Dynamical pairs with an absolutely continuous bifurcation measure

In this article, we study algebraic dynamical pairs $(f,a)$ parametrized by an irreducible quasi-projective curve $\Lambda$ having an absolutely continuous bifurcation measure. We prove that, if $f$ is non-isotrivial and $(f,a)$ is unstable, this is equivalent to the fact that $f$ is a family of Latt\`es maps. To do so, we prove the density of transversely prerepelling parameters in the bifucation locus of $(f,a)$ and a similarity property, at any transversely prerepelling parameter $\lambda_0$, between the measure $\mu_{f,a}$ and the maximal entropy measure of $f_{\lambda_0}$. We also establish an equivalent result for dynamical pairs of $\mathbb{P}^k$, under an additional assumption.


Introduction
Let Λ be an irreducible quasi-projective complex curve.An algebraic dynamical pair (f, a) parametrized by Λ is an algebraic family f : Λ × P 1 → P 1 of rational maps of degree d ≥ 2, i.e. f is a morphism and f λ is a degree d rational map for all λ ∈ Λ, together with a marked point a, i.e. a morphism a : Λ → P 1 .
Recall that a dynamical pair (f, a) is stable if the sequence {λ → f n λ (a(λ))} n≥1 is a normal family on Λ.Otherwise, we say that the pair (f, a) is unstable.Recall also that f is isotrivial if there exists a branched cover X → Λ and an algebraic family of Möbius transformations M : X × P 1 → P 1 so that M λ • f λ • M −1 λ : P 1 → P 1 is independent of the parameter λ and that the pair (f, a) is isotrivial if, in addition, M λ (a(λ)) is also independent of the parameter λ.A result of DeMarco [De] states that any stable algebraic pair is either isotrivial or preperiodic, i.e. there exists n > m ≥ 0 such that f n λ (a(λ)) = f m λ (a(λ)) for all λ ∈ Λ.
When a dynamical pair (f, a) is unstable, the stability locus Stab(f, a) is the set of points λ 0 ∈ Λ admitting a neighborhood U on which the pair (f, a) the sequence {λ → f n λ (a(λ))} n≥1 is a normal family.The bifurcation locus Bif(f, a) of the pair (f, a) is its complement Bif(f, a) := Λ \ Stab(f, a).If a is the marking of a critical point, i.e. f ′ λ (a(λ)) = 0 for all λ ∈ Λ, it is classical that the bifurcation locus Bif(f, a) has empty interior, [MSS].
The bifurcation locus of a pair (f, a) is the support of natural a positive (finite) measure: the bifurcation measure µ f,a of the pair (f, a), see Section 1 for a precise definition.The properties of this measure appear to be very important for studying arithmetic and dynamical properties of the pair (f, a), see e.g.[BD1,BD2,De,DM,DMWY,FG1,FG2,FG3].Note also that the entropy theory of dynamical pairs has been recently developed in [DGV].In the present article, we study algebraic dynamical pairs having an absolutely continuous bifurcation measure.
Assume that for some parameter λ 0 ∈ Λ, the marked point a eventually lands on a repelling periodic point x, that is f n λ 0 (a(λ 0 )) = x.Let x(λ) is the (local) natural continuation of x as a periodic point of f λ .We say that a is transversely prerepelling at λ 0 if the graphs of λ → f n λ (a(λ)) and λ → x(λ) are transverse at λ 0 .Finally, recall that a rational map f : P 1 → P 1 is a Lattès map if there exists an elliptic curve E, an endomorphism L : E → E and a finite branched cover p : E → P 1 such that p • L = f • p on E. Such a map has an absolutely continuous maximal entropy measure, see [Z].On the other hand, when f is a family of Lattès maps and the pair (f, a) is unstable, then Bif(f, a) = Λ, see e.g.[DM,§6].
Our main result is the following.
Theorem A. -Let (f, a) be a dynamical pair of P 1 of degree d ≥ 2 parametrized by an irreducible quasi-projective curve Λ. Assume that f is non-isotrivial and that (f, a) is unstable.The following assertions are equivalent: 1.The bifurcation locus of the dynamical pair (f, a) is Bif(f, a) = Λ, 2. Transversely prerepelling parameters are dense in Λ, 3. The bifurcation measure µ f,a of the pair (f, a) is absolutely continuous, 4. The family f is a family of Lattès maps.
Note that the hypothesis that f is not isotrivial is necessary to have the equivalence between 1. and 4. (see Proposition 4.3).
The first step of the proof consists in proving that transversely prerepelling parameters are dense in the support of µ f,a .Using properties of Polynomial-Like Maps in higher dimension and a transversality Theorem of Dujardin for laminar currents [Duj], under a mild assumption on Lyapunov exponents, we prove this property holds for the appropriate bifurcation current for any tuple (f, a 1 , . . ., a m ), where f : Λ×P k → P k is any holomorphic family of endomorphisms of P k and a 1 , . . ., a m : Λ → P k are any marked points.
In a second time, we adapt the similarity argument of Tan Lei to show that, if λ 0 is a transversely prerepelling parameter where the bifurcation measure is absolutely continuous, the maximal entropy measure µ f λ 0 of f λ 0 is also non-singular with respect to the Fubini-Study form on P 1 .As Zdunik [Z] has shown, this implies f λ 0 is a Lattès map.
We can see Theorem A as a partial parametric counterpart of Zdunik's result.However, the comparison with Zdunik's work ends there: Rational maps with P 1 as a Julia sets are, in general, not Lattès maps.Indeed, Lattès maps form a strict subvariety of the space of all degree d rational maps, and maps with J f = P 1 form a set of positive volume by [R].In a way, Theorem A is a stronger rigidity statement that the dynamical one.
Recall that an endomorphism of P k of degree d ≥ 2 has a unique maximal entropy measure µ f which Lyapunov exponents χ 1 , . . ., χ k all satisfy χ j ≥ 1 2 log d, by [BD4].We say that the Lyapunov exponents of f resonate if there exists 1 ≤ i < j ≤ k and an integer q ≥ 2 such that χ i = qχ j .
Recall also that, as in dimension 1, an endomorphism f of P k is a Lattès map if there exists an abelian variety A, a finite branched cover p : A → P k and an isogeny Berteloot and Dupont [BD3] generalized Zdunik's work to endomorphisms of P k : f is a Lattès map of P k if and only if the measure µ f is is not singular with respect to ω k P k , see also [Dup].As an important part of our arguments applies in any dimension, we have the following.
Theorem B. -Fix integers d ≥ 2 and k ≥ 1 and let (f, a) be any holomorphic dynamical pair of degree d of P k parametrized by a Kähler manifold (M, ω) of dimension k.Assume that for all λ ∈ B, any J-repelling periodic point of f λ is linearizable and that the Lyapunov exponents of f λ do not resonate for all λ ∈ M .Assume in addition that µ f,a : Then µ f,a is absolutely continuous with respect to ω k if and only if f is a family of Lattès maps of P k .
The paper is organised as follows.In section 1, we recall the construction of the biurcation currents of marked points and properties of Polynomial-Like Maps.Section 2 is dedicated to proving the density of transversely prerepelling parameters.In section 3, we establish the similarity proprety for the bifurcation and maximal entropy measures.Finally, in section 4 we prove Theorem A and B and list related questions.

The bifurcation current of a dynamical tuple
For this section, we follow the presentation of [DF, Duj].Even though everything is presented in the case k = 1 and for marked critical points, the exact same arguments give what we present below.
Let Λ be a complex manifold and let f : Λ × P k → P k be a holomorphic family of endomorphisms of P k of algebraic degree d ≥ 2: f is holomorphic and f λ := f (λ, •) : P k → P k is an endomorphism of algebraic degree d.
Definition 1.1.-Fix integers m ≥ 1, d ≥ 2 and let Λ be a complex manifold.A dynamical (m + 1)-tuple (f, a 1 , . . ., a m ) of P k of degree d parametrized by Λ is a holomorphic family f of endomorphisms of P k of degree d parametrized by Λ, endowed with m holomorphic maps (marked points) a 1 , . . ., a m : Λ → P k .
Let ω P k be the standard Fubini-Study form on P k and π Λ : Λ × P k → Λ and π P k : Λ × P k → P k be the canonical projections.Finally, let ω := (π For any n ≥ 1, we have , where ( u n ) n is a locally uniformly bounded sequence of continuous functions.
Pick now a dynamical (m + 1)-tuple (f, a 1 , . . ., a m ) of degree d of P k .Let Γ a j be the graph of the map a j and set a := (a 1 , . . ., a m ).
Definition 1.2.-For 1 ≤ i ≤ m, the bifurcation current T f,a i of the pair (f, a i ) is the closed positive (1, 1)-current on Λ defined by and we define the bifurcation current T f,a of the (m + 1)-tuple (f, a 1 , . . ., a m ) as For any ℓ ≥ 0, write , λ ∈ Λ.Let now K ⋐ Λ be a compact subset of Λ and let Ω be some compact neighborhood of K, then (a ℓ ) * (ω P k ) is bounded in mass in Ω by Cd ℓ , where C depends on Ω but not on ℓ.
Applying verbatim the proof of [DF, Proposition-Definition 3.1], we have the following.
Lemma 1.3.-For any 1 ≤ i ≤ k, the support of T f,a i is the set of parameters λ 0 ∈ Λ such that the sequence {λ → f n λ (a i (λ))} is not a normal family at λ 0 .Moreover, writing a i,ℓ (λ) := f ℓ λ (a i (λ)), there exists a locally uniformly bounded family (u i,ℓ ) of continuous functions on Λ such that As a consequence, for all j ≥ 1, we have ) on compact subsets of Λ.In particular, one sees that T k+1 f,a i = 0 on Λ. (1) Let us still denote π Λ : Λ × (P k ) m → Λ be the projection onto the first coordinate and for 1 ≤ i ≤ k, let π i : Λ × (P k ) m → P q be the projection onto the i-th factor of the product (P k ) m .Finally, we denote by Γ a the graph of a: Following verbatim the proof of [AGMV,Lemma 2.6], we get 1.2.Hyperbolic sets supporting a PLB ergodic measure Definition 1.4.-Let W ⊂ C k be a bounded open set.We say that a positive measure ν compactly supported on W is PLB if the psh functions on W are integrable with respect to ν.
We aim here at proving the following lemma in the spirit of [Duj,Lemma 4.1]: Proposition 1.5.-Pick an endomorphism f : P k → P k of degree d ≥ 2 which Lyapunov exponents don't resonate.There exists an integer m ≥ 1 a f m -invariant compact set K contained in a small ball B ⊂ P k and an integer N ≥ 2 such that f m | K is uniformly expanding and repelling periodic points of f m are dense in K, there exists a unique probability measure ν supported on K such that (f m | K ) * ν = N ν which is PLB.
To prove Proposition 1.5, we need the notion of polynomial-like map.We refer to [DS].Given an complex manifold M and an open set V ⊂ M , we say that V is S-convex if there exists a continuous strictly plurisubharmonic function on V .In fact, this implies that there exists a smooth strictly psh function ψ, whence there exists a Kähler form ω := dd c ψ on V .
The filled-Julia set of f is the set The set K f is full, compact, non-empty and it is the largest totally invariant compact subset of V , i.e. such that f The topological degree d t of f is the number of preimages of any z ∈ V by f , counted with multiplicity.Let k := dim V .We define where Λ := d −1 t f * .According to Theorem 3.2.1 and Theorem 3.9.5 of [DS], we have the following.
Theorem 1.7 (Dinh-Sibony).-Let f : U → V be a polynomial-like map of topological degree d t ≥ 2. There exists a unique probability measure µ supported by ∂K f which is ergodic and such that Fix m ≥ n 0 large enough so that Cd mk > d (k−1)m ≥ 2 and set The map g : U → V is polynomial-like of topological degree N , whence its equilibrium measure ν is the unique probability measure which satisfies g * ν = N ν by the first part of Theorem 1.7.We let K := supp(ν).Since the g i 's are uniformly contracting, the compact set K is f m -hyperbolic.
To conclude, it is sufficient to verify that N > d * k−1 .Fix n ≥ 1 and ϕ psh on V .Let ω be the (normalized) restriction to V of the Fubini-Study form of P k .Then where (u n ) n is a uniformly bounded sequence of continuous functions on P k .In particular, by the Chern-Levine-Niremberg inequality, if Taking the n-th root and passing to the limit, we get by assumption.The second part of Theorem 1.7 allows us to conclude.

The support of bifurcation currents
Pick a complex manifold Λ and let m, k ≥ 1 be so that dim Λ ≥ km.Let (f, a 1 , . . ., a m ) be a dynamical (m + 1)-tuple of P k of degree d parametrized by Λ.
Definition 2.1.-We say that a 1 , . . ., a m are transversely J-prerepelling (resp.properly J-prerepelling) at a parameter λ 0 if there exists integers n 1 , . . ., n m ≥ 1 such that f Recall that Lyapunov exponents of an endomorphism f resonate if there exists 1 ≤ i < j ≤ k and an integer q ≥ 2 such that χ i = qχ j , see 1.2.In this section, we prove Theorem 2.2.-Let (f, a 1 , . . ., a m ) be a dynamical (m + 1)-tuple of P k of degree d parametrized by Λ with km ≤ dim Λ. Assume that the Lyapunov exponent of f λ don't resonate for all λ ∈ Λ.
Then the support of coincides with the closure of the set of parameters at which a 1 , . . ., a m are transversely J-prerepelling.
Remark.-The hypothesis on the Lyapunov exponents is used only to prove the density of transversely prerepelling parameters and we think it is only a technical artefact.

Properly prerepelling marked points bifurcate
In a first time, we give a quick proof of the fact that properly J-prerepelling parameters belong to the support of , without any additional assumption.
Theorem 2.3.-Let (f, a 1 , . . ., a m ) be a dynamical (m + 1)-tuple of P k of degree d parametrized by Λ with km ≤ dim Λ. Pick any parameter λ 0 ∈ Λ such that a 1 , . . ., a m are properly J-prerepelling at λ 0 .Then The proof of this result is an adaptation of the strategy of Buff and Epstein [BE] and the strategy of Berteloot, Bianchi and Dupont [BBD], see also [G, AGMV].Since it follows closely that of [AGMV, Theorem B], we shorten some parts of the proof.
Before giving the proof of Theorem 2.3, remark that our properness assumption is equivalent to saying that the local hypersurfaces Proof of Theorem 2.3.-According to [G,Lemma 6.3], we can reduce to the case when Λ is an open set of C km .Take a small ball B centered at λ 0 in Λ.Up to reducing B, we can assume z j (λ) can be followed as a repelling periodic point of f λ for all λ ∈ B. Up to reducing B, our assumption is equivalent to the fact that j X j = {λ 0 }.
Our aim here is to exhibit a basis of neighborhood {Ω n } n of λ 0 in B with µ(Ω n ) > 0 for all n.For any m-tuple n := (n 1 , . . ., n m ) ∈ (N * ) m , we let . For a m-tuple n = (n 1 , . . ., n m ) of positie integers, we set We also denote As in [AGMV], we have the following.
Lemma 2.4.-For any m-tuple n = (n 1 , . . ., n m ) of positive integers, we let Γ n be the graph in Λ × (P k ) m of A n .Then, for any Borel set B ⊂ Λ, we have Suppose that the point z j is r j -periodic.For the sake of simplicity, we let in the sequel A n := A q+nr , where q = (q 1 , . . ., q m ), r = (r 1 , . . ., r m ) are given as above and q + nr = (q 1 + nr 1 , . . ., q m + nr m ).Again as above, we let Γ n be the graph of A n .
Let z := (z 1 , . . ., z m ) and fix any small open neighborhood Ω of λ 0 in Λ. Set and let δ be given by the expansion condition as above.Let S n be the connected component of Γ n ∩ Λ × B m δ (z) containing (λ 0 , z).Since z j (λ) is repelling and periodic for f λ for all λ ∈ B (if B has been choosen small enough), there exists a constant K > 1 such that for all (z, w) ∈ B(z j (λ 0 ), ǫ) and all λ ∈ B for some given ǫ > 0. In particular, the current [S n ] is vertical-like in in Λ × B m δ (z) and there exists n 0 ≥ 1 and a basis of neighborhood Let S be any weak limit of the sequence , where M −1 > 0 is the volume of B m δ (z) for the volume form j (ω k j ), where ω j = (p j ) * ω P k and p j : (P k ) m → P k is the projection on the j-th coordinate.
As a consequence, [S n ]/ [S n ] converges weakly to S as n → ∞ and, since the (mk, mk)current m j=1 (π j ) * ( T k ) is the wedge product of (1, 1)-currents with continuous potentials, we have By the above, this gives In particular, there exists n 2 ≥ n 1 such that for all n ≥ n 2 , Finally, since [S n ] is a vertical current, up to reducing δ > 0, Fubini Theorem gives Up to increasing n 0 , we may assume [S n ] ≥ (cδ 2k ) m /2 for all n ≥ n 0 .Letting α = M (cδ 2k ) m /4 > 0, we find To conclude the proof of Theorem 2.3, we rely on the following purely dynamical result, which is an immediate adaptation of [AGMV,Lemma 3.5].

Density of transversely prerepelling parameters
To finish the proof fo Theorem 2.2, it is sufficient to prove that any point of the support of an be approximated by transversely J-prerepelling parameters.We follow the strategy of the proof of Theorem 0.1 of [Duj] to establish this approximation property.Precisely, we prove here the following.
Theorem 2.6.-Let (f, a 1 , . . ., a m ) be a dynamical (m + 1)-tuple of P k of degree d parametrized by Λ with km ≤ dim Λ. Assume that the Lyapunov exponent of f λ don't resonate for all λ ∈ Λ.
Then, any parameter λ ∈ Λ lying in the support of the current T k f,a 1 ∧ • • • ∧ T k f,am can be approximated by parameters at which a 1 , . . ., a m are transversely J-prerepelling.
We rely on the following property of PLB measures (see [DS]): and pick λ 0 ∈ supp(Ω).According to Proposition 1.5, there exists an integer m ≥ 1 and a f m λ 0 -compact set K ⊂ P k contained in a ball and N ≥ 2 such that f m λ 0 | K is uniformly hyperbolic and repelling periodic points of f m λ 0 are dense in K, -there exisys a unique probailibty measure ν supported on K such that (f m λ 0 | K ) * ν = N ν which is PLB.Since K is hyperbolic, there exists ǫ > 0 and a unique holomorphic motion h : B(λ 0 , ǫ) × K → P k which conjugates the dynamics, i.e. h is continuous and such that for all λ ∈ B(λ 0 , ǫ), the map h λ := h(λ, •) : K → P k is injective and h λ 0 = id K , -for all z ∈ K, the map λ ∈ B(λ 0 , ǫ) → h λ (z) ∈ P k is holomorphic, and [dMvS,Theorem 2.3 p. 255].For all z := (z 1 , . . ., z m ) ∈ K m , we denote by Γ z the graph of the holomorphic map λ → (h λ (z 1 ), . . ., h λ (z m )).
Recall that we have set a n (λ) := (f n λ (a 1 (λ)), . . ., f n λ (a m (λ))).We define a * n ν by a * n ν := (π 1 ) * (ν ∧ [Γ an ]) , where π 1 : B(t 0 , ǫ) × (P k ) m → B(t 0 , ǫ) is the canonical projection onto the first coordinate.According to the claim, locally we have ν = dd c V , for some bounded (km − 1, km − 1)current V .In particular, we get a * n ν = a * n (dd c V ), as expected.Let ω be the Fubini-Study form of P k and Ω : On the other hand, we have 1 We now use [Duj,Theorem 3.1]: as (2km, 2km)-currents on B(λ 0 , ǫ) and only the geometrically transverse intersections are taken into account.In particular, this means there exists a sequence of parameters λ n → λ 0 and z n ∈ K m such that the graph of a n and Γ zn intersect transversely at λ n .Now, since repelling periodic points of f m λ 0 are dense in K, there exists z n,j → z n as j → ∞, where z j,n ∈ K m and (f m λ 0 , . . ., f m λ 0 )periodic repelling.Since z j,n (λ) := (h λ , . . ., h λ )(z j,n ) remains in (h λ , . . ., h λ )(K m ) and remains periodic, it remains repelling for all λ ∈ B(λ 0 , ǫ).By persistence of transverse intersections, for j large enough, there exists λ j,n where Γ an and Γ z j,n intersect transversely and λ j,n → λ n as j → ∞ and the proof is complete.
To finish this section, we prove the Claim.
Proof of the Claim.-Since the compact set K is contained in a ball, we can choose an affine chart C k such that K ⋐ C k and, up to reducing ǫ > 0, we can assume K λ = h λ (K) ⋐ C k for all λ ∈ B(λ 0 , ǫ).Let (x 1 1 , . . ., x 1 k , . . ., x m 1 , . . ., x m k ) = (x 1 , . . ., x m ) be the coordinates of (C k ) m and let h λ,i be the i-th coordinate of the function h λ .
For all 1 ≤ i ≤ k and 1 ≤ j ≤ m, we define a psh function Ψ j i on B(λ 0 , ǫ) × (C k ) m by letting According to Lemma 2.7 and Proposition 1.5, we have Since the functions Ψ j i are locally bounded, this ends the proof.

A renormalization procedure
Pick k, m ≥ 1 and let B(0, ǫ) be the open ball centered at 0 of radius ǫ in C km and let (f, a 1 , . . ., a m ) be a dynamical (m + 1)-tuple of degree d of P k parametrized by B(0, ǫ).

Now our tranversality assumption implies that
T z j (0) P k is invertible.As a consequence, the linear map Up to reducing again ǫ, we assume h is a biholomorphism onto its image and let r : The main goal of this paragraph is the following.
Proposition 3.2.-In the weak sense of measures on Ω, we have To simplify notations, we let Proof.-Note first that By definition, the sequence (r n ) n≥1 converges uniformly and exponentially fast to 0 on Ω, since we assumed z 1 (0), . . ., z m (0) are repelling periodic points and since r(0) = 0.Moreover, Λ rn → Λ 0 and φ rn(x) → φ 0 exponentially fast.In particular, and the convergence is uniform on Ω. Fix x ∈ Ω.Then and the conclusion follows.
Proof of Proposition 3.2.-Recall that we can assume there exists a holomorphic family of non-degenerate homogeneous polynomial maps For 1 ≤ j ≤ m, pick a open set U j ⊂ P k such that φ 0,j (B C k (0, δ j )) ⋐ U j and such that there exists a section σ j : In other words, for any x ∈ Ω, any 1 ≤ j ≤ m and any n ≥ n 0 , a n,j (x) := f In particular, there exists a holomorphic function u n,j : Ω → C * such that for all x ∈ Ω.Since log |u n,j | is pluriharmonic on Ω, the above gives Using again Lemma 3.3 gives This ends the proof since φ 0,j = π j • φ 0 by definition of φ 0 .

Families with an absolutely continuous bifurcation measure
Fix integers k, m ≥ 1 and d ≥ 2. The following is a consequence of the above renormalization process.
Proposition 3.4.-Let (f, a 1 , . . ., a m ) be a dynamical (m + 1)-tuple of degree d of P k parametrized by the unit Ball B of C km .Assume that a 1 , . . ., a m are transeversely Jprerepelling at 0 to a J-repelling cycle of f 0 which moves holmorphically in B as a Jrepelling cycle of f λ which is linearizable for all λ ∈ B. Assume in addition that the measure µ : is absolutely continuous with respect to the Lebesgue measure on B.
Then the measure µ f 0 is non-singular with respect to ω k P k . Proof.
-By assumption, we can write µ = h • Leb where h : r n and φ 0 be given as in Section 3.1.We can apply Proposition 3.2: has (finite) strictly positive mass in Ω.In particular, the measure converges to a non-zero finite mass positive measure on Ω.As r is a local holomorphic diffeomorphism, there exists a neighborhod of 0 in B such that we have r * Leb = v • Leb for some smooth function v > 0. Whence By the change of variable formula and Fubini, For all n, define a measurable function α n : B → R + by letting By assumption, the measure α n • Leb converges weakly on Ω to a non-zero finite positive measure, whence α n → α ∞ , as n → ∞, where α ∞ : Ω → R + is not identically zero.As a consequence, Using again Fubini, on Ω, we find Finally, since as positive measures on φ 0 (Ω), we have the measure µ f 0 is absolutely continuous with respect to Leb in an open set.
We now want to deduce Theorem B from the above, using [Z] when k = 1 and [BD3] when k > 1.In fact, they prove that f is a Lattès map if and only if the sum of its Lyapunov exponents L(f We use this characterization to prove Theorem B. Proof of Theorem B. -Assume firt that µ f,a is absolutely continuous with respect to ω k and let T be the set of parameters λ ∈ M such that a is transversely prerepelling at λ.The set T is dense in M by Theorem 2.2.Applying Proposition 3.4 at all λ ∈ T gives that µ f λ is non-singular with respect to ω k P k for all λ ∈ T .We then apply Zdunik or Berteloot-Dupont Theorem we have proven there exists a countable subset T which is dense in M such that the map f λ is a Lattès map for all λ ∈ T .In particular, L(f λ ) = k 2 log d for all λ ∈ T .As the function λ ∈ M → L(f λ ) is continuous, this implies L(f λ ) = k 2 log d for all λ ∈ M , i.e. f λ is a Lattès map for all λ ∈ M .
To conclude, we assume f is a family of Lattès maps and the pair (f, a) satisfies µ f,a > 0.
Let ω P k be the Fubini-Study form on P k .For all λ ∈ M , there exists a function u λ : for all (λ, z) ∈ M × P k .The above can be expressed as where ω = π * P k (ω P k ) and π P k : M × P k → P k is the canonical projection.Pick a local chart U ⊂ M and a local chart V ⊂ P k so that a(U ) ⊂ V and ω P k = dd c v on V where v is smooth.In U × V , the above gives Letting h(λ) := u(λ, a(λ)) and w(λ) := v • a(λ), we find Since w is smooth, the conclusion follows.
4. Proof of the main result and concluding remarks 4.1.J-stability and bifurcation of dynamical pairs on P 1 We say that a family f : Λ × P 1 −→ Λ × P 1 of degree d rational maps of P 1 is J-stable if all the repelling cycles can be followed holomorphically throughout the whole family Λ, i.e. if for all n ≥ 1, there exists N ≥ 0 and holomorphic maps z 1 , . . ., z N : Λ → P 1 such that {z 1 (λ), . . ., z N (λ)} is exactly the set of all repelling cycles of f λ of exact period n for all λ ∈ Λ.
Recall that an endomorphism of P 1 has a unique measure of maximal entropy µ f and let L(f ) := P 1 log |f ′ |µ f be the Lyapunov exponents of f with respect to µ f .By a classical result of Mañé, Sad and Sullivan [MSS], it is also locally equivalent to the existence of a unique holomorphic motion of the Julia set which is compatible with the dynamics, i.e. for λ 0 ∈ Λ, there exists h : Λ × J f λ 0 −→ Λ × P 1 such that for any λ ∈ Λ, the map h λ := h(λ, •) : Proof.-Since Bif(f, a) = supp(µ f,a ) = ∅, the set D of parameters λ 0 ∈ D such that a is transversely prerepelling at λ 0 is a non-empty countable dense subset of Bif(f, a).As J-repelling points of f λ 0 are contained in J f λ 0 , this gives Bif(f, a) ⊂ {λ ∈ D ; a(λ) ∈ J f λ }.
Using Montel theorem, one can deduce the following.Proposition 4.2.-Let (f, a) be a dynamical pair of degree d of P 1 parametrized by a one-dimensional complex manifold Λ. Assume that Bif(f, a) = Λ.Then f is J-stable and if f has a persistent attracting cycle, then it has period at most 2.
Proof.-Assume first f has a persistent attracting cycle of period at p ≥ 3. Pick a topological disk D ⊂ Λ.Then there exists holomorphic functions z 1 , . . ., z p : D → P 1 which paramerize this attracting cycle.In particular, z i (λ) = z j (λ) for all i = j and all λ ∈ D. Since we assumed Bif(f, a) = Λ, the sequence {λ −→ f n λ (a(λ))} n≥1 is not a normal family on D. By Montel Theorem, there exists n ≥ 1, 1 ≤ i ≤ p and λ 0 ∈ D such that By Lemma 4.1, since λ 0 ∈ Bif(f, a) this implies z i (λ 0 ) ∈ J f λ 0 .This is a contradiction with the fact that z i is attracting.If f is not J-stable, by Montel Theorem, there exists a non-empty open set U of Λ such that (f λ ) λ∈U is J-stable with an attracting periodic z 1 , . . ., z p of period p ≥ 3, and we reduce to the previous case (see e.g.[DF,Proposition 2.4]).

Proof of the main result
Proof of Theorem A. -Firs, remark that points 1. and 2. are equivalent by Theorem 2.2.Assume Bif(f, a) = Λ.By Proposition 4.2 the family f is J-stable.By [M,Theorem 2.4], since f is not isotrivial, f is a family of Lattès maps.Assume now f is a non-isotrivial family of Lattès and that µ f,a is non-zero.Recall that, since f is a family of Lattès maps, it is stable.We want to prove that supp(µ f,a ) = Λ.Assume it is not the case, then there exists a non-empty open set U ⊂ Λ such that U ⊂ Λ \ supp(µ f,a ).The pair (f, a) being stable in U , a(λ) cannot be a repelling periodic point of f λ for any λ ∈ U .From the uniqueness of the holomorphic motion, it follows that there exists z 0 ∈ P 1 such that a(λ) = h λ (z 0 ) for all λ ∈ U .By analytic contuinuation, this gives a(λ) = h λ (z 0 ) for all λ ∈ Λ.This contradicts the fact that µ f,a is non-zero.
As in the proof of Theorem 4.5, let T be the set of parameters λ ∈ Λ such that a is transversely prerepelling at λ.The set T is dense in supp(µ f,a ) by Theorem 2.2.Applying Proposition 3.4 at all λ ∈ T gives that µ f λ is non-singular with respect to ω P 1 for all λ ∈ T .By [Z], the map f λ is a Lattès map for all λ ∈ T .Consider now the morphism ϕ : λ ∈ Λ → f λ ∈ Rat d and let X ⊂ Rat d be the set of Lattès maps of degree d.X is a strict subvariety of Rat d .By the above, the algebraic curve ϕ(Λ) has an intersection with the subvariety X which admits accumulation points.By the Isolated Zeros Theorem, we have ϕ(Λ) ⊂ X, i.e. f is a family of Lattès maps.
The converse implication follows immediately from Theorem B.
Recall that when f is isotrivial, either J f λ = P 1 for all λ, or J f λ = P 1 for all λ.We conclude this section with the following easy proposition, which clarifies the case when f is isotrivial.
Proposition 4.3.-Let f be an isotrivial algebraic family parametrized by an irreducible quasiprojective curve Λ and let a : Λ → P 1 be such that the pair (f, a) is unstable.

Concluding remarks and questions
Dynamical pairs with a non-singular bifurcation measure.-First, when k > 1, the statement of Theorem B holds only if the Lyapunov exponents of f don't resonate and if all repelling J-cycles are linearizable.
This results raises several questions: Question 4.4.-Can we generalize Theorem B to the cases when 1.At some parameter, Lyapunov exponents do resonate?2. There exists J-repelling cycles that are non-linearizable?3. T k a is just non-singular with respect to a smooth volume form?
In fact, Zdunik [Z] completely classifies rational maps with a maximal entropy measure which is not singular with respect to a Hausdorff measure H α : either α = 1 and the rational map is conjugated to a monomial map z ±d or to a Chebichev polynomial T d , i.e. the only polynomial satisfying T d z + 1 z = z d + 1 z d for all z ∈ C, or α = 2 and the rational map is a Lattès map.
We expect the following complete parametric counterpart to [Z] to be true: As in the case of families of Lattès maps, we can expect the proof to generalize to the case when k > 1.This raises the following question.
Question 4.6.-Classify endomorphisms of P k which maximal entropy measure is not singular with respect to some Hausdorff measure H α on P k (and possible values of α).
As seen above, the case α = 2k has been treated by Berteloot and Dupont [BD3].Of course, there are also easy examples where α = k: take f : P 1 → P 1 which maximal entropy measure has dimension 1, then the endomorphism F : P k −→ P k making the following diagram commute (P 1 ) k (f,...,f ) where η k is te quotient map of the action by permutation of coordinates of the symmetric group S k , satisfies dim(µ F ) = k (see [GHK] for a study of symmetric products).
J-stability and dynamical pairs, when k ≥ 2. - We say that a family f : Λ × P k −→ P k of degree d ≥ 2 endomorphisms of P k is weakly J-stable if all the Jrepelling cycles can be followed holomorphically throughout the whole family Λ, i.e. if for all n ≥ 1, there exists N ≥ 0 and holomorphic maps z 1 , . . ., z N : Λ → P k such that {z 1 (λ), . . ., z N (λ)} is exactly the set of all repelling J-cycles of f λ of exact period n for all λ ∈ Λ.

Definition 1 .
6. -Given a connected S-convex open set and a relatively compact open set the measure µ is PLB and repelling periodic points are dense in supp(µ).Proof of Proposition 1.5.-First, we fix an open set O ⊂ P k with µ f (O) > 0. According to[BB, Proposition 3.2], there exists a ball B ⊂ O, a constant C > 0 depending only on O and n 0 ≥ 1 such that for all n

Lemma 2 .
7. -Let ν be PLB with compact support in a bounded open set W ⊂ C k and let ψ be a psh function on C k .The function G ψ defined by G ψ (z) := ψ(z − w)dν(w), z ∈ C k , is psh and locally bounded on C k .Proof of Theorem 2.6.-We follow the strategy of the proof of [Duj, Theorem 0.1].Write
Question 4.5.-Let (f, a) be any holomorphic dynamical pair of P 1 of degree d ≥ 2 parametrized by the unit disk D of C. Assume that (f, a) is unstable.Assume also there exists α > 0 and a function h : D → R + such that µ f,a = h • H α on D. Ca we prove that either α = 2 and f is a family of Lattès maps, -or α = 1, f is isotrivial and all f λ 's are conjugated to z ±d or a Chebichev polynomial?