On the regularity of the Green current for semi-extremal endomorphisms of $ {P}^2 $

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On the regularity of the Green current for semi-extremal endomorphisms of P

Christophe Dupont, Axel Rogue

To cite this version:

Christophe Dupont, Axel Rogue. On the regularity of the Green current for semi-extremal endo- morphisms of P². Discrete & Continuous Dynamical Systems - A, 2020, Conference on Dynamics, Equations and Applications (DEA), 40 (12), pp.6767-6781. �10.3934/dcds.2020163�. �hal-02936357�

DYNAMICAL SYSTEMS

Volume40, Number12, December2020 pp.6767–6781

ON THE REGULARITY OF THE GREEN CURRENT FOR SEMI-EXTREMAL ENDOMORPHISMS OF P²

Christophe Dupont and Axel Rogue

Universit´e de Rennes, CNRS, IRMAR - UMR 6625 F-35000 Rennes, France

Abstract. We study the regularity of the Green current for semi-extremal endomorphisms ofP². Under suitable assumptions, we show that the point- wise lower Radon-Nikodym derivative of stable slices with respect to the one dimensional Lebesgue measure is bounded at almost every point for the equi- librium measure. This provides a weak amount of metric regularity for the Green current along holomorphic discs.

1. Introduction. Let f be a holomorphic endomorphism ofP² of degree d≥2.

Let T be the Green current and let µ:=T ∧T be the equilibrium measure of f, see [13,26]. The measureµis invariant and ergodic, we denoteJ its support. The critical set off satisfiesµ(C) = 0. The Lyapunov exponentsλ2≤λ1ofµwill play a central role in this article. They are bounded below by ¹₂logd, see [8].

1.1. Extremal and semi-extremal endomorphisms. We say thatf isextremal ifλ₁=λ₂=¹₂logd. Those endomorphisms are characterized by the four equivalent properties: µLeb_P2, dimHµ= 4,Tis a positive smooth (1,1)-form on some open subset ofP², andf is a Latt`es map, see [6,4,10]. We recall that a Latt`es map on P^k is the projection of an affine dilation on a complexk-torus by means of a finite galoisian covering σ, see [15] for examples. Similar characterizations were proved for rational maps onP¹, in which caseλ= ¹₂logdandT =µ, see [23,24,27].

We say thatf issemi-extremal ifλ1> λ2= ¹₂logd. Examples are provided by endomorphismsf ofP²having an invariant pencil of lines on whichf induces a one dimensional Latt`es map, see [18]. A natural question is to find characterizations of semi-extremal endomorphisms, as we have for extremal ones. We already know that they satisfy dimHµ = 2 +^log_λ^d

1 , see [7, 10, 17], and that f is semi-extremal onceµT ∧ω, see [14]. Here ω stands for the Fubini-Study (1,1)-form on P², so thatT∧ωis the trace measure of the Green current. Dujardin asked in [14, Section 3] the following questions.

Question 1.1. Does every semi-extremal endomorphism satisfy µ T ∧ω? If µT∧ω, doesf contain (in some sense) a one-dimensional Latt`es map?

A strategy for the first question would to show that semi-extremal endomor- phisms have an invariant pencil of lines. Indeed, in this case the radial Lyapunov exponent of f is ≥logd(by [18, Theorem 1.7]), hence the Lyapunov exponent of the action off on the pencil of lines (which belongs to the spectrum of µby [18,

2020Mathematics Subject Classification. 37F10, 32U40, 28A15.

Key words and phrases. Holomorphic dynamics, Lyapunov exponents, Green current.

6767

Theorem 1.5]) must be equal to ¹₂logd(semi-extremality is assumed). This implies that f induces a Latt`es map on the pencil of lines, which ensures thatµT ∧ω by [18, Corollary 1.3].

We can make a similar observation for the second question. IfµT∧ω then

1

2logdis one of the Lyapunov exponents of µ by [14]. Using the same arguments as before (relying on [18]), the Lyapunov exponent of the action off on the pencil of lines must be ¹₂logd, ensuring that f induces a Latt`es map on this pencil.

In the general case (without assuming the existence of an invariant pencil of lines), the conditionµT∧ω, equivalent toT∧T T ∧ω, formally says thatT ω with respect to T. A first step to exhibit such regular properties for T consists in establishing regular properties for slices. Theorems 1.3 and 1.4 below provide results in that direction: they bound the pointwise lower Radon-Nikodym derivative of stable slices of T at µ-almost every point. In Section 1.4 we shall examine the guideline given by polynomial lifts of Latt`es mappings onP¹: such regular slices of course exist, moreover there exist regular slices which do not intersect the support ofµ(see Remark1.6).

1.2. Statements of the results.

Definition 1.2. Let f be a holomorphic endomorphism of P² of degree d ≥ 2.

Assume that the Lyapunov exponents of the equilibrium measure off are different, henceλ₂ < λ₁. Let v_s(x)∈ P(T_xP²) denote the stable Oseledec direction, which is defined for every xin an invariant µ-generic Borel subset A. Let R be the set of points x∈A such that there exists a holomorphic discξx : D→ P² satisfying ξx(0) =x, [ξ_x⁰(0)] =vs(x) and

lim inf

r→0

ξ_x^∗T(D(r))

Leb(D(r)) <∞. (1.1)

In that definition [v]∈P(T_xP²) stands for the line of T_xP² directed by v. Since µ(C) = 0, one can assume that A does not intersect ∪_n∈Zfⁿ(C). The relation f^∗T =dT then implies thatRis totallyf-invariant, in particular,µ(R)∈ {0,1}by ergodicity. We prove thatµ(R)>0 under two different sets of assumptions related to semi-extremality.

Theorem 1.3. Let f be a holomorphic endomorphism of P² of degree d ≥ 2.

Assume thatµT∧ω andλ2< λ1<2λ2. Thenµ(R) = 1.

Theorem 1.4. Let f be a holomorphic endomorphism of P² of degree d ≥ 2.

Assume thatf is semi-extremal and thatλ₂< λ₁<2λ₂. Assume also thatJ ∩C=∅ and that vs is H¨older continuous onA. Then µ(R) = 1.

For the second result, we shall see in Section2.2that the H¨older continuity ofvs

onAis satisfied whenf is partially hyperbolic onJ.

1.3. Outline of the proofs. To prove Theorems 1.3 and 1.4, a first step will consist in showing

d^nje^−2M1 ≤ kDxf^nj(~vs(x))k²≤d^nje^2M1 (1.2) for everyxin a subset of positiveµ-measure, where (nj)j is an increasing sequence of integers depending onxand~vs(x)∈TxP²is a unitary vector invs(x)∈P(TxP²) defined in Definition 1.2. It is crucial that no exponential error term e^±n occur in (1.2). Such error terms are actually inherent to (our context of) non uniform hyperbolicity, a difficulty is thus to get rid of them. A second step will be to integrate

(1.2), namely to construct a holomorphic disc on whichf^nj is a multiplication by d^nj/2e^±M1. We will use for that purpose a normal form Theorem for the inverse branches of f. Theorem 3.1 (see Section 3) summarizes both steps, it provides workable sufficient conditions ensuring that a point belongs toR.

Let us specify how the estimates (1.2) will be obtained. The lower bound is actually true for every endomorphism ofP² satisfyingλ2≤λ1 <2λ2, it does not need any semi-extremality assumption, see Theorem2.7. The upper bound in (1.2) needs semi-extremality: for Theorem1.3 we use arguments developed by Dujardin [14] to construct Fatou directions (see Section 4), and for Theorem1.4 we use the Central Limit Theorem for the logarithm of the tangent map on the stable direction vs(x) (see Section5).

Let us notice that the Central Limit Theorem was used in [27] to prove regularity properties for the equilibrium measure of rational maps onP¹and in [16] to provide a new proof of the fact thatµis absolutely continuous whenf is extremal onP^k. 1.4. Study of the lifts of one dimensional Latt`es mappings. The polyno- mial lifts of Latt`es mappings onP¹are the most simple examples of semi-extremal endomorphisms ofP². In this section we verify for themµ(R) = 1 and µT ∧ω by using normal forms for their Green function.

Proposition 1.5. Let f[z : w : t] := [P(z, w) : Q(z, w) :t^d], where [P : Q] is a Latt`es mapping ofP¹ of degreed≥2. Thenµ(R) = 1.

Proof. The mappingf belongs to the class of regular polynomial endomorphisms of C² studied by Bedford-Jonsson [2]. LetL:= (P, Q) and letLⁿ= (Pⁿ, Qⁿ) denote then-th iterate ofL. The Green function of the homogeneous polynomial mapping Lis

G(z, w) := lim

n

1

dⁿ log||(Pⁿ(z, w), Qⁿ(z, w))||.

Let us denote by Ω :={G <0} the basin of attraction of the origin inC², and by

∂Ω ={G= 0}the boundary of Ω. In the affine chart{t= 1}, the Green currentT off is equal todd^cG⁺, whereG⁺:= max{G,0}. Since [P :Q] is a Latt`es mapping, the support of the equilibrium measureµ=T∧T off coincides with ∂Ω.

The results [3, Theorem 1.2 and Lemma 4.2] due to Berteloot-Loeb provide normal forms for the function G when [P : Q] is a Latt`es map of P¹ of degree d.

Precisely, ifxdoes not belong to the union S of a finite number of complex lines in C² passing through the origin (which has zero µ-measure), then there exist a biholomorphismp: (C²,(0,0))→(C², x) andδ∈Rsuch that

G◦p(Z, W) =|Z|²+<(W) and p⁻¹◦L◦p(Z, W) = (e^iδ√

d Z, d W).

In particular, up to a holomorphic change of coordinates near x, ∂Ω is a piece of the 3-euclidian sphere. Now observe that the functionG◦pis strictly subharmonic on the Z-axis (the unique complex line in the tangent space at (0,0) of the real hypersurface {G◦p= 0}), which is included in{G◦p≥0}. HenceG⁺ is strictly subharmonic along the holomorphic disc

ξx(u) :=p(u,0).

The limit of ^ξ_Leb(^∗xT(D(r))

D(r)) whenrtends to 0 then exists and is equal todd^c(G⁺◦ξx)(0)∈ R^∗+. The fact that [ξ_x⁰(0)] =vs(x) comes from [3, Lemma 4.2], which lifts the Latt`es commuting diagram to line bundles, the local coordinate in the fibers beingW. We thus have verified thatµ-almost everyxbelongs toR, as desired.

Remark 1.6. In the preceding proof, for everyW0∈(C,0) such that<(W0)>0, the function G⁺ is strictly subharmonic along the holomorphic disc ξ(u, W0) :=

p(u, W0). Such a disc is included in {G◦p > 0}, which does not intersect the support ofµ.

Proposition 1.7. Let f[z : w : t] := [P(z, w) : Q(z, w) :t^d], where [P : Q] is a Latt`es mapping ofP¹ of degreed≥2. ThenµT∧ω.

Proof. We take the notations of the proof of Proposition1.5. It suffices to verify µ T ∧ω at every point x∈ ∂Ω\S, since µ(S) = 0. We work with the local coordinates provided by p. Let G0(Z, W) := |Z|²+<(W), T0 := dd^cG⁺₀ and ω0:= ₂ⁱdZ∧dZ+₂ⁱdW∧dW. For every test functionϕon (C²,(0,0)), we have by using and adapting [22, Proposition 6.5.5]:

Z

(C²,(0,0))

ϕ T₀∧ω₀= Z

{G₀=0}

ϕ d^cG₀∧ω₀+ Z

{G₀≥0}

ϕ dd^cG₀∧ω₀, (1.3) Z

(C²,(0,0))

ϕ T₀∧T₀= Z

{G0=0}

ϕ d^cG₀∧dd^cG₀, (1.4) where the orientation on{G0= 0} is induced by the one of{G0≤0}. A straigth- forward computation shows that

d^cG₀∧ω₀=d^cG₀∧dd^cG₀, (1.5) on the spherical 3-manifold {G0 = 0}. Finally, one obtains T0∧T0 T0∧ω0 by using (1.5) and comparing (1.3) and (1.4).

Remark 1.8. The measure given by (1.5) is equal to d=W ∧ ₂ⁱdZ ∧dZ, which belongs to the Lebesgue measure class of the 3-manifold{G0= 0}.

2. Oseledec theorem, stable directions, normal forms.

2.1. Oseledec theorem. Let us state the Oseledec Theorem in our context.

Theorem 2.1. Let f be a holomorphic endomorphism ofP² of degree d≥2. Let λ2≤λ1 be the Lyapunov exponents of the equilibrium measureµ off.

1. Ifλ1=λ2=λ, then there exists an invariant Borel subsetAof fullµ-measure and disjoint from∪_n∈Zfⁿ(C)such that for everyx∈A:

∀~v∈T_xP²\ {0} , lim

n→+∞

1

nlogkD_xfⁿ(~v)k=λ.

2. If λ1 > λ2, then there exists an invariant Borel subset A of full µ-measure and disjoint from∪_n∈Zfⁿ(C) such that for everyx∈A, there existsvs(x)∈ P(TxP²)satisfying:

∀~v∈TxP²\vs(x) , lim

n→+∞

1

nlogkDxfⁿ(~v)k=λ1.

∀~v∈vs(x)\ {0} , lim

n→+∞

1

nlogkDxfⁿ(~v)k=λ2.

Moreover vs is measurable and satisfies [Dxf](vs(x)) = vs(f(x)) for every x∈A, where[D_xf] is the projectivization of the tangent mapD_xf.

Let us assume thatλ2< λ1. For everyx∈A, we denote ψ(x) := logkDxf(~vs(x))k,

where~vs(x)∈ {~v∈vs(x)⊂TxP² , k~vk= 1}. We haveψ∈L¹(µ), see [26, Section 3.7].

Lemma 2.2. For every x∈A and everyn≥1, we have 1

n

n−1

X

i=0

ψ(fⁱ(x)) = 1

nlogkDxfⁿ(~vs(x))k. This implies R

ψ dµ=λ2.

Proof. The first formula follows from the definition of ψ and the relation [Dxf] (vs(x)) =vs(f(x)). We obtain the second formula by taking limitsµ-almost every- where: the left hand side converges toR

ψ dµby Birkhoff ergodic Theorem, and the right hand side converges toλ2 by Theorem2.1.

2.2. H¨older continuity of the stable direction. The following result, due to Brin, is [25, Proposition 3.9] adapted to our setting, see also [1, Section 5.3]. We use the notationdist for the distances onP² andP(TP²).

Theorem 2.3. Let f be a holomorphic endomorphism of P² of degree d ≥ 2.

Assume thatf is partially hyperbolic onJ: that means that there exist0< λ₋ < λ+

andc≥1such that for everyx∈ J, there exist unitary vectorsw~s(x), ~wu(x)∈TxP² whose angle is uniformly bounded from below onJ and which satisfy

∀n≥1 , kDxfⁿ(w~s(x))k ≤cλⁿ₋ , kDxfⁿ(w~u(x))k ≥c⁻¹λⁿ₊. (2.1) Then the directionws= [w~s] :J →P(TP²) is H¨older continuous. More precisely, if b:=max{1,kfk_C1} then for everya > b² there existsDa >1 such that:

dist(x, y)< D_a⁻¹ ⇒ dist(ws(x), ws(y))≤3c²λ₊

λ₋ (Dadist(x, y))

log(λ+/λ−) log(a/λ−) . Remark 2.4. Iff is partially hyperbolic onJ and ifλ2< λ1, then one verifies that the stable Oseledec directionv_s(x) provided by Theorem 2.1 coincides withw_s(x) for everyx∈A, hencev_sis H¨older continuous onAby Theorem2.3. In particular, every partially hyperbolic endomorphism satisfying λ₂ = ¹₂logd, λ₂ < λ₁ < 2λ₂ andJ ∩ C=∅ fulfills the assumptions of Theorem1.4.

Actually it is not easy to find examples of semi-extremal endomorphisms partially hyperbolic onJ. For instance the liftf[z:w:t] = [z²−2w²:z²:t²] of the Latt`es mapping L[z: w] = [z<sup>2</sup>−2w<sup>2</sup> : z<sup>2</sup>] is not partially hyperbolic onJ. To see this, observe that the Lyapunov exponents of (f, µ) are <sup>1</sup><sub>2</sub>log 2 (coming from the Latt`es mappingL) and log 2 (coming fromt²). In particular, the modulus of the multiplier of every n-repulsive cycle of L not intersecting the critical values of σ is equal to (√

2)ⁿ. On the other hand, that modulus is equal to 4 for the fixed point [1 : 1]

(this is a critical value ofσ, see [3, Section 3]). Hencef hasn-periodic points inJ with multipliers{(√

2)ⁿ,2ⁿ} and{4ⁿ,2ⁿ} respectively. It follows that noλ₋< λ+

satisfy (2.1) overJ.

Examples of partially hyperbolic endomorphisms should be found among uni- formly expanding endomorphisms. Jonsson [20] obtained examples in the family of polynomial skew products. For instance,f(z, w) = (z², w²+zw+cz) is uniformly expanding on J if c is large enough (see Theorems 4.8, 8.2 and Examples 9.2 of

this article). Corollary 8.3 of [20] moreover characterizes the Axiom A property in terms of the dynamics of the critical set: f(z, w) = (z²−6, w²+ 3−z) is uniformly expanding on J. We refer to [19] for results on a family of Axiom A endomor- phisms ofP² (calleds-hyperbolic): basic sets, attractors and invariant currents are investigated there in Sections 3, 4 and 5.

To deduce partial hyperbolicity from uniform expansivity is also not straight- forward. This should involve the Lyapunov exponents of µ: the stable Oseledec directionv_s(x) is a natural candidate forw_s(x). Nonetheless, that direction is only definedµ-a.e. onJ, and an other difficulty comes from the fact that the Lyapunov exponents only give an asymptotic growth rate. Hence, even if the Lyapunov ex- ponents are distinct, the existence ofc≥1 andλ₋< λ+ satisfying (2.1) for every n≥1 is not obvious.

Before stating the next Proposition, let us recall Remark 2.4: if f is partially hyperbolic onJ and ifλ2< λ1, then the stable Oseledec directionvs provided by Theorem2.1is H¨older continuous onA.

Proposition 2.5. Let f be a holomorphic endomorphism of P² of degree d≥ 2.

Assume thatλ₂< λ₁ and that v_s is H¨older continuous onA.

1. Thenvs extends to a H¨older continuous map onJ, still denotedvs. 2. If J ∩ C=∅, then ψ(x) = logkDxf(~vs(x))k is H¨older continuous on J. Proof. The first item is classical by using Cauchy sequences and the fact thatAis dense in J. Let us writedist(vs(x), vs(y))≤c⁰d(x, y)^αonJ and prove the second item. By working in charts of TP², let us fix a section of the unit tangent bundle x7→~vs(x) such that~vs(x)∈vs(x) andk~vs(x)−~vs(y)k ≤c⁰dist(x, y)^α. We get

kDxf(~vs(x))−Dyf(~vs(y))k

≤ kD_xf(~v_s(x))−D_xf(~v_s(y))k+kD_xf(~v_s(y))−D_yf(~v_s(y))k

≤ kDxfk k~vs(x)−~vs(y)k+kDxf −Dyfk k~vs(y)k

≤ kfk_C2(1 +c⁰)d(x, y)^α.

This implies that x7→ kDxf(~vs(x))k is H¨older continuous. Since J ∩ C =∅, that function is bounded below by a constant ρ > 0 on J. By using that log isρ⁻¹- Lipschitz on [ρ,+∞[, we get thatψ(x) = logkDxf(~vs(x))k is H¨older continuous on J.

2.3. Normal forms. Recall that the subsetAdefined in Definition1.2is invariant, does not intersect∪n∈Zfⁿ(C) and satisfiesµ(A) = 1 . Let

Aˆ:=

ˆ

x= (x_n)_n∈Z∈A^Z, x_n+1 =f(x_n)

and let ˆf : ˆA →Aˆ be the left shift. We denote ˆx_n := ˆfⁿ(ˆx) for every n ∈Z. A functionϕ : ˆA →]0,+∞[ is -tempered if e⁻ϕ(ˆx)≤ϕ( ˆf(ˆx))≤eϕ(ˆx). Let f_xˆ⁻ⁿ

n

be the inverse branch offⁿ sending a neighbourhood ofxn to a neighbourhood of x0. The following result provides normal forms for those mappings, see [5,21]. The measure ˆµ stands for the unique ˆf-invariant measure on ˆA such that (π₀)_∗µˆ =µ, whereπ₀(ˆx) :=x₀. LetB_x(r) denote the ball centered at xof radiusrin P². Theorem 2.6. [5, Proposition 4.3]For every >0there exist-tempered functions η, ρ: ˆA→]0,1],β, L, M: ˆA→[1,+∞[and a functionN : ˆA→Nsatisfying the following properties. There exist injective holomorphic mappingsξxˆ:Bx₀(η(ˆx))→ D²(ρ(ˆx))satisfying for every xˆ∈A:ˆ

1. ξˆx(x0) = 0 andDx₀ξxˆ(vs(x0))is the vertical axis inC²,

2. ∀p, q∈Bx₀(η(ˆx)), ¹₂dist(p, q)≤ kξxˆ(p)−ξˆx(q)k ≤β(ˆx)dist(p, q), and such that the following diagram commutes for everyn≥N(ˆx):

B_x0(η(ˆx))

ξxˆ

B_xn(η(ˆx_n))

f⁻ⁿ_ˆ

oo xn

ξ_xnˆ

D²(ρ(ˆx)) D²(ρ(ˆxn))

Rn,ˆxn

oo

The mappingsRn,ˆx_n have the following form depending on (λ1, λ2):

1. If λ1=λ2=λ, thenRn,ˆx_n is a linear mapping satisfying e^−n(λ+)k(z, w)k ≤ kRn,ˆxn(z, w)k ≤e^−n(λ−)k(z, w)k.

2. Ifλ1=kλ2for somek≥2thenRn,ˆx_n(z, w) = (αn,ˆx_nz, βn,ˆx_nw)+(γn,ˆx_nw^k,0).

3. If λ16∈ {kλ2, k≥1}, thenRn,ˆx_n(z, w) = (αn,ˆx_nz, βn,ˆx_nw).

Moreover, in the cases 2 and 3, we have

e^−n(λ1+)≤ |αn,ˆx_n| ≤e^−n(λ1−) , |γn,ˆx_n| ≤M(ˆx)e^{−n(λ1−2)} and

e^−n(λ2+)≤ |βn,ˆxn| ≤e^−n(λ2−). (2.2) In particular, ifλ2< λ1, the second coordinate ofRn,ˆx_n has the formw7→βn,ˆx_nw.

2.4. Estimates from pluripotential theory. The following result holds for ev- ery endomorphism of P² satisfyingλ1 < 2λ2. The proof relies on Briend-Duval’s strategy [8] by taking into accountλ2≥ ¹₂logd: the arguments use the facts thatT has continuous potentials, satisfiesf^∗T =dT and definesµ by the Monge-Amp`ere equationµ=T∧T. Let us set

Bn(ρ) :=

x∈P² , u7→fⁿ◦(x+ (Dxfⁿ)⁻¹(u)) :D²(ρ)→P²is injective , Rn(τ) :=n

x∈P² ,

(Dxfⁿ)⁻¹

−1≥τ⁻¹d^n/2o .

For the definition ofB_n(ρ), we used charts ofTP²so that the mappingx+(D_xfⁿ)⁻¹ is defined onD²(ρ) and takes its values in a neighbourhood ofxin P².

Theorem 2.7. [6, Propositions 1 and 2]Let f be a holomorphic endomorphism of P² of degreed≥2. If λ1<2λ2, then for everyβ >0,

1. ∃ρ >0,∀n≥1,µ(Bn(ρ))≥1−β,

2. ∃ρ >0,∀n≥1,∀τ >0,µ(Bn(ρ)∩Rn(τ))≥1−β−(ρτ)⁻². 3. In particular,∀a >0,∃τ >0,∀n≥1,µ(Rn(τ))≥1−a.

The following Corollary will be used to prove Proposition5.2.

Corollary 2.8. Let f be a holomorphic endomorphism ofP²of degree d≥2satis- fyingλ2< λ1<2λ2. For everyβ >0letρ >0given by Theorem2.7. There exists a Borel set H of full µ-measure satisfying for every x∈H, there exists n(x) ≥1 such that

∀n≥n(x), x6∈B_n(ρ)or kD_xfⁿ(~v_s(x))k ≥ d^n/2 n .

Proof. According to Theorem2.7, we haveP

n≥1µ(Bn(ρ)∩Rn(n)^c)≤ρ⁻²P

n≥1 1 n². By Borel-Cantelli’s Lemma, there existsH such thatµ(H) = 1 and

∀x∈H, ∃n(x)≥1, ∀n≥n(x), x∈Bn(ρ)^c∪Rn(n).

Letx∈H. Ifx∈B_n(ρ)^c, the proof is complete. If not,x∈R_n(n), and since~v_s(x) is unitary we getkDxfⁿ(~vs(x))k ≥

(Dxfⁿ)⁻¹

−1≥ ^dn/2_n .

3. A criterion for bounded Radon-Nikodym derivatives. The following the- orem gives sufficient conditions ensuring that a pointxbelongs to the setRdefined in Definition1.2. The integerN(ˆx) is defined in Theorem2.6.

Theorem 3.1. Let f be a holomorphic endomorphism of P² of degree d ≥ 2.

Assume thatλ1> λ2and letxˆ∈A. If there existˆ η0, M1, M2>0and an increasing sequence (nj)j∈N of integers such that n0= 0 andn1≥N(ˆx)satisfying:

1. η(ˆxn_j)≥4η0,β(ˆxn_j)≤e^M2,

2. e^−M1d^nj/2≤ kDxf^nj(~vs(π0(ˆx)))k ≤e^M1d^nj/2,

then x := π0(ˆx) ∈ R: there exists a holomorphic disc ξx : D → P² satisfying ξx(0) =x,[ξ_x⁰(0)] =vs(x)and

lim inf

r→0

ξ_x^∗T(D(r))

Leb(D(r)) <∞. (3.1)

Proof. Let us apply Theorem2.6which gives normal forms for the inverse branches of f. Since η(ˆxn_j) ≥4η0, the image of Bx_nj(η(ˆxn_j)) byξxˆ_nj contains D²(2η0).

Similarly, the image of B_x0(η(ˆx)) by ξ_xˆ contains D²(2η₀). Let V be the vertical disc

V : D(2η₀) → D(2η₀)×D(2η₀)

w 7→ (0, w) ,

and let us set

ξ˜₀:= (ξ_ˆx)⁻¹◦V and ξ˜_nj := (ξ_xˆnj)⁻¹◦V.

By pulling back (f^nj)^∗T =d^njT by ˜ξ0, we get

(f^nj ◦ξ˜0)^∗T =d^njξ˜₀^∗T on D(η0). (3.2) Let us observe that

f^nj◦ξ˜₀=f^nj◦(ξ_xˆ)⁻¹◦V = (ξ_xˆnj)⁻¹◦R⁻¹_n

j,ˆx_nj ◦V,

where the first equality comes from the definition of ˜ξ0 and the second one from Theorem 2.6. If βn_j : C→ Cdenotes the multiplication by βn_j,ˆx_nj, we also have the relation

R⁻¹_n

j,ˆx_nj ◦V =V ◦β_n⁻¹

j ,

since the second coordinate ofR_nj,ˆxnj is linear. Hence we get:

f^nj ◦ξ˜₀= ˜ξ_nj ◦β_n⁻¹

j onβ_nj(D(η₀)).

Equation (3.2) restricted toβn_j(D(η0))⊂D(η0) then implies:

ξ˜^∗_njT =d^nj(β_n⁻¹_j)_∗ξ˜₀^∗T onD(η0), which gives

ξ˜_n^∗_jT(D(η0)) =d^njξ˜₀^∗T(D(η0· |βn_j,ˆx_nj|)). (3.3) Now we bound from below the right hand side of Equation (3.3). This is where we use the second hypothesis of Theorem3.1.