Dynamics of fibered endomorphisms of $\mathbb P^k$

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Dynamics of fibered endomorphisms of P

Christophe Dupont, Johan Taflin

To cite this version:

Christophe Dupont, Johan Taflin. Dynamics of fibered endomorphisms of P^k. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Scuola Normale Superiore 2021, 22 (1), pp.53-78. �hal- 01925373�

Dynamics of fibered endomorphisms of P

Christophe Dupont and Johan Taflin^∗ November 16, 2018

Abstract

We study the structure and the Lyapunov exponents of the equilibrium measure of endomorphisms of P^k preserving a fibration. We extend the decomposition of the equilibrium measure obtained by Jonsson for polynomial skew products of C². We also show that the sum of the sectional exponents satisfies a Bedford-Jonsson formula when the fibration is linear, and that this function is plurisubharmonic on families of fibered endomorphisms. In particular, the sectional part of the bifurcation current is a closed positive current on the parameter space.

Key words : Equilibrium measure, Lyapunov exponents, Bifurcation current.

MSC 2010 : 32H50, 37F10.

1 Introduction

Let f be a holomorphic endomorphism of P^k of degree d≥ 2 which preserves a rational fibration parametrized by a projective space i.e. there exist a dominant rational map π:P^k 99KP^r and a holomorphic mapθ:P^r →P^r such that

π◦f =θ◦π. (1.1)

The generic fiber of π has dimension q := k−r. Another way to express (1.1) is that f permutes the fibers of π and this permutation is given by θ. We are interested in the relationships between the dynamics off and the one ofθ.

This type of maps has been recently used to exhibit interesting dynamical phenomena inP²(see [Duj16], [ABD⁺16], [BT17], [Duj17], [Taf17]). All these examples, except [BT17], come from polynomial skew products ofC², whose dynamical properties have been studied by Jonsson in [Jon99]. It is therefore interesting for future examples to extend the results of [Jon99] to a broader framework. Our initial motivation was to study the particular case where π is the standard linear fibration defined by π[y :z] = [y] withy := (y₀, . . . , y_r)∈ C^r+1 and z = (z0, . . . , zq−1) ∈ C^q. However, some of the techniques can be extended to a more general setting. In what follows, we choose the setting of each result in order to avoid unnecessary technical details. We refer to the end of this introduction for the possible scope of the techniques, in particular whenk = 2 thanks to the works of Dabija-Jonsson ([DJ08], [DJ10]) and Favre-Peirera ([FP11], [FP15]) on endomorphisms ofP² preserving a fibration, a foliation or a web.

∗Research partially supported by ANR project Fatou ANR-17-CE40-0002-01.

Green currents and Lyapunov exponents – The maps f, π and θ are given by homogeneous polynomials and a simple computation shows that f and θ have the same degree. Both maps have a Green current,T_f andT_θ respectively, which are positive closed (1,1)-currents with continuous local potentials. Their self-intersections are well-defined and their supports define dynamically meaningful filtrations

Ji(f) := supp(T_fⁱ) and Jj(θ) := supp(T_θ^j),

for i∈ {1, . . . , k} and j ∈ {1, . . . , r}.The equilibrium measures of f and θ are defined by µ_f := T_f^k and µ_θ := T_θ^r. Since f and θ are semi-conjugated by π, a natural question is whether there exists a relation between T_fⁱ and the pull back ofT_θ by π. Our first result gives such a relationship if i > q. More precisely, we will see in Section 2 how to define π^∗T_θ and ifS denotes the result normalized by its mass,

S:= π^∗T_θ kπ^∗T_θk, then we have the following result.

Theorem 1.1. Let f:P^k → P^k and θ: P^r → P^r be two endomorphisms of degree d≥ 2.

Assume there exists a dominant rational map π:P^k 99K P^r whose indeterminacy set I(π) is disjoint from Jq(f) and such that θ◦π =π◦f. Then for j ∈ {1, . . . , r}, the current S^j is well-defined, satisfiesS^j 6=T_f^j and T_f^q+j =T_f^q∧S^j. In particular, µ_f =T_f^q∧S^r and π_∗µ_f =µ_θ.

Let us emphasize that the proof only relies on the properties of the currents T_f and T_θ and is coordinate free. Using the classification obtained in [DJ08], one can check easily that the assumptionJq(f)∩I(π) =∅is always satisfied whenk= 2,in which caseq= 1.

This is also the case for the standard linear fibration in any dimension (see Lemma 5.1).

In general, we know no example where this assumption does not hold.

The main point in Theorem 1.1 is the formula µ_f =T_f^q∧S^r which can be seen as a generalization of the decomposition ofµ_f obtained by Jonsson [Jon99] for polynomial skew products ofC². Indeed, for µ_θ-almost everya∈P^r the fiber La :=π⁻¹(a) has dimension q and we can define the probability measure

µ_a:= T_f^q∧[L_a] kπ^∗T_θk^r .

Corollary 1.2. Let φ: P^k → R be a continuous function. Under the assumptions of Theorem 1.1 we have

Z

P^k

φ(x)dµ_f(x) = Z

P^r

Z

La

φ(x)dµ_a(x)

dµ_θ(a).

The other results in this paper can also be seen as consequences of the formula µ_f = T_f^q∧S^r and the main technical difficulties come from the fact that the currentsSand [L_a] are singular. As a direct consequence of Theorem 1.1, we obtain in the following result that if µ_θ is absolutely continuous with respect to Lebesgue measure (i.e. θ is a Lattès mapping of P^r, see [BD05]) then µ_f is absolutely continuous with respect to the trace measureσ_T^q

f :=T_f^q∧ω^r_Pk.Here, ωP^k (resp. ωP^r) is the Fubini-Study form onP^k (resp. P^r) normalized such thatω^k_Pk (resp. ω^r_Pr) is a probability measure.

Corollary 1.3. Under the assumptions of Theorem 1.1, if µ_θ<< ω^r_Pr then µ_f << σ_T^q

f.

That applies for Desboves mappings ofP², studied in [BDM07, Section 4] and [BT17], they indeed induce a Lattès mapping on a pencil of lines. Let us note that whenk= 2, the propertyµ_f << σ_Tf implies that the smallest exponent of µ_f is minimal, equal to ¹₂logd, see [Duj12, Theorem 3.6]. In particular the Lyapunov exponents of Desboves mappings areλ₁> λ₂ = ¹₂logd, with d= 4. The following Theorem generalizes that semi-extremal property to fibered endomorphisms satisfyingµ_θ << ω_P^rr. It is a consequence ofπ_∗µ_f =µ_θ and holds for more general smooth dynamical systems.

Theorem 1.4. Let f, π and θ be as in Theorem 1.1. If Λ is a Lyapunov exponent of multiplicitym for µ_θ then Λ is a Lyapunov exponent of multiplicity at least m of µ_f. Standard linear fibration – For the next results, we restrict ourselves to the cases whereπ is the standard linear fibration where the result below is already new whenk= 2 and r = 1. The indeterminacy set I(π) of π corresponds to {y = 0} ≃ P^q−1 and each fiberL_a=π⁻¹(a) is a linear projective spaceP^q in which I(π) can be identified with the hyperplane at infinity, i.e. L_a\I(π)≃C^q.Iff preserves the fibration defined byπ thenf acts on each periodic fibers as a regular polynomial endomorphism ofC^q.This class of maps have been studied by Bedford-Jonsson in [BJ00]. In particular, they obtained a formula for the sum of the Lyapunov exponents of the equilibrium measure. More precisely, letR be a regular polynomial endomorphism ofC^q of degree di.e. R extends to an endomorphism of P^q of degree d. We denote by T_R, Crit_R and G_R respectively the Green current, the critical set and the Green function inC^q of R. The restriction of R to the hyperplane at infinity P^q\C^q≃P^q−1 is an endomorphism of P^q−1 and we denote byΛ₀ the sum of the Lyapunov exponents of its equilibrium measure.

Theorem 1.5(Bedford-Jonsson [BJ00]). Let R be a regular polynomial endomorphism of C^q of degreed. The sumΛ_Rof the Lyapunov exponents of its equilibrium measure satisfies

Λ_R= logd+ Λ₀+hT_R^q−1∧[Crit_R], G_Ri.

We give a generalization of this formula in the fibered setting. To this aim, we introduce some notations. Ifπ◦f =θ◦π then we denote byΛ_f (resp. Λ_θ) the sum of the Lyapunov exponents ofµ_f (resp. µ_θ). Theorem 1.4 implies that Λ_f = Λ_θ+ Λ_σ where Λ_σ is the sum of the Lyapunov exponents of µ_f in the direction of the fibers. The indeterminacy set I(π)≃P^q−1 is invariant byf thus f_I(π) can be seen as an endomorphism of P^q−1 and we denote by Λ₀ the sum of the Lyapunov exponents of this restriction.

Sincef preserves the fibration, the setCrit_f is not irreducible. Some irreducible com- ponents ofCrit_f are foliated by fibers of π and constitute the “fibered” part ofCrit_f.The remaining part is its “sectional” part. Indeed, as the standard linear fibration π is a sub- mersion outsideI(π),we have a decomposition in terms of currents [Crit_f] = [C_∞] + [C_σ] where[C∞] :=π^∗[Crit_θ]and[Cσ]is the current of integration on the sectional part ofCrit_f (see Lemma 5.3).

Finally, we define therelative Green function as the unique lower semicontinuous func- tionG: P^k→[0,+∞]such that dd^cG=T_f −S and minG= 0.

Theorem 1.6. Let f be an endomorphism of P^k of degree d ≥ 2 which preserves the standard linear fibration. Then

Λ_σ = logd+ Λ₀+hT_f^q−1∧S^r∧[C_σ], Gi.

In particular, Λ_σ ≥ ^q+1₂ logd.

The idea of the proof is to apply the formula of Bedford-Jonsson to each n-periodic fiber of f.Since the current S^r can be seen as the limit of the average of the currents of integration on the n-periodic fibers, we obtain the above formula at the limit. However, in order to implement that idea, we need some additional care since the currents involved are singular.

Families of fibered endomorphisms – We now consider a family (f_λ)_λ∈M of endo- morphisms ofP^k which preserves the standard linear fibration π i.e. there exists a family (θ_λ)_λ∈M of endomorphisms ofP^r such that π◦f_λ =θ_λ◦π. The family(f_λ)_λ∈M induces a dynamical systemf(λ, x) := (λ, f_λ(x)) onM×P^k whose critical setCrit_f is the gluing of the critical sets off_λ, λ∈M. In the same way, there exists a positive closed (i, i)-current T_fⁱ onM×P^kwhose slices are equal toT_fⁱ

λ.In [BB07] Bassanelli and Berteloot established a formula between the currents [Crit_f]and T_f^k and the dd^c of Λ_f:λ7→ Λ_fλ where Λ_fλ is the sum of the Lyapunov exponents ofµ_fλ (see also [Pha05]). They proved that

dd^cΛ_f =p∗(T_f^k∧[Crit_f]),

where p:M ×P^k → M is the projection. This current is called the bifurcation current T_Bif(f)and its support coincides with several bifurcation phenomena in the family(f_λ)_λ∈M (see [BBD18]).

Similar objects can be defined for the family(θ_λ)_λ∈M and again, it is natural to inquire into their interplays with the ones defined for (f_λ)_λ∈M. Since this family preserves the fibration, as above the setCrit_f is not irreducible and we have[Crit_f] = [C_∞] + [C_σ]where [C_∞] := Π^∗[Crit_θ] withΠ(λ, x) = (λ, π(x)).This decomposition induces a decomposition ofT_Bif(f) in a fibered part and a sectional part. The result below states that the fibered part ofT_Bif(f) coincides with T_Bif(θ).

Theorem 1.7. Let M be a complex manifold and consider two holomorphic families (f_λ)_λ∈M and(θ_λ)_λ∈M of endomorphisms of P^k andP^r respectively such thatπ◦f_λ =θ_λ◦π where π is the standard linear fibration. The (1,1)-current T_Bif,σ(f) := T_Bif(f)−T_Bif(θ) is positive. Moreover, ifS^r := Π^∗T_θ^r then

T_Bif(θ) =p_∗(T_f^q∧S^r∧[C_∞]), T_Bif,σ(f) =p_∗(T_f^q∧S^r∧[C_σ]).

A different way of seeing this result is the following. By Theorem 1.4, we know that Λ_σ = Λ_f−Λ_θ is the sum of the Lyapunov exponents ofµ_f which are not in the Lyapunov spectrum ofµ_θ.Theorem 1.7 yields thatΛ_σ is a plurisubharmonic function onM and gives a formula ofdd^cΛ_σ in terms of currents onM×P^k,

dd^cΛ_σ =p_∗(T_f^q∧S^r∧[C_σ]).

Notice that Astorg-Bianchi initiated in [AB18] the study of bifurcations for skew products ofC² and proved in that setting that Λ_σ is plurisubharmonic.

Following an idea coming from [AB18], for each n≥1 we can consider the bifurcation current associated to the dynamics of the family (f_λ)_λ∈M on the n-periodic fibers. To be more precise, if λ ∈M and a ∈ P^r are such that θ_λ(a) = athen we denote by Λ(f_λ|Lⁿ

a) the sum of Lyapunov exponents off_λ|Lⁿ

a seen as a polynomial endomorphism of L_a≃P^q. Then we define

Λ_σ,n(λ) := 1 nd^rn

X

θⁿ_λ(a)=a

Λ(f_λ|Lⁿ _a) and T_Bif,n(f) :=dd^cΛ_σ,n.

It is easy to see that if all the cycles of the family(θ_λ)_λ∈M can be followed holomorphically on M then Λ_σ,n is plurisubharmonic. Actually, this holds in general and we can express the currentT_Bif,σ(f) in termsT_Bif,n(f).

Corollary 1.8. Let (f_λ)_λ∈M, (θ_λ)_λ∈M and π be as in Theorem 1.7. For each n≥ 1 the function Λ_σ,n is plurisubharmonic and

T_Bif,σ(f) = lim

n→∞T_Bif,n(f).

Moreover, if[Per_θ,n]denotes the current of integration on{(λ, a)∈M×P^r|θⁿ_λ(a) =a}by taking into account multiplicities, then

T_Bif,n = p_∗(T_f^q∧Π^∗[Per_θ,n]∧[C_σ])

d^rn .

The first part of this result was obtained in [AB18, Corollary 4.8] in the special case of skew products ofC² under the hypothesis thatdd^cΛ_θ= 0.And, as observed by Astorg- Bianchi, a consequence of Corollary 1.8 is that if the dynamics on(f_λ)_λ∈M bifurcates on one periodic fiber then, asymptotically whenn→ ∞,it bifurcates on a positive proportion of then-periodic fibers.

Final remarks and outline of the paper – To conclude this introduction, let us explain in which setting results similar to Theorem 1.1 and Theorem 1.4 could be obtained.

First, observe that the assumption that the base space is P^r is unnecessary as long as dim(I(π))≤q−1.Indeed, ifπis a dominant meromorphic map betweenP^kand a compact complex manifoldXof dimensionrwithdim(I(π))≤q−1,(q=k−r), then the restriction ofπ to a generic linear subspace of dimensionr inP^k gives a surjective holomorphic map from P^r to X. Then by results in [DHP08, Section 2 & 3], X is projective and then by [Laz84], X is isomorphic to P^r. Observe that this argument uses the smoothness of the base. The case with a singular base might appear naturally but goes beyond the scope of this paper.

Another natural setting is the following. Assume thatf is an endomorphism ofP^kwhich preserves a family(L_a)_a∈X of algebraic sets of dimensionq and of degreeα parametrized by a complex manifoldX of dimension r, i.e. there exists an endomorphism θ of X such that f(L_a) = L_θ(a). It is natural to expect that under some assumptions on the family (L_a)_a∈X and if θpossesses an equilibrium measure µ_θ, the measure µ_f can be written as µ_f =T_f^q∧S^r where

S^r:=

Z

X

[L_a]

α dµ_θ(a).

Indeed, it is easy to check, using the classifications in [DJ10] and [FP15] and the proof of Theorem 1.1, that this is the case when k = 2 and the family (L_a)_a∈X defines a web with algebraic leaves. However, in some of these examplesS =T_f.This holds for (ii)-(iv) in [DJ10, Theorem A] and for (ii) in [FP15, Theorem E]. Otherwise, S 6= T_f in these theorems.

Finally, let us mention that results of Dinh-Nguyên-Truong [DNT12], [DNT15] suggest that one might expect some of the results above (asπ_∗µ_f =µ_θand Theorem 1.4) to extend to the case of dominant meromorphic self-maps of compact Kähler manifolds with large (or dominant) topological degree which preserve meromorphic fibrations.

The paper is organized as follows. In Section 2, we give some technical results on the pull-back and the intersection of currents. In Section 3 and Section 4, we prove Theorem

1.1, Corollary 1.3 and Theorem 1.4 respectively. In Section 5, we restrict ourselves to the cases where π is the standard linear fibration and we establish our generalized Bedford- Jonsson formula in that context. Section 6 is devoted to bifurcations of such maps.

Acknowledgements – The authors would like to thank Charles Favre for useful com- ments on the preliminary version of this paper.

2 Basics on pluripotential theory

In Section 5 and Section 6, we will intersect currents supported by fibers of a rational map π: P^k 99K P^r and singular currents or functions. Moreover, we will need that the result depends continuously on the fiber. It seems complicated to obtain these statements for an arbitrary dominant rational map π. The aim of the first part of this section is to give two results in that direction whenπ is the standard linear fibration. In a second part, we explain how to define the pull-back π^∗τ of a positive closed (1,1)-current τ on P^r by a rational map and how to define its self-intersections (π^∗τ)^j in the setting of Theorem 1.1.

2.1 Continuous families of currents

Let π:P^k 99K P^r be the standard linear fibration defined by π[y : z] = [y] where y :=

(y₀, . . . , y_r)∈C^r+1 andz= (z₀, . . . , z_q−1)∈C^q.We recall that the indeterminacy setI(π) of π corresponds to {y = 0} ≃ P^q−1 and each fiber La :=π⁻¹(a) is a projective space in whichI(π) can be identified with the hyperplane at infinity, i.e. L_a\I(π)≃C^q.

In the following two results, we consider an integer 0 ≤ l ≤ k and a family (R_a)_a∈P^r

of positive closed (k−l, k−l)-currents in P^k such that a 7→ R_a is continuous. We also consider an open setU ⊂P^k and an upper semicontinuous function v:U →[−∞,0]such thatdd^cv=T₁−T₂,where T₁ and T₂ are two positive closed (1,1)-currents where T₂ has continuous local potentials.

Lemma 2.1. Assume there exist two analytic subsets X, Y ⊂P^k such thatv is continuous onU\Xand such that for alla∈P^rwe havesupp(Ra)⊂La∩Y anddim(La∩X∩Y)≤l−1.

Then, for alla∈P^r the current vR_a is well-defined on U and depends continuously on a.

Proof. The facts thatvRa is well-defined and that its mass is locally uniformly bounded with respect toafollow easily from the Oka inequality obtained by Fornæss-Sibony [FS95].

In order to give some details, we freely use the terminology coming from [FS95]. Leta∈P^r andx∈U.Sincedim(La∩X∩Y)≤l−1,there exists an(k−l, l)Hartogs figureHdisjoint fromL_a∩X∩Y such that its hullHb is a neighborhood ofxinU.By continuity ofa7→L_a, there exists a neighborhoodV ofasuch thatL_a^′∩X∩Y∩H =∅for alla^′ ∈V.Sincev≤0 anddd^cv =T₁−T₂ where T₂ has continuous local potential, up to a continuous function v is equal to a plurisubharmonic function on Hb and [FS95, Proposition 3.1] implies that vR_a^′ is well-defined for alla^′ ∈V.Moreover, again possibly by exchanging v byv+φwith φcontinuous, we can assume that vR_a^′ ≤0anddd^c(vR_a^′)≥0onH.b Hence, we can apply the Oka inequality [FS95, Theorem 2.4]. IfK is a compact set contained in the interior of Hb then there exists a constant C >0 such that for alla^′ ∈V

kvR_a^′k_K≤CkvR_a^′k_H.

Sincea7→Rais continuous andv is continuous on∪_a^′_∈Vsupp(R_a^′)∩H, we obtain that the mass ofvR_a^′ is uniformly bounded onKfora^′ ∈V and we conclude using the compactness ofP^r.

To prove the continuity ofa7→vRa,let(an)n≥1 be a sequence in P^r converging to a0. Since vR_a has locally uniformly bounded mass, we can assume that vR_an converges to a currentR^′.We must have

suppR^′ ⊂lim sup

n→∞ (supp(R_an))⊂lim sup

n→∞ (L_an∩Y)⊂L_a∩Y.

On the other hand, by continuity of a7→R_a and sincev is continuous on U \X, we have thatR^′ =vRa0 outsideLa∩Y ∩X which has dimension smaller than or equal to l−1.

Hence, the support theorem of Bassanelli [Bas94] for currents T such that T and dd^cT have order 0implies that R^′ =vR_a0 on U.

Lemma 2.2. Let (ν_n)_n≥1 a sequence of probabilities in P^r which converges to ν. Let us define R := R

R_adν and R_n := R

R_adν_n. If vR_a is well defined on U and a 7→ vR_a is continuous then vR_n and vR are well-defined on U and satisfy vR_n = R

vR_adν_n, vR = RvR_adν and lim_n→∞vR_n=vR.

Proof. Letφbe a (l, l) smooth form with compact support inU.We can assume that the support of φ is contained in a small ball B ⊂ U on which v = u₁−u₂ where u₁, u₂ are plurisubharmonic on B and u₂ is continuous. Hence, there exists a decreasing sequence (u_1,j)_j≥1of continuous plurisubharmonic functions onBconverging pointwise tou₁.Define v_j := u_1,j−u₂. Since vR_a is well defined then hv_jR_a, φi decreases to hvR_a, φi by [FS95, Corollary 3.3]. In particular, if we define ψ_j(a) := hv_jR_a, φi and ψ(a) := hvR_a, φi then ψj decreases pointwise to ψ which is a continuous function since a7→ vRa is continuous.

Hence, by Dini’s theoremψ_j converges uniformly to ψ.

On the other hand, by monotone convergence theorem hv_jR, φi decreases to hvR, φi which is potentially equal to−∞.But, by definition of R we have

j→∞limhv_jR, φi= lim

j→∞

Z

P^r

hv_jR_a, φidν(a) = lim

j→∞

Z

P^r

ψ_jdν = Z

P^r

ψdν= Z

P^r

hvR_a, φidν(a), i.e. vR=R

vR_adν. The same holds forR_n, and then lim_n→∞vR_n=vRsince a7→ vR_a is continuous.

2.2 Pull-back of (1,1)-currents by rational maps

In this subsection,π:P^k99KP^r is a dominant rational map with dim(I(π))≤q−1.

Asπ is not supposed to be a submersion on P^k\I(π),the definition of the pull-back operatorπ^∗ on currents requires some work. However, we will only consider currents given by wedge products of positive closed(1,1)-currents with continuous local potentials, which greatly simplifies the problem. Ifτ is a positive closed (1,1)-current onP^r which is equal locally todd^cuthenπ_|^∗_Pk\I(π)τ can be defined locally onP^k\I(π)asdd^cu◦π. Méo [Méo96]

proved that τ 7→ π_|^∗_Pk\I(π)τ is continuous. Moreover, since I(π) has codimension at least 2,the trivial extension of π_|^∗_Pk\I(π)τ to P^k is again a positive closed (1,1)-current that we denote by π^∗τ.We summarize in the following proposition the properties about pull-back we shall need in the sequel. Recall that if R₁ and R₂ are two positive closed currents on P^kof bidegree(1,1)and (j, j) respectively then the wedge productR₁∧R₂ is well-defined if the local potentials of R₁ are integrable with respect toR₂∧ω^k−j_Pk (see e.g. [BT82]).

Proposition 2.3. Let π:P^k99KP^r be a dominant rational map whose indeterminacy set has a dimension smaller than or equal to q−1. If τ is a positive closed (1,1)-current of mass1 on P^r then kπ^∗τk is equal to the algebraic degree deg(π) of π. If τ has continuous

local potentials then the self-intersections (π^∗τ)^j are well-defined for j ∈ {1, . . . , r}. The currents (π^∗τ)^j coincide with the trivial extension of the standard pull-back of τ^j if τ is smooth. Moreover, if(u_n) is a sequence of continuous functions which converges uniformly to0 then the currents τn:=τ+dd^cun satisfylimn→∞(π^∗τn)^j = (π^∗τ)^j.

Proof. Letτ be a positive closed(1,1)-current of mass1onP^r.As we have said, in [Méo96]

Méo proved thatπ^∗_|Pk\I(π)τ depends continuously onτ.On the other hand, since I(π) has codimension at least 2, the trivial extension, denoted by π^∗τ, is a positive closed (1,1)- current on P^k. Moreover, as in the proof of Lemma 2.1, the Oka inequality obtained in [FS95] implies that the mass ofπ^∗τ is bounded independently ofτ.Hence,τ 7→π^∗τ is also continuous. Indeed, if(τ_n)_n≥1 is a sequence of positive closed(1,1)-currents converging to τ then(π^∗τ_n)_n≥1 has uniformly bounded mass onP^k and using the continuity ofπ^∗_|Pk\I(π)

each limit value R has to be equal to π_|^∗_Pk\I(π)τ on P^k\ I(π). Finally, R = π^∗τ since I(π) has codimension at least 2. To see that kπ^∗τk is in fact independent of τ, observe that if τ = ωP^r +dd^cu where u is a continuous function, then u◦ π is in L¹(P^k) and hdd^c(u◦π), ω_P^k−1k i= 0. Moreover, (π^∗τ)−(π^∗ωP^r) =dd^c(u◦π) sokπ^∗τk=kπ^∗ωP^rk.The general case follows by continuity since smooth forms are dense in the space of positive closed(1,1)-currents on P^r.Sincekπ^∗τk is independent ofτ,we obtain that it is equal to deg(π) by taking an hyperplane inP^r.

We now assume as in the statement thatdim(I(π))≤q−1.LetRbe a positive closed (j, j)-current onP^k withj ∈ {1, . . . , r}.If τ has continuous local potentials then π^∗τ has continuous local potentials except onI(π),i.e. the set of points where these local potentials are unbounded is contained in I(π). Hence, using the assumption on dim(I(π)) we can deduce from [FS95] that these local potentials are integrable with respect toR∧ω_P^k−jk and thus(π^∗τ)∧Ris well-defined. In particular,(π^∗τ)^j is well-defined forj∈ {1, . . . , r+1}.The fact that these currents coincide with the trivial extension of π_|^∗_Pk\I(π)(τ^j) if τ is smooth andj∈ {1, . . . , r}follows exactly as above. Observe however that forj=r+ 1, π^∗(τ^r+1) vanishes whereas(π^∗τ)^r+1has massdeg(π)^r+1and thus differs from0.This implies that the support of(π^∗τ)^r+1 is contained inI(π)and thusdim(I(π))≥q−1i.e. dim(I(π)) =q−1.

We prove the last assertion by induction. The case j= 1 follows from the first part of this proof. Assume the assertion is true forj−1 withj ∈ {2, . . . , r}.Observe that since (u_n) converges uniformly to 0, P^k\I(π) is covered by open sets Ω where we can write π^∗τ =dd^cvand π^∗τn=dd^cvn where (vn)is a sequence of continuous functions converging uniformly to v.Hence, if φis a smooth form with compact support on Ωthen

h(π^∗τ_n)^j−(π^∗τ)^j, φi=hdd^c(v_n(π^∗τ_n)^j−1−v(π^∗τ)^j−1), φi

=hv((π^∗τ_n)^j−1−(π^∗τ)^j−1), dd^cφi+h(v_n−v)(π^∗τ_n)^j−1, dd^cφi.

The inductive hypothesis implies that the first term in this sum converges to0.The second term also converges to 0 since (v_n−v) converges uniformly to 0. Hence, any limit value of (π^∗τ_n)^j has to be equal to (π^∗τ)^j on P^k\I(π) and this equality extends to P^k since dim(I(π))≤q−1.

Remark 2.4. The degreedeg(π)ofπis not necessary equal to1.Iff preserves the fibration defined byπ then the fibration defined by π◦f is still preserved by f and if f has degree dthen deg(π◦f) =ddeg(π). Another example in our setting is the binomial pencil of P² considered in [DJ08] and [FP11] where the degree is an arbitrary integer.

Remark 2.5. During the above proof, we have shown that dim(I(π)) ≥q−1. This fact also follows from the definition of I(π) as the common zeros of the r + 1 homogeneous polynomials definingπ.

3 Structure of the Green currents

This section is devoted to the proofs of Theorem 1.1, Corollary 1.2 and Corollary 1.3.

Letf be an endomorphism of P^k of degreed≥2.Recall that theGreen current T_f of f can be defined as

T_f = lim

n→∞

1

dⁿf^n∗ωP^k.

We refer to [DS10] for a detailed study of this current. In what follows, we will use that T_f has Hölder local potentials and if l ∈ {1, . . . , k} then its self-intersection T_f^l satisfies T_f^l = lim_n→∞d^−lnf^n∗ω^l_Pk.

Proof of Theorem 1.1. First observe that since I(π)∩Jq(f) =∅,a cohomological argu- ments implies that the dimension ofI(π) is at most q−1 and π satisfies the assumption of Proposition 2.3. Using this proposition, we define

R:= deg(π)⁻¹π^∗ωP^r and S:= deg(π)⁻¹π^∗T_θ

which are two positive closed(1,1)-currents of mass1.We also deduce fromI(π)∩Jq(f) =

∅that there exists a neighborhoodU ofI(π)such thatJq(f)∩U =∅.SinceRis a smooth form on P^k\I(π), there is a constant C > 0 such that R ≤ CωP^k on P^k \U and thus T_f^q ∧R^j ≤ C^jT_f^q ∧ω_P^jk on P^k for 1 ≤ j ≤ r. Applying the operator d^−n(q+j)f^n∗ to this inequality gives

T_f^q∧ 1

d^njf^n∗R^j

≤C^jT_f^q∧ 1

d^njf^n∗ω^j_Pk

.

The equidistibution results for f and the fact that T_f has continuous local potentials (see [DS10]) imply that the right-hand side converges to C^jT_f^q+j. On the other hand, as π◦fⁿ=θⁿ◦π we have by Proposition 2.3

T_f^q∧ 1

d^njf^n∗R^j

=T_f^q∧

1

deg(π)^jd^njf^n∗π^∗ω^j_Pr

=T_f^q∧π^∗

1

deg(π)^jd^njθ^n∗ω^j_Pr

.

Recall that d⁻ⁿθ^n∗ωP^r = T_θ+dd^cu_n where u_n are continuous functions converging uni- formly to0. Hence, by Proposition 2.3 the sequence above converges toT_f^q∧S^j and thus T_f^q∧S^j ≤C^jT_f^q+j. Moreover, T_f^q∧S^j is invariant by d^−(q+j)f^∗ and T_f^q+j is extremal in the cone of such currents (see [Sib99] when q+j= 1 and [DS09] for the general case) so T_f^q∧S^j =T_f^q+j.

The proof of S^j 6=T_f^j for j ∈ {1, . . . , r} simply comes from the fact (observed in the proof of Proposition 2.3) that S^r+1 is supported in I(π) which has dimension at most q−1.On the other hand, since T_f has Hölder local potentials, it follows from [Sib99] that T_f^j ∧S^r+1−j gives no mass to analytic sets of dimension q−2 +j≥q−1.

Finally, in order to prove the last assertion, observe that since I(π)∩Jq(f) =∅,the currentT_f^q+j =T_f^q∧S^j satisfies

π∗(T_f^q∧S^j) = deg(π)^−jπ∗(T_f^q∧π^∗T_θ^j) = deg(π)^−j(π∗T_f^q)∧T_θ^j. (3.1) The currentπ_∗T_f^qis positive and closed of bidegree(0,0)onP^rthus it is a positive multiple of [P^r]. Since π_∗ preserves the mass of measures, the equation (3.1) with j = r implies π_∗T_f^q= deg(π)^r[P^r]and thusπ_∗(T_f^q+j) = deg(π)^r−jT_θ^j.In particular, π_∗µ_f =µ_θ.

Proof of Corollary 1.2. We shall use the formula µ_f = T_f^q∧S^r. The proof would have been straightforward ifπ were a submersion onP^k\I(π)and the current T_f^q were smooth.

However, sinceT_f has continuous local potentials, we can use regularization as follows.

Let φ:P^k → R be a continuous function. The critical set Critπ of π is by definition the union of I(π) with the set of points in P^k\I(π) where the differential of π has rank strictly less thanr. This set is algebraic so the measureµ_f gives no mass to it. Hence, if χn:P^k → [0,1] are smooth functions with compact support in P^k\Critπ which converge locally uniformly to 1 on P^k\Crit_π then hµ_f, φi = lim_n→∞hµ_f, χ_nφi. Moreover, for µ_θ- almost alla∈P^r the algebraic set L_a :=π_|⁻¹_Pk\I(π)(a) has dimension q and L_a∩Crit_π has dimension q−1. In particular, the current[La]has mass deg(π)^r and coincides with the trivial extension ofπ^∗_|Pk\Critπδ_a.This implies that if Ψis a smooth (q, q)-form on P^k then the functionπ_∗(χ_nφΨ)on P^r is equalµ_θ-everywhere to a7→ hΨ∧[L_a], χ_nφi.

On the other hand, the current T_f has continuous local potentials so there exists a continuous function g such that T_f = ωP^k +dd^cg. Let (g_l)_l≥1 be a sequence of smooth functions converging uniformly togand defineT_l:=ωP^k+dd^cg_l.The uniform convergence implies that ifR is a positive closed(r, r)-current thenT_l^q∧R converges toT_f^q∧R.Hence, µ_f =T_f^q∧S^r implies

hµ_f, φi= lim

n→∞hµ_f, χ_nφi= lim

n→∞ lim

l→∞hT_l^q∧S^r, χ_nφi= lim

n→∞ lim

l→∞hµ_θ, π_∗(χ_nφT_l^q)i/deg(π)^r

= lim

n→∞ lim

l→∞

Z

P^r

hT_l^q∧ [L_a]

deg(π)^r, χ_nφidµ_θ(a) = lim

n→∞

Z

P^r

hT_f^q∧ [L_a]

deg(π)^r, χ_nφidµ_θ(a)

= Z

P^r

hT_f^q∧ [L_a]

deg(π)^r, φidµ_θ(a),

where the last equality comes from the fact that forµ_θ-almost alla, L_a∩Crit_π has dimension q−1and T_f^q∧[L_a]gives no mass to such sets.

By the Radon-Nikodym theorem, the following result implies Corollary 1.3.

Corollary 3.1. Under the assumptions of Theorem 1.1, if there exists a positive ω_P^rr- integrable function h such that µ_θ =hω^r_Pr then there exists A > 0 such that µ_f ≤A(h◦ π)σ_T^q

f.In particular, µ_f << σ_T^q

f.

Proof. LetR := deg(π)⁻¹π^∗ωP^r as in the proof of Theorem 1.1. The same theorem gives thatµ_f = deg(π)^−rT_f^q∧(π^∗µ_θ) = _deg(π)^h◦πrT_f^q∧R^r. On the other hand, we have seen in the proof of Theorem 1.1 thatT_f^q∧R^r ≤C^rσ_T^q

f thus µ_f ≤

C deg(π)

r

(h◦π)σ_T^q

f.

4 Lyapunov exponents

This section is dedicated to the proof of Theorem 1.4. The main ingredients are the formula π_∗µ_f =µ_θ,the fact that these measures put no mass on proper analytic sets and the local uniform convergence in Oseledec Theorem. We refer the reader to [BP13, Chapter 5–6] for the details on Oseledec theorem we shall need.

Proof of Theorem 1.4. First, let us set some notations. Let Crit_θ be the critical set of θ. We denote by bP^r := {(y_n)_n∈Z ∈ P^r|θ(y_n) = y_n+1} the natural extension and by θb the left-shift on Pb^r. The projection proj : Pb^r → P^r defined by proj((yn)n∈Z) = y0 sat- isfies θ◦proj = proj◦θ.b The measure µ_θ has a unique lift µb_θ which is invariant by θb and such that proj_θ∗µb_θ = µ_θ. In what follows, if by is in Pb^r we will write y₀ instead of

proj(y).b The measure µb_θ inherits several properties from µ_θ.Since µ_θ is ergodic and inte- grates the quasi-plurisubharmonic functions, it follows thatbµ_θ is ergodic, gives no mass to

∪_p∈Zθˆ^p(proj⁻¹(Crit_θ))and integrates the functionslogkDθ^±1k.In particular, forµb_θ-almost allyb= (yn)n∈Z the differentials

D_ybθ⁻ⁿ:= (D_y−nθⁿ)⁻¹, D_byθⁿ:=D_y0θⁿ

are well defined forn≥1. Moreover, by Oseledec theorem, there exist distincts numbers Λ₁, . . . ,Λ_s∈Rand aθ-invariant setb Y ⊂bP^r,included in(P^r\Crit_θ)^Z,such thatbµ_θ(Y) = 1 and for eachyb∈Y the tangent space T_y0P^r admits a splitting T_y0P^r =Ls

i=1V_i(by) which satisfies

D_y0θ(V_i(y)) =b V_i(bθ(y))b and lim

n→±∞

1

nlogkD_ybθⁿ(u)k= Λ_i

uniformly foru∈Vi(y)b withkuk= 1.By definition, the multiplicity ofΛi is the dimension m_i of V_i(by). Moreover, the subspaces V_i(y)b can be characterized by V_i(by)\ {0} = {u ∈ T_y0P^r\ {0} | lim_n→±∞n⁻¹logkD_ybθⁿ(u)k= Λ_i}.

The natural extensionbP^k,the mapfband the measureµb_f are defined in the same way with respect to f and since π ◦f = θ◦π, the map π lifts to a map bπ: bP^k → Pb^r such thatbπ◦fb=θb◦bπ. The uniqueness of the lift µb_θ of µ_θ and the fact that π_∗µ_f =µ_θ imply thatπb_∗µb_f =µb_θ. In particular, the setπb⁻¹(Y) has full bµ_f-measure. The measure µb_f also admits a Oseledec decomposition on afb-invariant setZ of full µb_f-measure. And since the measureµ_f gives no mass to proper analytic sets, if Crit_π denotes the critical set ofπ (i.e.

the union of I(π) with the set of points in P^k\I(π) where the differential of π has rank strictly less thanr) then the fb-invariant set

X:=Z∩bπ⁻¹(Y)∩(P^k\(Crit_f∪Critπ))^Z also has fullµb_f-measure.

Now, fix i∈ {1, . . . , s}and for bx∈X consider the subspace of T_x0P^k defined by W_i(x) = (Db _x0π)⁻¹(V_i(π(b bx))).

AsX is disjoint from the critical sets of f andπ, these subspaces have dimension m_i+q and define aDf-invariant distribution i.e. D_x0f(W_i(bx)) =W_i(fb(x)).b Therefore, Oseledec theorem applies to the measurable cocycle defined by the action of Df on W_i(bx) and induces µb_f-almost everywhere a decomposition W_i(bx) = Ll

j=1F_ij(bx) for some integer l≥1. Since this cocycle is a sub-cocycle of the standard one, each Fij(x)b is associated to a Lyapunov exponentλ_j of µ_f and satisfies

Fij(bx)\ {0}={v∈Wi(x)b \ {0} | lim

n→±∞n⁻¹logkD_xbfⁿ(v)k=λj}.

In particular,dim(Fij(x))b is bounded by the multiplicity ofλj.Now we show that ifFij(bx) is not contained in kerD_x0π then λ_j = Λ_i, that property will be sufficient to conclude.

So let bx ∈ X and j ∈ {1, . . . , l} such that F_ij(x)b 6⊂ kerD_x0π. In particular, there exists v∈Fij(bx)\kerDx0π and so

Λ_i = lim

n→+∞

1

nlogkD_xb(θⁿ◦π)(v)k= lim

n→+∞

1

nlogkD_bx(π◦fⁿ)(v)k

≤ lim

n→+∞

1

nlogkD_bxfⁿ(v)k=λ_j.

Here, the inequality comes from I(π)∩supp(µ_f) = ∅and thus, there exists C > 0 such thatkD_xπ(v)k ≤Ckvk for allx∈supp(µ_f) andv∈T_xP^k.

For the converse inequality, observe that since the subspacesW_i(bx)andF_ij(x)b depends measurably onx,b there exist a constantc >0andB ⊂Xwithµb_f(B)>0such that for each b

x∈Bthere is a subspaceE(x)b ⊂F_ij(bx)of positive dimension satisfyingE(ˆx)∩kerD_x0π = {0}andkD_x0π(v)k ≥ckvkfor allv∈E(x). By Poincaré recurrence theorem, forb µb_f-almost allbx∈X there exists an increasing sequence (k_n)_n≥1 of integers such thatfb^kn(bx)∈B for alln≥1. Therefore, ifv_n∈F_ij(bx)is such thatkv_nk= 1andD_bxf^kn(v_n)∈E(fb^kn(bx))then

λ_j = lim

n→+∞

1

k_nlogkD_bxf^kn(v_n)k ≤ lim

n→+∞

1

k_nlogkD_xb(π◦f^kn)(v_n)k

= lim

n→+∞

1

k_nlogkD_xb(θ^kn◦π)(v_n)k ≤Λ_i.

Here, the first equality comes from the fact that the convergence in Oseledec theorem is uniform on the unit sphere of F_ij(x).b The last inequality uses a similar argument as well as the uniform boundkD_xπ(v)k ≤Ckvkfor x∈supp(µ_f) and the fact thatv_n∈/kerD_x0π sinceD_bxf^kn(v_n)∈E(fb^kn(x)).b

We have shown that if F_ij(x)b 6⊂ kerD_x0π then λ_j = Λ_i. Thus there is a unique j ∈ {1, . . . , l} with this property. As dim(W_i(bx)) = m_i +q and dim(kerD_x0π) = q, the dimension ofFij(bx) has to be at least mi.

5 Generalized Bedford-Jonsson’s formula

In this section, we assume that the fibrationπis the standard linear fibrationπ:P^k99KP^r defined byπ[y :z] = [y]where y := (y₀, . . . , y_r) ∈C^r+1 and z = (z₀, . . . , z_q−1) ∈ C^q. We first analyse some basic properties of maps preserving such a fibration and then we give a proof of Theorem 1.6.

As we have said in the introduction, the indeterminacy set of π corresponds to {y = 0} ≃ P^q−1 and each fiber L_a := π⁻¹(a) is a projective space P^q in which I(π) can be identified with the hyperplane at infinity, i.e. L_a\I(π)≃C^q.Iff:P^k →P^kpreserves such a fibration then it lifts to a polynomial endomorphism F ofC^k+1 of the form

F(y, z) = (Θ(y), R(y, z)),

where y ∈ C^r+1, z ∈C^q and Θ, R are homogeneous polynomials. The map Θ is a lift of the endomorphismθ ofP^r such that θ◦π=π◦f.The inequality kΘ(y)k_∞ ≤ kF(y, z)k_∞ implies that the functions

G_Θ(y) := lim

n→∞

1

dⁿlogkΘⁿ(y)k_∞ and G_F(y, z) := lim

n→∞

1

dⁿlogkFⁿ(y, z)k_∞, satisfyG_Θ(y)≤G_F(y, z).The difference of these functions goes down toP^k and we define it as therelative Green function of f, G[y:z] :=G_F(y, z)−G_Θ(y).It is the unique lower semicontinuous function such that

dd^cG=T_f −S and minG= 0.

Here, T_f is the Green current off and S :=π^∗T_θ.It is easy to check that G is invariant (i.e. d⁻¹G◦f = G), continuous on P^k\I(π) and {G = +∞} =I(π). As we will see in the proof of the next lemma, I(π) is an attracting set for f and G encodes the speed of convergence toward it. In particular, the assumptions of Theorem 1.1 are satisfied.