > 0 such that T − c[V ] is again a positiveclosed current. We denote this last property by the inequality T ≥ c[V ].

In this article, we will work in the complex projective space P2. We willneed the following representation theorem for positive closed current of bidegree(1, 1). We let π : C3 − {0} → P2 be the natural projection onto P2.

4 CHARLES FAVRE

Proposition 1.7. (see for instance [Si98])Given any positive closed current T of bidegree (1, 1) on P2, one can find a

psh function G on C3, its potential, such that

(1) there exists a constant c > 0 for which ∀Z ∈ C3, and ∀λ ∈ C,

G(λZ) = c log |λ| + G(Z) ;

(2) π∗T = ddcG .

Conversely, given a psh function G on C3 satisfying the homogeneity condition(1), one can find a unique positive closed current T of bidegree (1, 1) such that(2) holds.

1.3. Lelong number of a positive closed current. The Lelong number ofa positive closed current is a quantitative way of characterizing its singularities.Proposition-definition 1.8.

• Let T be a positive closed current of bidegree (1, 1) in X, fix a pointp ∈ X and some local coordinates mapping p to the origin in C2. Takea local psh potential u of T defined around 0 in these coordinates. Thefunction r → sup|z|=r u(z) is an increasing convex function of log r. Wecan hence define the Lelong number of u at 0 by setting

ν(u, 0) := max{c > 0, such that u(z) ≤ c log |z| + O(1)}

which is a finite non-negative real number. We define then,

ν(T, p) := ν(u, 0),

which does not depend on any choice we made.• For a positive measure µ on X, we define the Lelong number of T at p

by

ν(µ, p) := µ{p},

the atomic mass of µ at the point p.

We will use the following Theorem of Siu:

Theorem 1.9. Siu TheoremLet T be a positive closed current of bidegree (1, 1) in P2, and V ⊂ P2 a

complex curve. If there exists α > 0 such that the set {z ∈ V, ν(z, T ) ≥ α} isinfinite, then T charges V , and T ≥ α[V ].

1.4. Intersection theory for positive closed currents of bidegree (1, 1).Let T1 and T2 be two positive closed currents of bidegree (1, 1) on X.

If T1 is smooth, one can naturally define the wedge product T1 ∧ T2 by< T1 ∧ T2, ω >:=< T2, ωT1 > for any test function ω. When T1 is no longersmooth, it is however still possible to give a sense to this wedge product undersome conditions.

Fix in a local chart two psh potentials u1, u2 for T1 and T2 respectively, andlet ω be a kahler form defined in this chart. We define the trace measure of T2

by σT2 := T2 ∧ ω.


Proposition-definition 1.10.

If u1 ∈ L1(σT2), one can define

T1 ∧ T2 := ddc(u1T2),

and it does not depend on the choices we made. It is a positive closed currentof bidegree (2, 2) hence a positive borelian measure on X.

In this paper, we will use the following criterium for the existence of thewedge product.Proposition 1.11.

Assume T1 is defined by a continuous form outside a discrete subset of X.Then for any local potential u1, and any positive closed current T2 of bidegree(1, 1), u1 ∈ L1(σT2).

We can also formulate the following characterization of multiplicities in termsof Lelong numbers.

Let I ⊂ O20 be a primary ideal, and (f1, · · · , fk) be a system of generators of

I. By 1.11, we can define the positive measure µI = (ddc)2(1/2 log(∑k

i=1 |fi|2).

Proposition 1.12.

e(I) = ν(µI , 0) = µI{0}.

2. Birkhoff Theorem for analytically rigid cocycle

This Section is devoted to the proof of Theorem 2. We will in fact show aslightly more general result, which makes the proof more transparent. Beforedefining precisely the objects we will deal with, let us recall first the usualBirkhoff Theorem (see [KH95]) in order to point out the deep similarity withTheorem 2.

Let us consider (A,µ) a probability space and f : A → A a measurableinvariant transformation.Definition 2.1.

A function τ : A × N → R∗+ is called a cocycle if it satisfies the property

τ(z, n + m) = τ(z, n)τ(fn(z),m)

for any z ∈ A and any integers n,m.

Theorem 2.2. Birkhoff TheoremAssume the function τ(., 1) is µ-integrable. Then there exists a unique f -

invariant function τ(.,∞) such that for µ-a.e. point

τ(z, n)1/n → τ(z,∞) .

We are going to prove an analogous Theorem in an analytic context. Wedrop the existence of an invariant measure and replace it by a strong analyticproperty on τ(., 1), which will make things rigid enough to force the convergence

of the sequence τ(z, n)1/n for any z.Fix X a compact irreducible analytic space (which eventually have singular

points) and f : X → X a holomorphic map.

6 CHARLES FAVRE

Definition 2.3. Analytically rigid cocycleA function τ : X×N → R∗

+ is called an analytically rigid cocycle if it satisfiesthe following two conditions:

• ∀n,m ∈ N, ∀z ∈ X, τ(z, n + m) = τ(z, n)τ(fn(z),m);• the function z → τ(z, 1) is upper semi-continuous (u.s.c.) with respect

to the Zariski topology on X, or equivalently ∀α ≥ 0, {τ(., 1) ≥ α} isanalytic.

We begin by pointing out some elementary properties of analytically rigidcocycle, we will constantly make use of in the sequel.Proposition 2.4.

Let τ be an analytically rigid cocycle.

- Let Y ⊂ X be an irreducible subspace. Then there exists a set Y ( Ywhich consists of an at most countable union of proper complex subspacesof Y such that τ(., 1)|Y −Y is constant equal to

τ(., 1)|Y −Y ≡ min{τ(z, 1), z ∈ Y } .

- The range of τ(., 1) is bounded above.- For each n ∈ N, τ(., n) is u.s.c. with respect to the Zariski topology

Proof. The first two assertions follow from the upper-semicontinuity property.Set α := inf{τ(z, 1), z ∈ Y }. Then {z ∈ Y, τ(z, 1) > α} =

⋃

n≥0{z ∈

Y, τ(z, 1) ≥ α + n−1} is an increasing union of proper subspaces of Y . Inparticular, we conclude that the infimum of τ(., 1) is actually attained, andthat for any point z outside {z ∈ Y, τ(z, 1) > α} we have τ(z, 1) = α.

The function τ(., 1) is bounded because any u.s.c. function attains its maxi-mum on a compact set.

The last proposition merely follows from the fact that a product of twopositive u.s.c. functions remains u.s.c. �

Assume we are given an analytically rigid cocycle, and fix Y ⊂ X an irre-ducible analytic subspace of X. We define

• τ(Y, k) := min{τ(z, k), z ∈ Y } the generic value of τ(., k) on Y .• deg(τ) := max{τ(z, 1), z ∈ X}.

Fix a real number α > τ(Y, 1). We define the analytic subspaces

• CY (τ, α) := {z ∈ Y, τ(z, 1) ≥ α > τ(Y, 1)}.• C(τ, α) := CX(τ, α).

We now state our main Theorem. We note O(z) =⋃

n≥0 fn(z) the orbit of

z, and GO(z) =⋃

n,k≥0 f−k{fn(z)} its grand orbit.Theorem 2.5.

(1) For all z ∈ X, τ(z,∞) := limn→∞ τ(z, n)1/n exists.(2) If τ(z,∞) > τ(X, 1), there exists an irreducible complex subspace V

and two integers k, l such that fk(z) ∈ V and f l(V ) = V . MoreoverV ⊂ C(τ, α) for any τ(X, 1) < α < τ(z,∞).

(3) Fro any z ∈ X, τ(z,∞) ∈ [τ(X, 1),deg(τ)].(4) The function τ(.,∞) is constant on any grand orbit GO(z).


(5) For any irreducible analytic subspace V ⊂ X, the function τ(.,∞) isconstant on V outside a countable union of proper subspaces, and it isequal to its minimum value, we denote by τ(V,∞).

Assume now that X is smooth, and f : X → X is finite-to-one i.e. f−1{z}is finite for any point z ∈ X.

Definition 2.6.

Let z ∈ X. We define e(z, f) to be the local degree of the germ defined by f atz onto f(z).

The application e(., f) defines thus a function of X to the set of integers[[1,deg(f)]] := {1, 2, · · · ,deg(f)}, where deg(f) denotes the topological degreeof f . It is well-known that the multiplicity behaves multiplicatively underiteration. That is, we have

e(z, fn ◦ fm) = e(z, fn)e(fn(z), fm) .

On the other hand, the following Theorem holds.Theorem 2.7.

The function e(., f) is u.s.c. with respect to the Zariski topology.

Proof. SketchAs the problem is local, we can assume that f is defined on a ball B ⊂ Cn

with values in B. Let V ⊂ B × B be the analytic set defined by the equationsf(x) = f(y). One checks that:

e(V, (x, x)) = e(x, f).

But for any integer k, the set {p ∈ V, e(V, p) ≥ k} is an analytic set by theorem3.8E of [W72] or proposition 4.1. of [L82]. Hence the set

{x ∈ B, e(x, f) ≥ k} = ı−1({p ∈ V, e(V, p) ≥ k}),

is also analytic, where ı(x) := (x, x) is the injection of B into the diagonal ofB × B. �

Combining these two facts, one gets:Proposition 2.8.

The function (z, n) → e(z, fn) is an analytically rigid cocycle.

We can thus define according to Theorem 2.5

e∞(z, f) := limn→∞

e(z, fn)1/n .

It is this function we will use in the next Sections. This function measures theasymptotic multiplicity of f at a point z.

The rest of this Section is devoted to the proof of Theorem 2.5.Remark on notations: given any two integers n < m, we denote by [[n,m]]

the set of integers [[n,m]] := {n, n + 1, · · · ,m − 1,m}; for a given real α, wedenote by [α] its integer part.

Proof. Theorem 2.5.Let us prove the first two assertions. In fact 2. implies 1. by an induction

argument as we will see. The assertions 3. and 4. are very easy to check, once

8 CHARLES FAVRE

we know that τ(z,∞) exists. We postpone the proof of 5. to the end of theSection.

We first note that by definition, we can compute τ(., n) in terms of τ(., 1)since τ is a cocycle. For all z ∈ X, n ∈ N,

τ(z, n) =n−1∏

k=0

τ(fk(z), 1) . (1)

The idea of the proof of 1. and 2. is that if lim supn→∞ τ(z, fn)1/n > τ(X, 1),then the iterates of the point z have to be in a complex subspace C(τ, α) for someα > τ(X, 1) so often that one of them belongs to a periodic complex subspaceincluded in C(τ, α). The periodicity is then given by a crucial arithmetic resulton sets of integers with positive upper-density (Szemeredi Theorem).

Define

τ(z,∞) := lim supn→∞

τ(z, n)1/n ,

and

τ(z,∞) := limn→∞

τ(z, n)1/n

as soon as it exists.We first remark that if τ(z0,∞) exists, and we pick any z ∈ GO(z0) then

τ(z,∞) exists also and τ(z,∞) = τ(z0,∞). In fact, assume fN(z) = fM(z0).Then for n ≥ N

τ(z, n)1/n = τ(z,N)1/n × τ(fN (z), n − N)1/n

= τ(z,N)1/n × τ(fM(z0), n − N)1/n

= τ(z,N)1/n ×τ(z0, n − N + M)1/n

τ(z0,M)1/n

→ τ(z0,∞) .

We proceed now by induction on the dimension of X. If X is a point theTheorem is trivial. Assume the following lemma:

Lemma 2.9. For any z ∈ X, if τ(z,∞) > α > τ(X, 1), there exist k, l ∈ N∗

and V an irreducible complex subspace of C(X,α), such that fk(z) ∈ V andf l(V ) = V .

Let l be the period of such a V . We apply the induction hypothesis to allcomplex subspaces fk(V ) ⊂ X for k = 1, · · · l− 1, with the map f l. We deduce

first that n → τ(fk(z), n)1/n converges, hence τ(z,∞) exists by the formerargument. Lemma 2.9 implies immediately assertion 2.

Proof. Lemma 2.9.First note that ∀z ∈ X,

τ(X, 1) ≤ τ(z, f) ≤ deg(τ).

Fix α > 0 such that τ(z,∞) > α > τ(X, 1). Define the set

E := {k ∈ N, τ(fk(z), 1) ≥ α > τ(X, 1)} = {k ∈ N, fk(z) ∈ C(τ, α)} .


We have the following inequalities:

τ(z, n)1/n =

n−1∏

k=0

τ(fk(z), 1)1/n ≤ α∏

k∈E∩[[0,n−1]]

τ(fk(z), 1)1/n

≤ α(1 + deg(τ))|E∩[[0,n−1]]|

n

By taking the supremum limit and the log, we get

lim supn→∞

|E ∩ [[0, n − 1]]|

n≥

log τ(z,∞) − log α

log(1 + deg(τ))> 0

Thus the upper density of E, the real number β := lim supn→∞ n−1|E∩[[0, n−1]]|is strictly positive. We now appeal to the following Theorem of Szemeredi onset of integers with positive upper-density.

In fact, for our proof we need a slightly improved version of the originalSzemeredi Theorem. We first recall the basic Theorem.

Theorem 2.10. Szemeredi Theorem (see [SZ74] or [FKO82] Theorem IF )Fix an integer k. For any n ∈ N, define rk(n) to be the least integer such

that everyset E ⊂ [[1, n]] satisfying |E| ≥ rk(n) admits (at least) one arithmeticsequence of length k. Then

rk(n) = o(n) .Corollary 2.11.

Let E be a set of positive numbers whose upper density is strictly positive. Forany k ∈ N∗, there exists l ∈ N, such that E contains infinitely many arithmeticsequences of length k and difference l.

Proof. Choose β a real number strictly less than the upper-density of E, andN ∈ N large enough such that rk(N) ≤ βN . Define

F := {j ∈ N, such that |E ∩ [[jN, (j + 1)N − 1]]| ≥ βN} .

Assume F is finite, and set M := 1 + maxF . For n ≥ NM + N + 1, we have

|E ∩ [[0, n − 1]]| ≤ |E ∩ [[0, NM − 1]]| + |E ∩ [[NM,N[

N−1(n − 1)]

− 1]]| +

+|E ∩ [[N[

N−1(n − 1)]

, n − 1]]|

≤ βN([

N−1(n − 1)]

− M) + N(M + 1).

And this yields for all n sufficiently large,

lim supn→∞

1

n|E ∩ [[0, n − 1]]| ≤ β,

which contradicts the assumption on the upper-density of E. For each j ∈ Fwe can thus apply Theorem 2.10: there exists an arithmetic sequence of lengthk and difference lj contained in E ∩ [[jN, (j + 1)N − 1]]. But the difference ljcan be obviously bounded by lj ≤ N/k. We can therefore extract an infinitesubsequence of terms j ∈ F for which the difference is the same. This concludesthe proof of corollary 2.11. �

By corollary 2.11, for any k ∈ N∗ fixed, we can find an integer l ∈ N∗,and a strictly increasing sequence of integers ni tending to infinity, such that

10 CHARLES FAVRE

fni(z), fni+l(z), · · · , fni+kl(z) ∈ C(τ, α). We conclude with the help of thefollowing lemma applied to V = C(τ, α).Lemma 2.12.

Let V ⊂ X be an irreducible complex subspace of X, k ≥ dim V and l beintegers, and z ∈ X.

If {n ∈ N, fn(z) ∈ V ∩ f−l(V ) ∩ · · · ∩ f−kl(V )} is infinite, there exists anirreducible complex subspace W in V such that

• f l(W ) = W ,• O(z) ∩ W 6= ∅.

We apply the lemma to the case V := C(τ, α) to conclude the proof of lemma2.9. �

Proof. Lemma 2.12.We prove it by induction on the dimension of V . If dimV = 0, let us say

V = {p} then z ∈ V ∩f−l(V ) implies z = p = f l(p). Now in the general case, weeither have V ⊂ f−l(V ), thus V = f l(V ) and we are done, or V1 = V ∩ f−l(V )has dimension strictly less than V . But from fn(z) ∈ V ∩f−l(V )∩· · ·∩f−kl(V )

follows fn+l(z) ∈ V1 ∩ f−l(V1) ∩ · · · ∩ f−k(l−1)(V ). We can therefore choose an

irreducible component V2 ⊂ V1 such that {n ∈ N, fn(z) ∈ V2∩· · ·∩fk(l−1)(V2)}is not finite, and apply the induction hypothesis. �

Let us deal now with assertion 5. Take V ⊂ X any irreducible analyticsubspace of X. Define for k ≥ 0

Vk := {z ∈ V, τ(fk(z), 1) > τ(fk(V ), 1)} .

Then Vk is a countable union of proper complex subspaces contained in V forτ(., 1) is u.s.c. with respect to the Zariski topology. For any z ∈ V −

⋃

k≥0 Vk,and any n ∈ N we have

τ(z, n)1/n = τ(V, n)1/n = minz∈V

τ(z, n)1/n .

Therefore, limn→∞ τ(V, n)1/n := τ(V,∞) exists, τ(V,∞) = minz∈V τ(z,∞),and ∀z ∈ V −

⋃

k≥0 Vk, τ(z,∞) = τ(V,∞).�

3. Basic properties of rational maps of P2

Let f : P299K P2 be a dominant rational map. In homogeneous coordinates,

f [z : w : t] = [P0(z,w, t) : P1(z,w, t) : P2(z,w, t)] where Pi are three homo-geneous polynomials of the same degree d without any common factor. Theinteger d is by definition the algebraic degree of f . We let I(f) be the indeter-minacy set of f which is finite, C(f) the Zariski closure of its critical set, andC(f) the Zariski closure of the set of points where f is not finite. We alwayshave C(f) ⊃ I(f) ∪ C(f).

We also define I(f∞) :=⋃

n≥0 I(fn) =⋃

n≥0 f−n(I(f)) the generalized in-

determinacy set, C(f∞) :=⋃

n≥0 C(fn), and C(f∞) :=⋃

n≥0 C(fn).


For a point in p ∈ I(f), we define its multiplicity m(p, f) to be the multiplicityof the ideal (P0◦σ, P1◦σ, P2◦σ) where σ is a local Section of π in a neighborhoodof p, and π : C3 − {0} → P2 is the natural projection.

We denote by deg(f) the topological degree of f , i.e. the number of preimagesof a generic point in P2. The relation between d and deg(f) is given by thefollowing formula.

Proposition 3.1. (See [F1-98])

d2 = deg(f) +∑

p∈I(f)

µ(p, f).

We make the following definition.

Definition 3.2.

Let f : X → X be a meromorphic map of a compact complex manifold. LetD(f) = X − I(f) be its domain of definition, Gr(f) the closure of its graph inX × X, and π1, π2 the two projections of Gr(f) respectively onto the first andthe second component. If V is any complex subspace of X, we define

• the total transform of V , f(V ) := π2π−11 (V ),

• the strict transform of V , f(V ) := f |D(f)(V ),

• the preimage of V , f−1(V ) := π1π−12 (V ).

We say that

• V is f -invariant if and only if f(V ) = V ;• V is totally invariant if and only if f−1(V ) ⊂ V

Proposition 3.3.

Let f : P299K P2 be a dominant rational map of degree d.

- Assume f∗[V ] = d[V ]. Then V is totally invariant.- Conversely, if V is a totally invariant irreducible curve, then f∗[V ] =

d[V ].- Any totally invariant irreducible curve V is contained in C(f), contains

I(f), and either deg(V ) ≤ 3 or f(V ) is reduced to a point.

Example 1. Take any polynomial P (z,w) homogeneous of degree d. Definef [z : w : t] = [zP (z,w) : wP (z,w) : td+1]. Then {P = 0} is totally invariant.

Example 2. Define f [z : w : t] = [z2t : w3 : t3]. Then V := {zwt = 0} istotally invariant but f∗[V ] = 2[V ] + [{w = 0}] + 2[{t = 0}].

Remark 1. In the case where f is holomorphic, one can actually classify com-pletely totally invariant curves (see [FS94] and a recent result by Shiffman-Shishikura-Ueda). Each of these is necessarily a complex line.

If there is one totally invariant line, f is conjugated to a map of the form[z : w : t] → [P (z,w, t) : Q(z,w, t) : td].

If there are two totally invariant lines, f is conjugated to a map of the form[z : w : t] → [P (z,w, t) : wd : td].

Finally, if three lines are totally invariant, f is conjugated to [z : w : t] →[zd : wd : td].

12 CHARLES FAVRE

Example 3. An example of rational maps with a totally invariant quadric isgiven by:

f [z : w : t] = [t2 − (zw − t2) : t2 + (zw − t2) : t2] .

The quadric Q := {zw = t2} is totally invariant. Note I(f) = {[1 : 0 : 0], [0 :1 : 0]}.

Proof. Proposition 3.3.Let P be a homogeneous polynomial defining V = π{P = 0} of minimal

degree. Then f−1(V ) = π{P ◦ F = 0}.If f∗[V ] = d[V ], we have P ◦F = C×P d for some constant C. Thus P ◦F = 0

implies P = 0, and V is totally invariant.If V is totally invariant, P ◦F = 0 implies P = 0. As V is irreducible, we can

assume P to be so. Thus P ◦ F = CP d′ for some integer d′ and some constantC. By looking at the respective degrees of each hand side, we get d′ = d.

The last statement follows from the functional equation P ◦ F = CP d. Theinclusion I(f) ⊂ V is immediate. By differentiating the equation, we getDPF (Z) ◦ DFZ = CdP d−1(Z)DPZ . As P is irreducible, the set of points Z

such that DPZ = 0 has codimension greater than 2. If f(V ) has codimension≥ 2, V ⊂ C(f). Otherwise, a generic point on Z ∈ V satisfies DPF (Z) 6= 0.This implies that DFZ has a non trivial kernel, hence V ⊂ C(f).

Now if codimf(V ) ≤ 1, we have DPF (Z) 6= 0 for a generic point on V .Locally, at such a point Z0 we can find 2 constant vectors u1, u2 such that thematrix defined column by column MZ := (tDPF (Z), u1, u2) is invertible for everyZ close to Z0. But

det(MZ) det(DFZ) = det(tDFZMZ)

= CdP (Z)d−1 det(tDPZ ,tDFZu1,tDFZu2) .

Therefore P d−1 divides detDF locally at Z0 hence globally. But detDF hasdegree 3d − 3. Thus deg(P ) ≤ 3. �

We would like to study the function e(., f) introduced in the first Section inthe context of rational mappings. We extend the definition of e(., f) to I(f)and C(f) by setting

• ∀z ∈ I(f), e(z, f) = 0;• ∀z ∈ C(f) − I(f), e(z, f) = ∞.

If V is an irreducible compact complex curve in P2, we define as in the formerparagraph the quantity e(V, f) to be the generic value of e(., f) restricted to V .

Proposition 3.4.

Let f : P299K P2 be a rational map of degree d, and V 6⊂ C(f) any irreducible

curve. We have

(1) e(V, f) ≤ d;(2) If V is f -invariant, e(V, f) = d implies that V is totally invariant.

Proof.We first prove 1. The proof follows [U97] and [FS94]. Fix a curve V 6⊂ C(f).Take a point z ∈ Vreg such that f(z) ∈ f(V )reg. Assume furthermore that z isa regular for the map f |Vreg . We can then choose coordinates where the map f


can be written under the form f(x, y) = (xl, y). The integer l is exactly e(V, f).To concluide we have to show l ≤ d. Take a line H passing through z whichintersects V transversally at this point, and another one L passing through f(z)and tangent to f(V ). As the degree of f∗L is exactly d, and the multiplicity ofthe intersection of f∗L with H is l, we obtain the desired inequality.

For the second assertion, we first prove for any irreducible curve V 6⊂ C(f)that f∗[V ] = e(V, f)[V ] + [V2] where V2 is an analytic cycle such that V2 ∩ V isfinite. We need the following lemma.

Lemma 3.5.

Let f : (Cn, 0) → (Cn, 0) be a holomorphic finite germ of degree l such thatf−1{z1 = 0} = {z1 = 0}, and f |z1=0 is an automorphism. Then

f∗[z1 = 0] = l[z1 = 0] .

Proof. Lemma 3.5.The condition f−1{z1 = 0} = {z1 = 0} implies that we can factorize the first

coordinate f1(z) = azl01 (1 + g(z)) where g is holomorphic and l0 ∈ N. Hence

f∗[z1 = 0] = l0[z1 = 0]. We want to show that l = l0. By definition l = µ(I, 0)is the multiplicity of the primary ideal I = (f1, f2, · · · , fn). We have µ(I, 0) =

µ(zl01 , f2, · · · , fn), hence we can assume that f(z1, · · · , zn) = (zl0

1 , f2, · · · fn). Asthe function f induces an automorphism on {z1 = 0}, in a neighborhood of 0each fiber z1 = c is mapped injectively into z1 = cl0 by f . This means that fora point p close to 0 not lying in {z1 = 0}, the cardinal of its preimages shouldbe exactly l0. Thus l = l0 which concludes the proof. �

Write f∗[V ] = k[V ] + [V2], where V2 is an analytic cycle whose support is theunion of all irreducible components of f−1(V ) different from V .

Take a point z ∈ V − V2 such that e(z, f) = e(V, f), V is smooth at z andf(z), and (f |V )′(z) 6= 0. We conclude k = e(V, f) by applying lemma 3.5 to frestricted to a small neighborhood of V around z onto a small one around f(z).

We have therefore

d deg(V ) = e(V, f) deg(V ) + deg(V2) ,

and

e(V, f) = d −deg(V2)

deg(V ). (2)

This finishes the proof of proposition 3.4. �

4. Asymptotic multiplicities for generic maps

Following Fornaess-Sibony [FS95], we introduce the following condition:

Definition 4.1.

A rational map f is said to be generic if and only if ∀n ∈ N, I(fn) = f−n(I(f))is finite.

We will use the following characterization of generic rational map.

Proposition 4.2. (see [Si98])Let f : P2 → P2 be a rational map. the following assertions are equivalent:

14 CHARLES FAVRE

(1) f is generic;(2) no curve is mapped into I(f∞);(3) for all n ∈ N, the algebraic degree of fn is dn.

In the sequel, we will always assume f to be generic. Under this assumption,we can define e(z, fn) for all points z ∈ X, and every n ∈ N. Otherwise,for a point z ∈ C(f) whose image is an indeterminacy point, we would havee(z, f2) = e(z, f) × e(f(z), f) = ∞× 0!

Theorem 2.5 can be rewritten in the following way.

Theorem 4.3.

Let f be a generic rational map of P2. Then for all z ∈ X, the sequencen → e(z, fn)1/n converges. Let us denote its limit by e∞(., f).

(1) e∞(z, f) < 1 ⇔ e∞(z, f) = 0 ⇔ z ∈ I(f∞).(2) e∞(z, f) = ∞ ⇔ z ∈ C(f∞) − I(f∞).(3) If ∞ > e∞(z, f) > 1, e∞(z, f) ≤ deg(f), and either z is preperiodic or

there exist k, l ∈ N∗ and V an irreducible component of C(f), such thatfk(z) ∈ V and f l(V ) = V .

Proof. Sketch.The fact that f is no longer holomorphic does not essentially affect the proof

we gave in the second Section. We nevertheless repeat it for convenience.For points belonging to either I(f∞) or C(f∞) the Theorem is obvious.The crucial fact is that we still have the u.s.c. of the function e(., f) outside

I(f). More precisely ∀k ≥ 0, {e(z, f) ≥ k}∪ I(f) is an analytic subspace of P2.We can now copy the proof we made in the holomorphic case. Take a point

z /∈ I(f∞)∪C(f∞) such that e∞(z, f) > 1. Either it is preperiodic, and we aredone, or they are infinitely many iterates fni(z), an integer L and an irreduciblecomponent V ⊂ C(f) − C(f∞) such that fni(z), fni+L(z) ∈ V . Note that indimension 2, we use Szemeredi Theorem for l = 1 (in the notation of 2.11)which is very easy. This implies that fL(V ) = V and we are done.

�

We now introduce the following set of exceptional points:

E(f) := {z ∈ P2 such that ∞ > e∞(z, f) ≥ d} .

Note that this set is automatically empty if deg(f) < d. This set will probablyplay a crucial role (in the holomorphic case at least) in proving volume estimate,and convergence Theorems for sequence of currents of the form d−nfn∗T whereT is a positive closed (1, 1) current (see the proof of Theorem 4.16 in [FS95]).

The proposition 3.4 allows us to describe precisely this set.Theorem 4.4.

Set VE the union of irreducible 1-dimensional components V ⊂ C(f) suchthat

∀n ≥ 0, e(fn(V ), f) = d .

Then there exist finitely many periodic points {pi}i=1,··· ,N , such that

E = VE ∪N⋃

i=1

GO(pi) .


Note that by propositions 3.4 and 3.3, we know that VE is totally invariant,and deg(VE ) ≤ 3.

Proof. Define F := {z ∈ P2 such that ∞ > e(z, f) > d}. By proposition 3.4,F is finite. Take now a point z0 ∈ E .

Assume first that z0 is not preperiodic. By Theorem 4.3, eventually byreplacing z0 by an iterate, z0 belongs to V a periodic component of C(f). Note

n its period. Define the finite set G :=⋃n−1

k=0{z ∈ fk(V ), e(z, f) > e(fk(V ), f)}.O(z0) is infinite, thus fk(z0) 6∈ G for k ≥ N for some sufficiently large integerN . But then

d ≤ e∞(z0, f) =

(

n−1∏

k=0

e(fk(V ), f)

)1/n

.

It follows from proposition 3.4 that ∀k ≥ 0, e(fk(V ), f) = d. Thus z0 ∈ VE .Assume now that there are infinitely many periodic orbits {pi} such that

(1) e∞(pi, f) ≥ d,(2) pi 6∈ VE .

Note ni the period of pi. Define the finite set

F ′ =

M⋃

k=0

{z ∈ Vk, e(z, f) > e(Vk, f)} ∪ F ,

where V1, · · · , VM denotes the irreducible components of C(f). By eventuallytaking a subset of {pi}, we can assume that

3. all pi belong to the same irreducible component V ⊂ C(f);4. O(pi) ∩ F ′ = ∅.

But by 4.,

e∞(pi, f) =

(

ni−1∏

k=0

e(fk(V ), f)

)1/ni

.

It follows from 3 that for all k ≤ ni, e(fk(V ), f) = d. If the sequence ni

is bounded, then some fN(V ) ∩ V contains the infinite set {pi}, hence V isperiodic. In any case, we obtain ∀k ≥ 0, e(fk(V ), f) = d, which forces V to beperiodic, and V ⊂ VE which contradicts 2. �

5. Lelong number of the invariant current

When studying the dynamics of a map, it is interesting to construct invari-ant objects. For a dominant generic f rational map of P2, one can naturallyassociate a positive closed (1, 1) current. Note ω the usual Kahler form on P2.

Definition 5.1. Let f : P299K P2 be a dominant rational map, and F : C3 →

C3 be a polynomial lift of f of minimal degree. Let T be a positive closed (1, 1)current on P2, and G(T ) : C3 → R ∪ {−∞} be a potential of T (see 1.7).

We define the pull-back of T by f by its potential

G(f∗T ) := G(T ) ◦ F.

16 CHARLES FAVRE

In all the sequel, we fix a dominant rational map f : P2 → P2 which weassume to be generic, and we let F be a polynomial lift fo f to C3 of minimaldegree.

Proposition-definition 5.2. (see [Si98])The sequence of current Tn(f) := d−nfn∗ω is converging. We can therefore

define T (f) := limn→∞ d−nfn∗ω.The current T (f) is a positive closed current of bidegree (1, 1) and mass 1,

and satisfies the functional equation f∗T (f) = dT (f).

Proof. We proceed as follows. We can assume that sup|Z|=1 |F | = 1, by even-

tually multiplying F by a constant. The potential of the current d−nfn∗ω isgiven by Gn(Z) := d−n log |Fn(Z)|. By homogeneity, we have the inequality

|F (Z)| ≤ |Z|d .

We pull it back by Fn, and we obtain

1

dn+1log |Fn+1(Z)| ≤

1

dnlog |Fn(Z)| ,

so that the sequence Gn(Z) of potential is decreasing to a function we note G.To conclude to the existence of T (f), it just remains to check that G can notbe degenerate.

Consider the Cesaro average σn = n−1∑n

k=1 d−kfk∗ω. They are all of mass1 so that we can extract a subsequence converging to a (1, 1) current σ. NoteH its potential. From d−1f∗σn = σn + n−1(d−n−1fn+1∗ω − d−1f∗ω), we getthe invariance f∗σ = dσ. It implies that H ≤ G + C for some constant. Inparticular, G is not degenerate. �

We are now in position to study the singularities of the invariant currentT (f) that is its Lelong numbers ν(T, z). We first give a basic Theorem due toFornaess and Sibony.

Theorem 5.3. (see [FS95])Let f : P2

99K P2 be a dominant generic rational map. T (f) does not chargeany compact complex curve .

Our Theorem can be then stated as follows.Theorem 5.4.

Under the same assumptions as Theorem 5.3, we have

ν(T (f), z) > 0 if and only if z ∈ I(f∞) .

Proof.The proof essentially relies on a proposition which gives us inequalities betweenthe Lelong number of a positive closed (1, 1) current T and its pull-back f∗T .This proposition is proved in [F2-98] and an independent proof has been givenby Kiselman [K98]. See also [D98] for the case of birational mappings.

Proposition 5.5.

Let T be a positive closed current of bidegree (1, 1), and f : (C2, 0) → (C2, 0) agerm of holomorphic map.

(1) ν(f∗T, 0) ≥ ν(T, 0) .


(2) If f is finite of degree l, we have

ν(f∗T, 0) ≤ l × ν(T, 0) .

Assume now f has a Jacobian determinant non vanishing identically in a neigh-borhood of the origin. Then there exists a constant Cf independent of T suchthat

ν(f∗T, 0) ≤ Cfν(T, 0) .

In particular,

3 if f is not finite at p, ν(T, q) = 0 ⇒ ∀p ∈ f−1{q}, ν(f∗T, p) = 0.

Proof. Proposition 5.5.We will give here a proof of the first two assertions only. The last one is more

technical, and we refer to [F2-98] for a proof. Take a local potential u of T .If f is a holomorphic mapping, |f(Z)| ≤ A|Z| for some constant A > 0 so

that the estimate

u(Z) ≤ ν(T, 0) log |Z| + O(1)

implies

u(f(Z)) ≤ ν(T, 0) log |Z| + O(1)

which give us the stated inequality.Let us assume that f is a finite morphism of degree l. In particular, f is

open. Define I to be the primary ideal of the ring O20 of holomorphic germs at 0

generated by the n components of f . We have l = dimC O20/I (see 1.12). Thus,

for i = 1, 2, the sequence of vectors zi, z2i , · · · , zl

i can not be independent. Hence

there exists a linear combination of such vectors lying in I: akizki

i +· · ·+alzli ∈ I,

with ki ≤ l and aki6= 0. It implies for i = 1, 2 the existence of a constant C > 0

such that |zki

i | ≤ C|f | . In particular we get, for Z small enough

|f(Z)| ≥ C|Z|l .

We have then the following inequalities:

u(f(Z)) ≤ ν(f∗T, 0) log |Z| + O(1)

≤ l−1 × ν(f∗T, 0) log |f(Z)| + O(1).

And this implies

l × ν(T, 0) ≥ ν(f∗T, 0) .

�

• First case: z ∈ I(f∞).The sequence of psh functions Gn is decreasing, thus

ν(T (f), z) ≥ ν(Tn(f), z) > 0 .

We conclude by noticing that ν(Tn(f), z) > 0 if and only if z ∈ I(fn). In fact,the potential of Tn(f) is given by Gn(Z) = d−n log |Fn(Z)| which is smoothoutside I(fn). Take now p ∈ I(fn), and look at the ideal of germs at p

I = (Fn0 ◦ σ, Fn

1 ◦ σ, Fn2 ◦ σ) ,

18 CHARLES FAVRE

where σ is a local Section of π : C3 − {0} → P2 in a neighborhood of p. Then

ν(Tn(f), p) = ν(Gn ◦ σ, p)

≥1

dnmax{k ∈ N,Mk ⊂ I} > 0 ,

where M denotes the maximal ideal of germs of function vanishing at the origin.Note, even if we do not need this fact, that we actually have the equalityν(Gn ◦ σ, p) = d−n max{k ∈ N,Mk ⊂ I}.

• Second case: z ∈ C(f∞) − I(f∞).Suppose that ν(T (f), z) > 0. We can assume that z ∈ V , where V is a

curve contracted by some iterate of f to some point z0 ∈ f(C(f)), and infact that f(V ) = z0 because T (fn) = T (f). By proposition 5.5 assertion 3.,ν(T (f), z0) > 0 and therefore ∀p ∈ V , ν(T (f), p) > 0. By an argument ofcountability, we deduce the existence of α > 0 such that {p ∈ V, ν(T, p) ≥ α}is infinite. By Siu Theorem T (f) charges V , which contradicts 5.3.

• Third case: z ∈ P2 − I(f∞) − C(f∞).For every n ∈ N the map fn is finite at z so that we can apply proposition

5.5. We obtain

ν(fn∗T (f), z) ≤ e(z, fn)ν(T (f), fn(z)) . (3)

Using the functional equation f∗T (f) = dT (f), we get

ν(T (f), z) ≤ C

(

e(z, fn)1/n

d

)n

,

where C := max{ν(T (f), p), p ∈ P2}. In particular, if e∞(z, f) < d we aredone. Otherwise, we can use Theorem 4.4. There are two possibilities: either zis preperiodic, or z belongs to a curve V which is totally invariant and satisfiese(V, f) = d. In the first case, the potential G(f) is finite on π−1(z) so that wehave ν(T (f), z) = 0. In the second one, the orbit O(z) is infinite and does nothit I(f∞). We necessarily have e(fn(z), f) = d for n > n0 large enough. Inthat case, if D =

∏

k<n0e(fk(z), f), we have

ν(T (f), fn(z)) ≥dn

e(z, fn)ν(T (f), z) ≥

dn0

Dν(T (f), z) .

If ν(T (f), z) > 0, it would imply that the set

{p ∈ V such that ν(T (f), .) > D−1dn0ν(T (f), z) > 0}

is infinite, hence by Siu Theorem that T (f) charges V which is impossible. �

6. Self-intersection of the invariant current.

This last Section is devoted to the study of the self-intersection of the invari-ant current T (f).

Let us begin by few remarks. In the holomorphic case, the local potentialfor T (f) is continuous, we can thus define the measure µ := T (f) ∧ T (f). Thismeasure is particularly interesting from a dynamical point of view. Let usmention the results of Fornaess-Sibony and Briend-Duval.


Theorem 6.1. [FS95]The measure µ is mixing and of maximal entropy.

Theorem 6.2. [B97]The smallest Lyapunov exponent λ of µ is strictly positive. We have in fact

the inequality

λ ≥log d

2.

The measure µ describes the distribution of the repulsive periodic points.

1

d2n

∑

fn(p)=p, p is repulsive

δp → µ .

In the rational (non-holomorphic) case, the situation changes drastically. Thefirst problem deals with the existence of the wedge product µ := T (f) ∧ T (f).There is no known counterexample where T (f) fails to be locally integrable withrespect to its trace measure, nor any proof of this fact in any great generality.However, for each n we can define µn := Tn(f) ∧ Tn(f), because the localpotential of these currents is smooth outside I(fn) which is finite. Now wecan ask for the behaviour of µn when n tends to infinity. It turns out, thatµn converges to a measure only supported on I(f∞) which is countable. Thismeasure does not give much insight into the dynamics of f , but collects theasymptotic algebraic structure on the indeterminacy points of the map.

Russakovskii and Shiffman (see [RS97] p.2) were the first to notice that anyweak limit of the sequence of self-intersection of Tn(f) would give an atomicmeasure on the indeterminacy points.

The method we use here is totally different, by proceeding to the explicitcomputation of the limit Tn(f) ∧ Tn(f). Note they also constructed an inter-esting measure for rational maps of P2 satisfying the condition deg(f) > d.Nevertheless it is still unknown if this measure is invariant or not. In the caseof birational mappings of special type, one can also construct mixing invariantmeasure in a natural way (see [D98]).

Let us first recall some results from [F1-98]. Recall we let π1 and π2 be therespective projections of the graph Gr(f) onto the first and the second factor.If z ∈ I(f), set f(z) := π2 ◦ π−1

1 {z}.

Proposition 6.3.

Let f , g: P299K P2, be two rational maps of respective degree d and l. Let

z ∈ P2, and assume that g−1(I(f)) is discrete in a neighborhood of z. Then

m(z, f ◦ g) = d2m(z, g) +

+∑

α∈g(z)∩I(f)

e(z,α)

(

I(Gr(g)) + I(π−12 {α})

)

m(α, f), (*)

where e(z,α)(I(Gr(g)) + I(π−12 {α})) denotes the multiplicity at the point (z, α)

of the intersection of the closure of the graph of g with the space π−12 {α}.

From this, we get the corollary:

20 CHARLES FAVRE

Corollary 6.4.

If f is generic, for any z ∈ P2, µ∞(z, f) := limn→∞ d−2nm(z, fn) exists. Thisfunction has the two basic properties:

(1) µ∞(z, f) > 0 if and only if z ∈ I(f∞);(2)

∑

z∈I(f∞) µ∞(z, f) = 1.

We can then formulate our Theorem.Theorem 6.5.

Let f be generic and non-holomorphic. We have weak convergence,

Tn(f) ∧ Tn(f) →∑

z∈I(f∞)

µ∞(z, f)δz .

Proof. Corollary 6.4.We apply proposition 6.3 with g = fn. Set Mn(z) = d−2nm(z, fn). We get

Mn+1(z) = Mn(z) +

+1

d2n+2

∑

α∈fn(z)∩I(f)

e(z,α)

(

I(Gr(fn)) + I(π−12 {α})

)

m(α, f) .

This implies that the sequence Mn(z) is increasing. From 3.1, one deduces

∑

z∈P2

m(z, fn) = d2n − deg(f)n.

As f is not holomorphic, we have deg(f) < d2. Hence

∑

z∈P2

Mn(z) =∑

z∈P2

1

d2nm(z, fn)

= 1 −

(

deg(f)

d2

)n

,

and

limn→∞

∑

z∈P2

Mn(z) = 1. (4)

Each term Mn(z) is bounded from above by 1, and it follows that Mn(z) con-verges. Assertion 1 is immediate, and assertion 2 follows from equation 4. �

Proof. Theorem 6.5.The existence of the wedge product µn := Tn(f) ∧ Tn(f) follows from 1.11

and the fact that the potential Gn of the current Tn(f) is smooth outside thefinite set I(fn) .

Let us compute the atomic part of µn. Outside I(fn), the potential of Tn(f) issmooth. In particular, µn is absolutely continuous with respect to the Lebesguemeasure on P2 − I(fn).


In a neighborhood of p ∈ I(fn), we can take as a local potential for Tn(f) acomposition gn(f) := Gn(f) ◦ σ where σ is a local section of π. Thus

µn{p} = ν((ddc)2gn(f), p)

= ν

(

(ddc)2(1

2dnlog(|Fn

0 ◦ σ|2 + |Fn1 ◦ σ|2 + |Fn

2 ◦ σ|2)), p

)

=1

d2nm(p, fn) = Mn(p),

where the before last equality follows from 1.12.We deduce from this and equation 4 that the mass of the non-atomic part of

µn tends to 0 as n tends to infinity. Hence

limn→∞

µn = limn→∞

∑

p∈I(f∞)

Mn(p)δp =∑

p∈I(f∞)

µ∞(p)δp.

�

22 CHARLES FAVRE

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[D97] J. DILLER. Dynamics of birational maps of P2, Indiana Univ. Math. J., n.

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Department of Mathematics, Royal Institute of Technology, S-100 44 Stock-

holm, Sweden

E-mail address: [email protected]

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