STABILITY AND BIFURCATIONS IN PROJECTIVE HOLOMORPHIC DYNAMICS

STABILITY AND BIFURCATIONS IN PROJECTIVE HOLOMORPHIC DYNAMICS

FRANÇOIS BERTELOOT AND FABRIZIO BIANCHI

Abstract. We survey the classical theory of stability for rational maps and give a new proof based on potential and ergodic theoretic tools. We then exploit this approach to present a generalization of this theory which is valid in any dimension. This text is partially based on the lectures given by the first author at IMPAN during the Simons SemesterDynamical Systemsheld in Banach Center.

In these lectures we will discuss stability and bifurcation phenomena for holomorphic families of rational maps and, more generally, of endomorphisms ofP^k. More precisely, we will describe the classical results independently obtained by Lyubich and Mañé, Sad and Sullivan about the stability of rational maps and then present a new approach leading to their generalization in any dimension.

The four first lectures all deal with the case ofP¹. In the first lecture we briefly introduce the framework and present the tools which will be used. The second lecture is devoted to the classical proof of Lyubich-Mañé-Sad-Sullivan theorem and emphasizes the aspects which fail in higher dimension. In the third lecture we present an alternative proof of the Lyubich-Mañé-Sad-Sullivan theorem which is based on pluripotential and ergodic theories.

This approach is the one we will follow in the higher dimensional setting and we discuss which are the difficulties related to that generalization. We also introduce in that lecture the notion of bifurcation current by means of Lyapunov exponents. The fourth lecture is a digression on some possibilities offered by the bifurcation currents techniques.

The remaining lectures deal with a generalization of Lyubich-Mañé-Sad-Sullivan theo- rem to P^k which has been obtained by C. Dupont and the authors of these notes. This generalization is based on new methods and ideas which, at least partially, have already been explained in the special case of P¹ in the third lecture. In the fifth lecture, we first introduce some tools and, in particular, some concepts to deal with holomorphic motions of Julia sets, we then describe the main result and present a general strategy for its proof.

In the two next lectures, admitting a fundamental lemma concerning Misiurewicz param- eters, we prove how to obtain stability (holomorphic motions) of Julia sets from stability of repelling cycles. The eighth lecture is devoted to the proof of the fundamental lemma.

In the last two lectures we cover some related aspects of our result and present a list of open questions.

As we have tried to stress the ideas and concepts rather than the technical aspects, our presentation is mostly informal and our style is that of Notes. In the subsection 3.2

1

however, we gather some material for the first time and thus give precise and complete proofs.

For background on dynamics in several complex variables, we refer to the lecture notes [DS10] by T.C. Dinh and N. Sibony. For background on bifurcation currents, we refer to the survey [Duj14] by R. Dujardin or the lecture notes [Ber13] by the first author. In particular, an extensive bibliography may be found in these three texts.

1. Holomorphic families 1.1. Definitions and examples.

Definition 1.1. A holomorphic family of endomorphisms of P^k is a holomorphic map f :M×P^k →M×P^k of the form f(λ, z) = (λ, fλ(z))such that:

(1) M is a complex manifold;

(2) the algebraic degreedeg_af_λ of f_λ is constant: deg_af_λ ≡d≥2.

We shall say that dis the degree of the holomorphic familyf.

Let us recall a few basic facts about holomorphic endomorphisms of projective spaces.

(1) Eachf_λcan be lifted toC^k+1through the canonical projectionπ:C^k+1\{0} →P^k: C^k+1\ {0} C^k+1\ {0}

P^k P^k

π π

Fλ

fλ

Here F_λ is a non-degenerate d-homogeneous polynomial map, i.e., F_λ⁻¹{0} ={0}

andF_λ(tz) =t^dF_λ(z),∀t∈C,∀z∈C^k+1 whered= deg_af_λ.

(2) The map fλ : P^k → P^k is a finite ramified covering whose topological degree deg_topfλ is equal tod^k.

(3) For every λ0 ∈M, the family f can be lifted to a family F: Ω×C^k+1 → Ω×C^k+1

(λ, z) 7→ (λ, F_λ(z)) for every sufficiently small ballΩ⊂M centered atλ₀. The following are basic examplesof holomorphic families.

(1) The space H_d(P^k) of all holomorphic endomorphisms of algebraic degree dof P^k. Actually H_d(P^k) ≈ P^Nd,k \Z where Z is an hypersurface in P^Nd,k and Nd,k =

(k+1)(d+k)!

d!k! .

(2) M is any complex submanifold of H_d(P^k). This is particularly interesting whenM is dynamically defined.

(3) The polynomial quadratic family; k= 1, M =Cand f_λ(z) =z²+λ.

(4) The degree d polynomial family; k = 1, M = C^d−1 and f_λ is defined by the condition: f_λ⁰ is unitary, the critical set C_fλ ={0, λ₁, . . . , λd−2}and f_λ(0) =λd−1. 1.2. The equilibrium measure of f_λ. A reference for this subsection and the next one is given by the lecture notes [DS10]. Letω denote the Fubini-Study form ofP^k. For eachλ, one has d⁻¹f_λ^∗ω=ω+dd^cv_λ, where v_λ is a smooth function on P^k. One may for instance take vλ(z) =d⁻¹log^kF_k˜^λ_zk^(˜z)k, wherek·kis the euclidean norm on C^k+1 and π(˜z) =z.

By induction one gets d⁻ⁿ(f_λⁿ)^∗ω=ω+dd^cg_n(λ, z), where the function g_n is given by g_n(λ, z) :=v_λ+· · ·+d⁻⁽ⁿ⁻¹⁾v_λ◦f_λⁿ⁻¹.

It is clear that gn is locally uniformly converging to some function g(λ, z). One thus gets lim_nd⁻ⁿ(f_λⁿ)^∗ω =ω+dd^c_zg(λ, z).One then sets

Tλ:=ω+dd^c_zg(λ, z).

This is a positive closed (1,1)-current on P^k, it is called the Green current of fλ. The function g(λ, z) is called theGreen function off_λ. By construction we have

f_λ^∗Tλ =dTλ.

Let us stress thatgn→glocally uniformly and that thegn’s are smooth. It turns out that g is Hölder continuous, a fact which was first proved by M. Kosek [Kos97].

The equilibrium measure µ_λ off_λ is defined by

µ_λ := (T_λ)^k=T_λ∧ · · · ∧T_λ. By construction µ_λ is a probability measure onP^k such that

f_λ^∗µ_λ =d^kµ_λ and (f_λ)_∗µ_λ =µ_λ.

Two crucial facts about the measure µ_λ are that it is ergodic and does not give mass to proper analytic subsets of P^k.

1.3. Two fundamental equidistribution theorems.

Theorem 1.2. Repelling periodic points equidistributeµλ: d^−knP

z∈Rn(λ)δz →µλ, where R_n(λ) :={n-periodic repelling points in J_λ}.

Theorem 1.3. Iterated preimages equidistributeµ_λ: there exists a proper algebraic subset E ofP^kwhich is contained in the postcritical set offλ and such thatd^−knP

f_λⁿ(z)=aδz →µλ

for all a∈P^k\ E.

Note that the exceptional set E is well understood when k= 1. For k≥2, one knows that E is the largest proper algebraic subset of P^k which is totally invariant by f_λ. We refer to the papers of Briend-Duval [BD99, BD09] and Dinh-Sibony [DS03] for the proofs and to the papers of Brolin [Bro65], Freire-Lopes-Mañé [FLM83] and Lyubich [Ly83b] for the one-dimensional case.

1.4. Goals. In these lectures, we aim to

• study the stability of the ergodic dynamical systems (J_λ, f_λ, µ_λ);

• understand howJλ depends on λ.

We will first review the classical results in dimension k = 1 and then discuss their generalization to higher dimension.

2. When k= 1: Lyubich-Mañé-Sad-Sullivan Theorem

2.1. The statement. Here we make the mild assumption that f :M ×P¹ → M×P¹ is a holomorphic family (of degree d) such that

C_f =Critf ={(λ, c_j(λ)) : λ∈M,1≤j ≤2d−2}, where the maps cj :M →P¹ are holomorphic.

The results obtained by Lyubich [Ly83a] and Mañé, Sad and Sullivan [MSS83] in the early 80’s are essentially summarized by the next theorem and its corollary.

Theorem 2.1. (Lyubich, Mañé-Sad-Sullivan) For every λ₀ ∈M the following asser- tions are equivalent:

(1) J_λ moves holomorphically on some neighbourhood of λ0; (1’) J_λ moves continuously on some neighbourhood ofλ₀;

(2) the repelling cycles of f_λ move holomorphically on some neighbourhood of λ0; (2’) f_λ has no unpersistent neutral cycle near λ₀;

(3) (f_λⁿ(cj(λ))) is normal at λ0 for all 1≤j≤2d−2 (stability of the critical orbits).

In view of that, one defines

Stability locus off := Stab(f) :={λ₀: (1). . .(3) occur}

Bifurcation locus off := Bif(f) :=M\Stab(f).

The above theorem has the following fundamental consequence.

Corollary 2.2. Stab(f) is dense in M.

Example: the Mandelbrot set. Consider the quadratic polynomial family which is given by p : C×P¹ → C×P¹, where p(λ, z) = (λ, z² +λ). The Mandelbrot set M2 is defined as:

M₂:={λ∈C: (pⁿ_λ(0))_n is bounded in C}.

It follows from the above theorem that

λ0∈Bif(p)⇔(pⁿ_λ(0))_n is not normal atλ0 ⇔λ0 ∈bM2. We now explain the concepts involved in the above Theorem.

2.2. Holomorphic motions.

Definition 2.3. A holomorphic motion ofK⊂P¹ over a complex manifoldM and centered at λ₀ ∈M is a map h:M ×K →P¹ such that

i) h(λ₀,·) =Id_K;

ii) h(λ,·) is one-to-one, for every λ∈M;

iii) h(·, z) is holomorphic on M, for every z∈K.

Remark 2.4. The λ-lemma says that such a motion can always be extended to M×K and is continuous on M×K, moreover this extension is unique. We shall later implicitely give the proof of this simple and basic fact.

It is useful to think about holomorphic motions in terms of laminations.

λ0

P¹

M

h(·, z) z₁ K

z₂

h(·, z₁)

h(·, z₂)

Definition 2.5. Let R_n(λ) := {n-periodic repelling points of f_λ}. We say that the re- pelling cycles of f_λ move holomorphically overΩ⊂M if for every n≥1 there exists a set of holomorphic mapsρ_j,n: Ω→P¹ such thatR_n(λ) ={ρ_j,n(λ) : 1≤j≤N_d(n)}, for every λ∈Ω. Here Nd(n) :=Card(R_n(λ)).

Remark 2.6. This is equivalent to say that there exists a holomorphic motion of R_n(λ0) over Ω and centered at λ₀ for every n≥1. The important fact here is that Ωis the same for all n. Indeed, if n is fixed and Ω is a sufficiently small neighbourhood of λ0, such a motion of R_n(λ₀) always exists by the implicit function theorem.

Definition 2.7. One says that J_λ moves holomorphically over M if there exists a holo- morphic motionhof Jλ which is centered atλ0 ∈M and such thathλ:=h(λ,·) conjugates the dynamics:

Jλ0 Jλ

J_λ0 J_λ

fλ₀ f_λ

hλ

h_λ

2.3. Continuity in the Hausdorff topology.

Comp^∗(P¹) :={non-empty compact subsets of P¹}.

ForK ∈Comp^∗(P¹)we setK_ε:=ε-neighbourhood ofK. For a mapE :M →Comp^∗(P¹) we have the following definitions (by ≈we mean: sufficently close):

E is upper semicontinuous (usc) atλ₀ ⇔ ∀ε >0 :E(λ)⊂(E(λ₀))_ε for λ≈λ₀ E is lower semicontinuous (lsc) atλ0⇔ ∀ε >0 :E(λ0)⊂(E(λ))_ε for λ≈λ0

E is continuous at λ₀ ⇔E is ucsand lscat λ₀.

2.4. Proof of Lyubich-Mañé-Sad-Sullivan Theorem. The structure of the proof is as follows:

(1) (3) (2) (1)

(1⁰) (2⁰)

We focus on the proof of (1) ⇒ (3) ⇒ (2) ⇒ (1) and the proof of the Corollary 2.2, stressing the arguments which cannot be adapted to P^k, withk≥2.

(1) ⇒(3) Assume that (f_λⁿ(c(λ)))_n is not normal at λ₀ but J_λ moves holomorphically near λ0. Then one has the following picture:

λ0

P¹

M f_λ^N(c(λ))

J_λ⊂P¹ P¹⊃J_λ0

hλ

this is a critical value off_λ^N

critical value

Picard-Montel⇒f_λ^N(c(λ)) cuts some leaf

Owing to the h_λ, this is a

1 2

3 conjugacy by

off_λ^N₀

This argument fails for k≥2 (J_λ too small) λ

The picture shows that the set of critical values of f_λ^N

0 contains an open piece of J_λ0: this is impossible.

(2)⇒(1)By assumption we have the following picture for every n.

λ P¹

M o

graphs of ρj,n

R_n(λ)

λ₀

One easily sees (using lifts to C² \ {0}) that the family {ρ_j,n: 1 ≤ j ≤N_d(n), n ≥ 1}

is normal. Since ∪_nR_n(λ) is dense in Jλ, the idea is to take limits to get the expected lamination. The only problem is that of uniqueness of limits. The picture below explains why two distinct limits cannot meet.

If the graphs of two limits γ₁, γ₂ meet, then for ρ_j,n close to γ₁ and ρjn close to γ₁...

ρ_j,n

M γ₁

P¹

γ₂ ρj,m

Impossible!

... the graphs of ρ_j,n and ρ_j,m must meet by Hurwitz lemma!

Fails when k≥2

(3) ⇒ (2) The idea is as follows. Assume that a repelling cycle of fλ0 does not move holomorphically over M. Then one finds a holomorphic map ρ : U₀ → P¹ such that f_λⁿ⁰(ρ(λ)) = ρ(λ), λ₀ ∈U₀ and ρ(λ₀) is repelling for f_λⁿ⁰

0 but ρ(λ₁) is attracting for f_λⁿ⁰

1. The orbit of a critical point c_i(λ) is contained in the basin of attraction ofρ(λ₁). In this situation, (f_λⁿ(ci(λ)))_n cannot be normal onU0 as the picture below shows.

λ₀ P¹

M c_i ρ

λ

f_λⁿ(c_i(λ)

Also fails for k≥2

2.5. Proof of Corollary 2.2. Let λ0 ∈ Bif(f). After a small perturbation (λ0 → λ1) one gets f_λ1, which has an attracting cycle of period ≥3. LetB(λ) be the basin of that cycle (exists for λ ≈ λ1). If for some 1 ≤ i ≤ 2d−2 the family (f_λⁿ(ci(λ)))_n were not normal atλ₁, then, using Picard-Montel Theorem, one sees that after a small perturbation (λ1 →λ2) one has

f_λⁿ⁰

2(ci(λ2))

∈B(λ2)and thus (f_λⁿ(ci(λ)))_n is normal atλ2.

One repeats this argument a finite (≤ 2d−2) number of times to get λ_k ∈ Stab(f) which is arbitrarily close toλ₀.

The fact that this argument cannot be extended to P^k, k≥2, is essentially due to the non-finiteness of C_fλ.

3. When k= 1: a potential theoretic approach to bifurcations

3.1. The main idea explained for polynomials families. We consider here a holo- morphic polynomial family

p:M×C → M ×C (λ, z) 7→ (λ, p_λ(z))

where p_λ is a unitary polynomial of degree d≥2. As in the previous lecture, we assume that the critical setC_p={(λ, c_j(λ)) : 1≤j≤d−1}, wherec_j :M →Care holomorphic.

The Green function Gof p onM ×Cis defined by G(λ, z) := lim

n d⁻ⁿlog⁺|pⁿ_λ(z)|.

As the convergence is locally uniform, the function G is psh and continuous. Moreover, one easily sees that

• G(λ, z) = 0⇔(pⁿ_λ(z))is bounded

• J_λ =b{G(λ, z) = 0}

• µλ=dd^c_zG(λ, z).

The Lyapunov exponent L(λ) of p_λ with respect to its equilibrium measure µ_λ is defined by

L(λ) :=

Z log

p⁰_λ(z) µλ(z).

Note that log|p⁰_λ| ∈L¹(µ_λ), since µ_λ has a continuous potential.

Proposition 3.1. L(λ) is the exponential rate of growth of (pⁿ_λ)⁰(z) for µλ-almost every point z.

Proof. Apply Birkhoff Theorem (µ_λ is ergodic). For µ_λ-almost every z, Z

log p⁰_λ(z)

µ_λ(z) = lim

n

1 n

n−1

X

k=0

log

p⁰_λ(p^k_λ(z)) = lim

n

1 nlog

(pⁿ_λ)⁰(z) .

The following formula [Prz85] relates the Lyapunov exponent with the Green function.

Theorem 3.2. Przytycki Formula. L(λ) = logd+Pd−1

j=1G(λ, cj(λ)).

Proof. Integrate by parts.

L(λ) = Z

log p⁰_λ

µλ = Z

log

dY

j

(z−cj(λ))

µλ

= logd+X

j

Z

log|z−cj(λ)|dd^c_zG(λ, z) = logd+X

j

D

δ_cj(λ), G(λ,·)E .

Corollary 3.3. The function L(λ) ispsh, continuous and bigger than logd.

Applying dd^c_λ to Przytycki Formula one gets the following fundamental formula:

(1)

dd^c_λL(λ) =

d−1

X

j=1

dd^c_λG(λ, c_j(λ))

deals with holomorphic motions of repelling cycles

detects the instability of the orbit of cj(λ)

Formula (1) offers a possibility to avoid Picard-Montel arguments in the proof of Lyubich- Mañé-Sad-Sullivan Theorem and, more generally, provides another way to look at stability questions in dimension one. Although this will be discussed in details in the next subsec- tion, the proposition below roughly indicates how this is possible for polynomial families.

Proposition 3.4. The following hold:

(1) G(λ, c_i(λ))is pluriharmonic on a ball B ⊂M ⇔(pⁿ_λ(c_i(λ)))_n is normal on B;

(2) if the repelling cycles of p move holomorphically over U ⊂M, then L is plurihar- monic on U.

Proof. (1) Set un(λ) := pⁿ_λ(ci(λ)), and recall that d⁻ⁿlog⁺|u_n(λ)| → G(λ, ci(λ)) lo- cally uniformly.

• (un)_nnormal onB ===^shrink⇒

B







if u_n−−→ ∞ ⇒^unif G(λ, c_i(λ))is a limit of pluriharmonic functions on B if unk

−−→unif h holomorphic⇒G(λ, ci(λ) = 0 on B

• Assume G(λ, c_i(λ))is pluriharmonic on B.

First case: G(λ, ci(λ0)) = 0, λ0 ∈ B. Then G(λ, ci(λ)) = 0 on B (maximum principle) and thus(u_n(λ))_n is uniformly bounded on B.

Second case: G(λ, ci(λ)) > 0 on B. Then pⁿ_λ(ci(λ)) → ∞ locally uni- formly onB.

(2) This is a direct consequence of the approximation formula which will be discussed in the next lecture.

3.2. A new approach to the Lyubich-Mañé-Sad-Sullivan theory. We aim here to give a proof of Theorem 2.1 which exploits the ideas described in the former subsection. As it turns out, this approach can be partially adapted to the higher dimensional setting and we shall also explain which are the main technical difficulties related to this achievement.

We first recall our notations and introduce some new concepts. Let us consider a holo- morphic family

f:M×P¹ →M×P¹ (λ, z)7→(λ, fλ(z))

of degree d ≥ 2 and whose critical set C_f is given by the graphs of 2d−2 holomorphic maps cj :M →P¹ as follows:

C_f =Critf ={(λ, c_j(λ)) : λ∈M,1≤j ≤2d−2}.

Definition 3.5. A parameter λ₀ is said Misiurewicz if there exist integers p₀, n₀ ≥1 and a holomorphic map σ defined on some neighbourhood of λ0 such that σ(λ)∈ R_n0(λ) and σ(λ₀) =f^p0◦c_j(λ₀) butσ 6=f^p0 ◦c_j for some 1≤j≤2d−2.

We recall that R_n(λ) = {n-periodic repelling points of f_λ} and that the cardinal of R_n(λ)is equivalent to dⁿ for every λ.

We endow O(M,P¹) :=

γ :M →P¹:γ holomorphic with the topology of local uni- form convergence; this is a metric space. The space of interest here is the (possibly empty) subspace

J :=

γ ∈ O(M,P¹) : γ(λ)∈J_λ,∀λ∈M . We have two natural maps. The first one is a self-map on J

F :J → J γ 7→ F(γ)

defined by F(γ)(λ) =f_λ(γ(λ))and the second one is a projection p_λ:J →P¹

γ 7→γ(λ).

Definition 3.6. An equilibrium web forf is a compactly supported probability measureM on J such that F_∗M=Mand (p_λ)_∗M=µ_λ for all λ∈M.

The existence of an equilibrium web is a weak form of holomorphic motion of Julia sets, it rather means that the measures µ_λ move holomorphically over M. As we shall see, the two equidistribution theorems described in Subsection 1.3 combined with the Banach- Alaoglu compactness theorem will allow us to construct such equilibrium webs.

For each λ∈M, the Lyapunov exponent of the system (Jλ, fλ, µλ) is L(λ) :=

Z log

f_λ⁰ µ_λ.

One may easily check that it does not depend on the choice of the metric |·|on P¹. We have to generalize Przytycki formula to rational maps. This has been done by DeMarco [dM03] and a different proof, more in the spirit of integration by part, has been given by Bassanelli and the first author [BB07] (see also [Ber13]). Applyingdd^c_λ, this formula yields:

Theorem 3.7. dd^cL-Formula. dd^c_λL(λ) = (πM)_∗

ω+dd^c_λ,zg(λ, z)

∧[C_f]

.

Recall that ω is the Fubini-Study form on P¹, g(λ, z) the Green function off and [C_f] the current of integration on the critical setC_f of f inM×P¹.

We shall also use an approximation formula forLwhich basically says that the Lyapunov exponent of f_λ (with respect to µ_λ) is the asymptotic limit of the averages of Lyapunov exponents of repellingn-cycles. This approximation formula was first proved for endomor- phisms of P^k by C. Dupont, L. Molino and the first author in [BDM08], a simpler proof adapted to the case of P¹ is written in [Ber13], Y. Okuyama [Oku12] has given another proof.

Theorem 3.8. Approximation Formula. L(λ) = lim_nd⁻ⁿP

z∈Rn(λ) 1

nlog|(f_λⁿ)⁰(z)|.

We are now ready to state our version of Theorem 2.1. The parameter space M will be assumed to be simply connected. Note that the equivalence (1)⇔ (3)is a new result, it says that either all repelling cycles move holomorphically or they asymptotically all bifurcate (see [Ber17]).

Theorem 3.9. For λ₀ ∈ M and n ∈ N we denote by N_d(n) the number of n-repelling periodic points of fλ0 and byτ0(n)Nd(n) the number of those which move holomorphically over M. Then the following assertions are equivalent.

(1) lim sup_nτ₀(n)>0

(2) f admits an equilibrium web M

(3) all repelling cycles of f_λ move holomorphically over M (i.e. τ₀(n) = 1,∀n) (4) Jλ move holomorphically over M

(5) dd^cL(λ) = 0 on M

(6) (f_λⁿ(cj(λ)))_n is a normal family for every1≤j≤2d−2 (7) there are no Misiurewicz parameters in M.

The next diagram describes the proof of the above Theorem. Continuous arrows indicate that the corresponding parts of the proof easily adapt in higher dimension, while for the implications marked with a dashed arrow one would need major modifications of the argument. Parenthesis indicate that the corresponding property will need to be modified when dealing with dimension k≥2.

3 5

1 (4) (6)

2 7

approximation formula

λ−lemma ^ddcLformula

equidistribution of repelling cycles

equidistribution of iterated preimages

Proof. (1)⇒ (2). By assumption, for every n ∈N^∗ there exists a collection of holomor- phic maps ρj,n : M → P¹ where 1 ≤ j ≤ τ0(n)Nd(n) =: N_d⁰(n) and such that one has {ρ_j,n(λ) : 1≤j ≤N_d⁰(n)} ⊂ R_n(λ) for allλ∈M.

We may thus define a sequence(M_n)_n of discrete measures on J by setting M_n:=d⁻ⁿ

N_d⁰(n)

X

j=1

δρj,n.

Since lim sup_n|M_n| = lim sup_nτ0(n)Nd(n) =: τ > 0, Banach-Alaoglu Theorem gives a subsequence M_nq

q such that M_nq → M and |M| = τ. Since by construction F_∗ M_nq

=M_nq we getF_∗M=M.

Let us now set σλ := pλ∗M for every λ∈ M. One has |σ_λ|=|M| =τ and it follows from p_λ◦ F =f_λ◦p_λ that fλ∗σ_λ =σ_λ. Moreover

σ_λ =pλ∗M= lim

q pλ∗M_nq = lim

q d^−nq

N_d⁰(nq)

X

j=1

ρ_j,nq(λ)≤lim

q d^−nq X

z∈R_nq(λ)

δ_z=µ_λ

where the last equality follows from the Equidistribution Theorem 1.2. Writing now µλ = τ ^σ_τ^λ

+ (1−τ)

µλ−σλ

1−τ

one deduces from the ergodicity of µλ that σλ = τ µλ

for every λ∈Ω0. Then ¹_τMis an equilibrium web forf.

(2)⇒(3). Let us first observe that for every(λ0, z0)∈M×Jλ0 there existsγ ∈suppM such thatz₀=γ(λ₀). Indeed, sinceJ_λ0 = suppµ_λ0 = supp (p_λ0∗M)there exist a sequence (γk)k insuppMsuch thatγk(λ0)→z0. Then, asMis compactly supported, we can take for γ any limit of(γ_k)_k.

Now assume thatz₀is a repellingn-periodic point forf_λ0. Then, by the implicit function theorem, there exists a neighbourhood V_λ0 of λ0 and a holomorphic map w : V_λ0 → P¹ such that w(λ₀) =z₀ and w(λ) is n-periodic for f_λ. Let us show that w coincides on V_λ0 with the above map γ. All we shall actually need is that γ(λ0) = w(λ0) and that the family f_λ^pn(γ(λ))

p = (F^pn(γ))_p is normal. Our argument is local and so we can work on C. Sincez0 is repelling, we can shrinkV_λ0 and find A >1,r >0 such that

(2) |w(λ)−f_λⁿ(z)|=|f_λⁿ(w(λ))−f_λⁿ(z)| ≥A|w(λ)−z|

whenλ∈V_λ0 and|w(λ)−z|< r. On the other hand, asf_λ^p(γ(λ)) = (F^p·γ) (λ)andMis compactly supported and F-invariant, one sees that f_λ^pn(γ(λ))

p is a normal family. We can therefore shrink again V_λ0 so that |w(λ)−f_λ^pn(γ(λ))|< rfor everyp≥1and λ∈V_λ0. Combining this with (2) we obtain r > |w(λ)−f_λ^pn(γ(λ))| ≥ A^p|w(λ)−γ(λ)| for every p≥1 and λ∈V_λ0. This impliesw(λ) =γ(λ) onV_λ0 sinceA >1.

By analytic continuation we havef_λⁿ(γ(λ)) =γ(λ) for everyλ∈M and it thus remains to see that all n-periodic points γ(λ) are repelling. Let m(λ) denote the multiplier of the n-cycle to which γ(λ) belongs, the function m is holomorphic on M. From γ ∈suppM we immediately deduce thatγ(λ)∈J_λ and thus|m(λ)| ≥1for everyλ∈M, by the maxi- mum modulus principle applied to1/mwe must therefore have|µ(λ)|>1for everyλ∈M.

(3)⇒(1). This is obvious.

(3)⇔ (4). To deduce the existence of a holomorphic motion of Julia sets from that of repelling cycles one uses the classical λ-lemma which has been discussed in subsection 2.2.

A holomorphic motion of Julia sets conjugates the dynamics (see Definition 2.7) and thus induces a holomorphic motion of all repelling cycles.

(3) ⇒ (5). By assumption, R_n(λ) = {ρ_j,n(λ) : 1≤j ≤N_d(n)} for every n∈N^∗ and every λ∈ M, the maps ρ_j,n : M → P¹ being holomorphic. Thus, by the Approximation Formula (Theorem 3.8), we have L(λ) = limnd⁻ⁿP

1≤j≤N_d(n) 1

nlog|(f_λⁿ)⁰(ρj,n(λ))|where the convergence is pointwise. The sequence on the right-hand side is a locally bounded sequence of pluriharmonic function on M. Then the convergence actually occurs in L¹_loc and L is pluriharmonic onM (i.e. dd^cL= 0).

(5) ⇒ (6). For simplicity we assume that M is one dimensional. Since the problem is local we can replace M by any small disc D of C. Let us set u_n,j(λ) := f_λⁿ ◦c_j(λ) and ϕj,n(λ) := (λ, un,j(λ)). We have to show that the families (un,j(λ))_n are normal on D. Since|f_∗ⁿ[C_f]|_D×P1 =P2d−2

j=1

R

Dϕ^∗_j,n ω+dd^c|λ|²

=P2d−2 j=1

R

Ddd^c|λ|²+u^∗_n,jω one sees that it suffices to show that |f_∗ⁿ[C_f]|_D×P1 is uniformly bounded. A different proof will be given in the next subsection.

We will actually establish the estimate |f_∗ⁿ[C_f]|_D×

P¹ = dⁿR

Ddd^cL+O(1) and then conclude by thedd^cL-Formula 3.7. We first note that the functorial relationf_λ^n∗Tλ =dⁿTλ

seen in subsection 1.2 can be rewritten as f^n∗(ω+dd^c_λ,zg) =dⁿ(ω+dd^c_λ,zg)which implies

that f^n∗ω =dⁿ(ω+dd^c_λ,zg)−dd^c_λ,z(g◦fⁿ). Then

|f_∗ⁿ[C_f]|_D×

P¹ = Z

D×P¹

[C_f]∧f^n∗(ω+dd^c|λ|²)

=dⁿ Z

D×P¹

[C_f]∧(ω+dd^c_λ,zg) + Z

D×P¹

[C_f]∧ dd^c|λ|²−dd^c_λ,z(g◦fⁿ)

=dⁿ Z

D

dd^cL+ Z

D×P¹

[C_f]∧dd^c|λ|²− Z

D×P¹

[C_f]∧dd^c_λ,z(g◦fⁿ)

= Z

D×P¹

[C_f]∧dd^c|λ|²− Z

D×P¹

[C_f]∧dd^c_λ,z(g◦fⁿ)

where the third equality comes from thedd^cL-Formula and the last one from our assump- tion that dd^cL vanishes onD.

To conclude it remains to check that R

D×P¹[C_f]∧dd^c_λ,z(g◦fⁿ) is uniformly bounded.

This follows from the fact that the functiongis bounded and can be seen using integration by parts.

(6) ⇒ (7). Assume that λ0 is a Misiurewicz parameter. Then, by definition, there exist a critical mapc_j and a holomorphic mapσ defined on some neighbourhoodV₀ ofλ₀ such that σ(λ) is n0-periodic and repelling for every λ∈ V0 and f_λ^p0

0(cj(λ0) = σ(λ0) but f^p0◦c_j 6=σ. In that situation, the family (f^n+p0 ◦c_j)_n cannot be normal at λ₀. Indeed, we have shown in(2)⇒(3)thatf^p0◦cj would coincide withσif(f^n+p0◦cj)nwere normal.

(7) ⇒ (2). We will show that for every parameter λ0 ∈ M there exists a small ball B₀ ⊂M centered at λ₀ and an equilibrium web for the restriction of f to B₀×P¹. This will imply that the repelling cycles off move holomorphically onB0(recall that we already know that(2)and(3)are equivalent). The holomorphic motion of repelling cycles overM then follows since M is simply connected. Again, as(2)and (3)are equivalent, we get the existence of an equilibrium web over M .

Let z0 be an0-periodic repelling point for fλ0. By the implicit function theorem, there exist a ball B₀ centered at λ₀ and a holomorphic map σ :B₀ → P¹ such that σ(λ₀) =z₀ and σ(λ) isn0-periodic for fλ. By assumption, there are no Misiurewicz parameters and thus the graphΓ_σ ofσcannot meet the post-critical set∪_p≥1f^p(C_f)unless one of the maps cj satisfies a relation of the form f^p0+n0 ◦cj =f^p0 ◦cj on M which means that cj(λ) is pre-periodic for every λ∈ M. As C_f is the union of the (2d−2)graphs of the maps c_j, we are sure to find a periodic repelling point z₀ for which Γ_σ∩(∪p≥1f^p(C_f)) =∅.

Now, as fⁿ: B0×P¹

\f⁻ⁿ(∪_1≤p≤nf^p(C_f))→ B0×P¹

\(∪_1≤p≤nf^p(C_f)) is a cov- ering of degree dⁿ, there exist dⁿ holomorphic graphs Γ_σj,n such that fⁿ Γ_σj,n

= Γ_σ. Note that by the invariance of Julia sets all the σ_j,n are in J and that Fⁿ·σ_j,n =σ by

construction. Moreover it is a classical fact that the family (σj,n)_j,n is normal. Let us set M_n:= 1

dⁿ

dⁿ

X

j=1

δ_σj,n.

Then F∗M_n+1 =M_nand, by the Equidistribution Theorem 1.3, we have limn pλ∗M_n= lim

n

1 dⁿ

dⁿ

X

j=1

δ_σj,n(λ)= lim

n

1 dⁿ

X

f_λⁿ(z)=σ(λ)

δz=µ_λ

for every λ ∈ B₀. It follows that every limit M of (M_n)_n is a compactly supported probability measure onJ such thatpλ∗M=µ_λ for everyλ∈B₀. IfMwere F invariant then it would be an equilibrium web for f|_B

0×P^k. In order to get equilibrium webs, it suffices to replace M_n by the Cesaro average _n¹ P

1≤k≤nM_k.

We will now describe the main difficulties which appear when adapting the above proof to the higher dimensional setting. We follow the same organization and do not discuss the implications which are easy to adapt.

(2)⇒(3). The proof follows the same lines but a serious difficulty appears for showing that all n-periodic points γ(λ) are repelling. One has to use some linearization argument and thus have to deal with delicate problems of resonances. This is why we work in di- mension k= 2 or assume that M =H_d(P^k).

(3)⇒(4). One of the main difficulties appears here. Theλ-Lemma fails because of the lack of Hurwitz theorem. We rather show that any equilibrium webMsomehow contains a measurable holomorphic motion of J. This is done by studying the dynamical system (J,F,M). The sections 6 and 7 are devoted to a description of that proof.

(7)⇒(2). The proof follows the same lines but it is not at all obvious that one can find a graphΓ_σ which is not contained in∪_p≥1f^p(C_f). We have to use a delicate entropy argu- ment here. This point is not discussed in these notes (see also the Question A in Section 10).

(5) ⇒ (7). We have to argue directly here since the normal family arguments fail and we cannot follow the route (5) ⇒ (6) ⇒ (7). We rather show directly that Misiurewicz parameters belong to the support of dd^cL by using a dynamical rescaling argument. This is discussed in section 8. Note however that the computation given in the above proof of (5)⇒(6)is valid in any dimension and yields some estimate on the growth of the volume of fⁿ(C_f).

3.3. The bifurcation current. Let us still consider a holomorphic family f:M×P¹ →M×P¹

(λ, z)7→(λ, f_λ(z))

whose degree isd≥2. It follows immediately from the results of the above subsection that the support of dd^cL(λ) is precisely the bifurcation locus Bif(f). This was first proved by DeMarco [dM03] who defined the bifurcation current Tbif of the familyf by

T_bif :=dd^c_λL(λ).

As shown by thedd^c-formula (Theorem 3.7), the currentT_bif is a(1,1)-current onMwhich is positive and closed.

We will now decompose the bifurcation currentT_bif as a sum of closed positive currents Ti, each of them detecting the instability of the critical orbits (f_λⁿ(ci(λ))n or, in other words, the activity of the critical points ci. These currents are calledactivity currents.

For every 1 ≤i≤ 2d−2 we denote by [C_i]the current of integration on the graph of c_i (note that this graph is also an hypersurface in M ×P¹) and define a closed positive (1,1)-currentTi by

Ti := (πM)_∗ ω+dd^c_λ,zg(λ, z)

∧[C_i] . By the dd^cL- formula (Theorem 3.7) we have

Tbif =

2d−2

X

i=1

Ti.

Let us show that(f_λⁿ(ci(λ))nis normal whenTi = 0(it will then be clear that the converse is also true). This is a local problem so we can replace M by any small ball Ω ⊂ M on which the familyf can be lifted to a familyF ofd-homogeneous non-degenerate polynomial maps on C^k+1:

F: Ω×C^k+1 → Ω×C^k+1 (λ, z) 7→ (λ, Fλ(z)).

We may also assume that the maps cj can be lifted to maps bcj : Ω → C^k+1 \ {0}. Let us consider the Green function G(λ,·) of F_λ. This is a plurisubharmonic and continuous function on Ω×C^k+1 which is defined by

G(λ, z) := lim

n d⁻ⁿlnkF_λⁿ(z)k.

We stress that G satisfies the following homogeneity property: G(λ, tz) = ln|t|+G(λ, z) for every t∈Cand every (λ, z)∈Ω×C^k+1 (see [Ber13] for details).

A straightforward computation shows that G_λ(bc_i(λ))is a potential forT_i which means that dd^cGλ(bci(λ)) = Ti, the vanishing of Ti thus means that the function Gλ(bci(λ)) is pluriharmonic onΩ. We shall see that this implies the normality of the family(f_λⁿ(c_i(λ)))_n (we proved a similar fact in Proposition 3.4 for polynomial families).

The function G_λ(bc_i(λ))being pluriharmonic, there exists a non-vanishing holomorphic function h_i on Ωsuch that G_λ(bc_i(λ)) = ln|h_i|which, owing to the homogeneity property

of Gλ, can be rewritten asGλ

bci(λ) hi(λ)

= 0. After replacingbci by _h^bci(λ)

i(λ) we thus have {F_λⁿ(bc_i(λ)) : n∈N, λ∈Ω} ⊂ ∪_λ∈ΩG⁻¹_λ {0}.

After shrinking Ω the set ∪_λ∈ΩG⁻¹_λ {0} becomes a relatively compact subset in C² and Montel’s theorem then tells us that the family(F_λⁿ(bc_i(λ)))_nis normal onΩ. It follows that (f_λⁿ(ci(λ)))_nis normal onΩ too.

4. When k= 1: application of the bifurcation currents techniques We consider here a holomorphic family f :M×P¹→M×P¹,(λ, z)7→(λ, f_λ(z))of de- greedrational maps onP¹. Our aim is to use bifurcation current techniques to prove that certain parameters in M (for a sufficently “big” parameter space M) can be accumulated either by hyperbolic parameters or by parameters for which f_λ has the maximal number of distinct neutral cycles (i.e. 2d−2 by the Fatou-Shishikura inequality). Let us recall that a parameter λ is hyperbolic if and only if f_λ has 2d−2 distinct attracting cycles.

A parameter for which f_λ has 2d−2 distinct neutral cycles will be called a Shishikura parameter. The results discussed in this section have been obtained by Bassanelli and the first author [BB07].

Let us recall that the equilibrium measure of f_λ is denoted by µ_λ. The Lyapunov exponent L(λ) of(Jλ, fλ, µλ) is given byL(λ) =R

P¹log|f_λ⁰|µλ.It follows from the Mañé- Manning ([Man84],[Mañ88]) formula dim_H µ_λ = ^htop_L(λ)^(fλ) and the Misiurewicz-Przytycki inequalityh_top(f_λ)≥logdthatL(λ)≥ ^log₂^d.We shall use here the Approximation formula (see Theorem 3.8).

Let us consider the following subsets of the parameter space M:

Per_n(w) :={λ∈M:f_λ has a n-cycle of multiplierw}.

One may show thatPer_n(w)is a (singular) hypersurface inM. More precisely, there ex- ists a collection of functionsp_n(λ, w)which are unitary polynomials of degreeN_d(n)∼ ^d_nⁿ in wand whose coefficients are holomorphic function inλ, such thatPer_n(w) ={p_n(·, w) = 0}

for w6= 1 (the case w= 1 is more delicate).

According to the Poincaré-Lelong formula, the integration current [Per_n(w)]on the hy- persurface Per_n(w) is given by

[Per_n(w)] =dd^c_λlog|p_n(λ, w)|.

We are interested in comparing the limits of d⁻ⁿ[Per_n(w)] and the bifurcation current T_bif =dd^c_λL(λ).

4.1. Averaging the multipliers. We aim here to establish the following formula for the bifurcation current.

Theorem 4.1. For every r ≥0 one has Tbif = limnd⁻ⁿ 2π

R2π 0

Per_n(re^iθ) dθ.

Proof. SetL^r_n(λ) := ^d_2π⁻ⁿR2π 0 log

p_n(λ, re^iθ)

dθ, for r ≥ 0. Denoting the roots of p_n(λ,·) by wn,j(λ),1≤j≤Nd(n) (taken with mutiplicity) we get

L^r_n(λ) = d⁻ⁿ 2π

Z 2π 0

log

Nd(n)

Y

j=1

re^iθ−wn,j(λ)

dθ

= d⁻ⁿ 2π

Nd(n)

X

j=1

Z 2π 0

log

re^iθ−w_n,j(λ) dθ

=d⁻ⁿ

Nd(n)

X

j=1

log max (r,|w_n,j(λ)|)dθ.

By Fatou Theorem, there exists n(λ) ∈ N such that all n-cycles of f_λ are repelling for n≥n(λ). Thus, for0≤r ≤1 andn≥n(λ) , we obtain

L^r_n(λ) =d⁻ⁿ

Nd(n)

X

j=1

log|w_n,j(λ)|=d⁻ⁿ X

Rn(λ)

1 nlog

(f_λⁿ)⁰(z) and by the approximation formula limnL^r_n(λ) =L(λ).

When r >1 we have (always for n≥n(λ)) L^r_n(λ) =d⁻ⁿ

Nd(n)

X

j=1

log|w_j,n(λ)|+d⁻ⁿ X

1≤|wj,n(λ)|<r

log r

|w_j,n(λ)|

and therefore 0 ≤ L^r_n(λ)−L⁰_n(λ) ≤ d⁻ⁿNd(n) logr. As d⁻ⁿNd(n) ∼ _n¹, we get again lim_nL^r_n(λ) = L(λ). Since the sequence (L^r_n(λ))_n is a sequence of psh functions which is locally uniformly bounded from above (easy to check), the convergence actually occurs in L¹_loc(M). Takingdd^cwe thus haveTbif =dd^cL= limndd^cL^r_n= limnd⁻ⁿ

2π

R2π 0

Per_n(re^iθ) dθ.

4.2. Perturbation of Lattès examples. We aim here to use the above formula and simple potential theoretic arguments to show that rigid Lattès examples are accumulated by hyperbolic parameters or by Shishikura parameters. Let us recall that a Lattès map is a rational functionf which is induced on the Riemann sphere from a dilation on a complex torus D:T →T by mean of some elliptic function p:T →P¹.

T T

P¹ P¹

p p

D

f

Such maps are of course very rare. They have been characterized by A. Zdunik [Zdu90]

by the minimality of their Lyapunov exponent.

Theorem 4.2. f_λ is a Lattès example ⇔L(λ) = ^log₂^d.

For simplicity, we consider the moduli space of quadratic rational maps which (accord- ing to Milnor [Mil93]) can be considered as a family f :M×P¹ →M×P¹ parametrized by M =C².

We shall use the Monge-Ampère measure associated to thepshfunctionL. By definition this positive measure is given by dd^cL∧dd^cL. The fact that this product of two closed positive current is well defined follows from the continuity of L. Indeed, for any closed positive currentS, the functionLis integrable with respect to the trace measure ofS and therefore the currentLS is well defined. One then setdd^cL∧S:=dd^c(LS)(recall that S closed means thatdd^cS= 0).

Thebifurcation measure of the familyf has been introduced by Bassanelli and the first author who gave its first properties. It is defined by

µ_bif := 1

2dd^cL∧dd^cL.

It is an elementary property that the strict minima of L belong to the support of dd^cL∧dd^cL. Since degree two Lattès example are rigid, they correspond to isolated parameters in M. The theorem of Zdunik thus tells us that they belong to the support of the bifurcation measure. This is also true for all Lattès example as it has been proved by X. Buff and T. Gauthier [BG13]. To achieve our goal it remains to show that any parameter λ0 in the support of the bifurcation measureµbif can be accumulated either by hyperbolic parameters or by Shishikura parameters. This is actually a direct consequence of the following approximation formula which is obtained from Theorem 4.1 by mean of elementary potential theoretic arguments:

µbif = lim

n

2^−(n+k(n)) 2(2π)²

Z

[0,2π]²

[Per_n(re^iθ1)]∧[Per_k(n)(re^iθ2)]dθ1dθ2.

In the above formula, k(n) is a suitable increasing sequence of integers. The choice of 0 < r <1 shows that the support of µ_bif is accumulated by hyperbolic parameters while the choice r= 1 shows that it can be accumulated by Shishikura parameters.

What we have proved is actually an equidistribution statement for the bifurcation mea- sure. Full details on the content of this section are given in [Ber13]. Sharp equidistribution results for the bifurcation measure have recently be obtained by T. Gauthier, Y. Okuyama and G. Vigny [GOV17].

5. When k≥1: extension of Lyubich-Mañé-Sad-Sullivan Theorem We consider here a holomorphic family

f:M ×P^k→M×P^k (λ, z)7→(λ, fλ(z))

whose degree isd≥2. Recall that, for everyλ∈M, we have an ergodic dynamical system (J_λ, f_λ, µ_λ), where µ_λ is the equilibrium measure of f_λ (see Lecture 1). The sum of the Lyapunov exponents of (J_λ, f_λ, µ_λ) is given by the following expression:

L(λ) :=

Z

P^k

log

detf_λ⁰(z) µλ(z).

Our aim in this lecture is to describe a recent result which extends Lyubich-Mañé-Sad- Sullivan theorem in higher dimension and is due to C. Dupont and the authors [BBD15].

5.1. A substitute to the notion of holomorphic motions of Julia sets. Let us recall a few notations and concepts which have been introduced in subsection 3.2 for families of rational maps. We endow O(M,P^k) :=

γ :M →P^k:γ holomorphic with the topology of local uniform convergence; this is a metric space. The space of interest here is the (possibly empty) subspace

J :=

n

γ ∈ O(M,P^k) :γ(λ)∈Jλ,∀λ∈M o

.

We have two natural maps. The first one is a self-map on F :J → J which is defined by F(γ)(λ) = f_λ(γ(λ)) and the second one is a projectionp_λ :J →P^k which is defined by p_λ(γ) =γ(λ).

Definition 5.1. An equilibrium web forf is a compactly supported probability measureM on J such that F_∗M=Mand (p_λ)_∗M=µ_λ for all λ∈M.

λ P^k

M

µλ graphsΓγ ofγ ∈SuppM

µ_λ = Z

J

δ_γ(λ)dM(γ) = (p_λ)_∗M Interpretation: