Collet, Eckmann and the bifurcation measure

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Collet, Eckmann and the bifurcation measure

Matthieu Astorg, Thomas Gauthier, Nicolae Mihalache, Gabriel Vigny

To cite this version:

Matthieu Astorg, Thomas Gauthier, Nicolae Mihalache, Gabriel Vigny. Collet, Eckmann and the

bifurcation measure. 2018. �hal-01888305�

by

Matthieu Astorg, Thomas Gauthier, Nicolae Mihalache & Gabriel Vigny

Abstract. — The moduli space M

d

of degree d ≥ 2 rational maps can naturally be endowed with a measure µ

bif

detecting maximal bifurcations, called the bifurcation measure. We prove that the support of the bifurcation measure µ

bif

has positive Lebesgue measure. To do so, we establish a general sufficient condition for the conjugacy class of a rational map to belong to the support of µ

bif

and we exhibit a large set of Collet-Eckmann rational maps which satisfy this condition. As a consequence, we get a set of Collet-Eckmann rational maps of positive Lebesgue measure which are approximated by hyperbolic rational maps.

Contents

1. Introduction. . . . 1

2. Basics on bifurcation currents. . . . 5

3. Generalized large scale condition and the bifurcation currents. . . . 8

4. Misiurewicz maps and the generalized large scale condition . . . 13

5. The large scale condition and good Collet-Eckmann maps. . . 17

6. The proofs of Theorem A and Corollary 1. . . 33

References. . . 34

1. Introduction

For an integer d ≥ 2, we let Rat

_d

be the space of rational maps of degree d of the Riemann sphere P

¹

. The space Rat

_d

of degree d rational maps is a quasi-projective subvariety of dimension 2d + 1. More precisely, there exists an (irreducible) variety V such that Rat

_d

' P

^2d+1

\ V . The J -stability locus in Rat

_d

is defined as the set of maps f which are structurally stable on their Julia set J

_f

. By the seminal work [MSS] of Ma˜ n´ e, Sad and Sullivan, it is an open and dense subset of Rat

_d

. The bifurcation locus in Rat

_d

is its complement.

One can give a measurable description of this bifurcation as follows: any rational map f ∈ Rat

_d

has a unique measure µ

_f

of maximal entropy log d. The Lyapunov exponent of f with respect to the measure µ

f

can be defined as L(f) := R

P¹

log |f

⁰

| µ

f

, where | · | is any her-

mitian metric on P

¹

. DeMarco [De] proved that the function L : f ∈ Rat

d

7→ L(f ) ∈ R

+

is

plurisubharmonic (p.s.h for short) and continuous and that the bifurcation locus is the support

of the positive closed (1, 1)-current T

_bif

:= dd

^c

L.

Moreover, M¨ obius transformations act by conjugacy on Rat

_d

and the quotient space is an orbifold affine variety of dimension 2d − 2 which is known as the moduli space M

_d

of degree d rational maps. As the function L is invariant by conjugacy, the function L induces a p.s.h and continuous function L : M

_d

−→ R. The support of dd

^c

L coincides with the image of the bifurcation locus under the quotient map Π : Rat

_d

→ M

_d

.

Definition 1.1. — The bifurcation measure of the space M

_d

is the positive measure µ

_bif

:= (dd

^c

L)

^2d−2

.

Notice that the support of µ

_bif

is strictly contained in the bifurcation locus. This measure has been introduced by Bassanelli and Berteloot in [BB]. Its support is exactly the accumulation set of the hyperbolic (conjugacy class of) rational maps (see [BB] and also Buff and Epstein [BE] and the second author [G2] for other dynamical characterizations). Recall that a rational map is hyperbolic if it is uniformly expanding on its Julia set J

_f

, i.e. there exist C > 0 and λ > 1 such that for any n ≥ 1 and any z ∈ J

_f

, |(f

ⁿ

)

⁰

(z)| ≥ C · λ

ⁿ

. Another way to define µ

_bif

is as follows: it is the only finite measure on M

_d

such that Π

^∗

(µ

_bif

) = T

_bif^2d−2

as closed positive (2d − 2, 2d − 2)-currents on Rat

d

(see [BB, §6]).

In the family {z

²

+ c}

_c∈C

of quadratic polynomials, this measure is exactly the harmonic measure of the Mandelbrot set and its support is exactly the boundary of the Mandelbrot set.

One of the important questions in complex dynamics which is still open is the following.

Question. — Does the boundary of the Mandelbrot set have positive area?

It is natural to ask similar questions in the moduli space M

_d

of degree d rational maps. For that, let us naturally endow M

_d

with a volume form. the moduli space M

_d

is a affine variety in some P

^N

. Let ω

_P^N

be the Fubini-Study form on P

^N

. Its restriction defines a (1, 1) form ω

d

on M

_d

and we consider the volume form on M

_d

Vol

M_d

:= ω

_d^2d−2

.

This is (up to renormalization) a probability measure on M

_d

. Moreover, in any (smooth) local chart of M

_d

it is smoothly equivalent to the Lebesgue measure, hence it is a non-degenerate volume form in M

_d

. For the bifurcation locus, Rees showed that it has positive Vol

M_d

-measure [R]. This leaves the question of the volume of the support of the measure µ

bif

in M

_d

.

Another major problem concerning the moduli space M

_d

is the Hyperbolicity Conjecture : Conjecture (Hyperbolicity). — Hyperbolic rational maps are dense in Rat

_d

.

In the present paper, we are interested in the following simpler but related problem: is any Collet-Eckmann rational map approximated by hyperbolic rational maps? Refining Rees’

results, Aspenberg [Asp2] showed that there is a set of positive Vol

M_d

-measure of (suitable) Collet-Eckmann rational map (such maps are in the bifurcation locus). A rational map f ∈ Rat

_d

is Collet-Eckmann if the critical set C(f) of f is contained in J

_f

and if there exist γ, γ

₀

> 0 such that

(CE(γ, γ

₀

)) |(f

ⁿ

)

⁰

(f (c))| ≥ e

^nγ−γ0

, for any c ∈ C(f ) and any n ≥ 0.

Our main result is the following

Theorem A. — Pick an integer d ≥ 2 and let [f] ∈ supp(µ

_bif

). Then for any open neighborhood Ω ⊂ M

_d

of [f ], there is a set CE

Ω

⊂ Ω ∩ supp(µ

_bif

) of Collet-Eckmann maps with:

Vol

M_d

(CE

_Ω

)) > 0.

This implies the following result reated to the above conjecture.

Corollary 1. — There exists a set of positive Vol

_Ratd

-measure of Collet-Eckmann rational maps f of Rat

d

with [f] ∈ supp(µ

bif

) such that f is approximated by hyperbolic rational maps.

We rely on the following strategy: we first give a very general sufficient condition for a conju- gacy class of rational maps to belong to the support of the bifurcation measure (see Theorem B).

Then we exhibit a large set of rational maps fulfilling this condition. Theorem A becomes a corollary of Theorem B, Theorem C and Theorem D below and of the Main Theorem of [BE].

When a rational map has simple critical points, one can follow them holomorphically as maps c

1

, . . . , c

2d−2

: V → P

¹

where V is an open neighborhood of f in Rat

_d

. For any (2d − 2)-tuple of positive integers n := (n

₁

, . . . , n

2d−2

), one also can define a holomorphic map V

_n

: V −→

( P

¹

)

^2d−2

by letting

V

_n

(g) := (g

ⁿ¹

(c

1

(g)), . . . , g

^n2d−2

(c

2d−2

(g))) .

This map detects, in a certain sense, the collective stability/instability of the critical points c

₁

, . . . , c

2d−2

. In the sequel, we will use the following definition in a crucial way.

Definition 1.2. — Let f ∈ Rat

d

with simple critical values and let Γ

n

denote the graph of V

_n

. We say that f satisfies the large scale condition if there exist a (local) complex submanifold Λ

f

⊂ Rat

d

of dimension 2d − 2, a sequence n

_k

= (n

k,1

, . . . , n

k,2d−2

) of (2d − 2)-tuples with n

_k,j

→ +∞ for all j, a basis of neighborhoods {Ω

_k

}

_k≥1

of f in Λ

_f

and δ > 0 such that for any k, the connected component of Γ

_nk

∩ Ω

_k

× ( D (f

^nk,1

(c

₁

(f )), δ) × · · · × D (f

^nk,2d−2

(c

2d−2

(f )), δ)) containing (f, V

_n

k

(f )) is contained in Ω

⁰_k

× ( D (f

^nk,1

(c

₁

(f )), δ) × · · · × D (f

^nk,2d−2

(c

2d−2

(f )), δ)) for some Ω

⁰_k

b Ω

_k

.

This means that a rational map f satisfies the large scale condition if, infinitely many times, the map V

_n

sends an arbitrarily small neighborhood of [f ] in the moduli space M

_d

to a polydisk of fixed size in ( P

¹

)

^2d−2

and its graph is vertical-like near [f]. The first step of our proof consists in proving that the large scale condition is actually a sufficient condition for maximal bifurcations to occur at f . More precisely, we prove the following.

Theorem B. — Pick f ∈ Rat

d

. Assume that ω(c) ⊂ J

_f

for all c ∈ C(f), that f has simple critical points and that f satisfies the large scale condition. Then [f ] ∈ supp(µ

_bif

).

This condition extracts the core of the ones previously used, such as having a uniform expan-

sion along the postcritical set, the famous Misiurewicz condition. To illustrate the interest of

this condition, we show in §4 that this condition is satisfied by Misiurewicz maps. The proof

drastically simplifies the proof of [G1, Theorem 1.4]. In particular, we now are able to avoid any

linearization process along the critical orbits which was a crucial step in the proof given by the

second author (see [G1, §5]). Indeed, we only use here the transversality and the holomorphic

motion of the hyperbolic set containing the post-critical set. The proof of Theorem B is based

on a phase-parameter transfer (giving a measurable version of Tan Lei’s work [Ta]). For clarity,

we give the proof in the easier case of one critical point in Section 3.1.

A rational map is strongly Misiurewicz if all its critical points are preperiodic to repelling cycles. Reformulating the main result of Aspenberg [Asp2] in terms of the conditions CE(γ, γ

0

), CE2(µ, µ

₀

), BA(α) and FA(η, ι) (defined at the beginning of Section 5) gives the following.

Theorem C. — Assume that f ∈ Rat

d

is strongly Misiurewicz, has simple critical points and is not a flexible Latt` es map. Then, there exist µ, µ

₀

, γ, γ

₀

> 0 and α > ˆ 0 such that for all α < min(

₂₀₀^γ

, α), there exist ˆ η > ˆ 0 and ˆ ι > 0 such that for all η < η ˆ and for all ι < ˆ ι, the map f is a Lebesgue density point of rational maps satisfying CE(γ, γ

₀

), CE2(µ, µ

₀

), BA(α) and FA(η, ι).

The last key ingredient to prove Theorem A can be formulated as follows (conditions (K1-6) are defined in Section 5.1).

Theorem D. — Let γ, γ

0

, µ, µ

0

, η, κ > 0 and α < γ/200. There exists ι > 0 such that any f ∈ Rat

_d

with simple critical points and satisfying CE(γ, γ

₀

), CE2(µ, µ

₀

), BA(α), FA(η, ι) and (K1-6) satisfies the large scale condition.

To prove Theorem D, we follow Tsujii’s generalization [Ts] of Benedicks and Carleson con- struction [BC] that we need to adapt to the complex setting. The strategy of the proof is summarized below.

• Take a Collet-Eckmann map f and a small ball in the dynamical plane centered at a critical value. That ball will go to the large scale with exponential growth and good distortion estimates under f

ⁿ

by (CE) as long as its orbit stays far away from the critical set C(f).

For each n we will choose such a starting ball B

n

.

• Passages of the critical orbit near C(f ) impose upper bounds of the size of B

_n

. The assertion (BA) gives lower bounds for the approach rate to C(f ). After a close visit near C(f ), the image is even closer to the visited critical value. Lemma 5.4 shows that the sequel of the critical orbit copies the good properties of a long prefix of the orbit of the visited critical value.

• Lemma 5.10 guarantees that just before going (again) near a critical point, we gain ex- pansivity by (CE2), so we have restored the exponential growth and the bound for the distortion on B

_n

. We use a Lemma ` a la M˜ an´ e (Proposition 5.11) for the suffix of a (finite) critical orbit that does not visit a neighborhood of C(f ).

• Lemma 5.13 gives a large density of times n for which B

_n

goes to the large scale ((FA) tells that the critical orbit does not go too often near the critical set). We intersect over all the critical values and still have positive density of times for which the starting balls go to the large scale for all the critical values.

• Using Lemma 5.15 allows to bound parametric distortion on complex lines passing through f thanks to the transversality of critical relations (using again (CE) and (BA)).

• Finally, we use a result of Sibony and Wong [SW] and the transversality to extend the distortion to a ball on a neighborhood of f to get the large scale condition (Theorem 5.16).

Structure of the paper. — In Section 2, we recall facts on bifurcation currents. In Sec-

tion 3, we define a (generalized) large scale condition and show that if a parameter satisfies it

for some m then it is in the support of T

_bif^m

. In Section 4, we apply this result to k-Misiurewicz

maps. In Section 5, we prove Theorems C and D. In Section 6, we prove Theorem A and a

strengthened version of Corollary 1.

2. Basics on bifurcation currents

This section is devoted to the bifurcation currents. We begin with giving a description of the bifurcation currents. We then give a new formula for the higher bifurcation currents.

2.1. The bifurcation currents of critical points

Let (f

_λ

)

_λ∈Λ

be a holomorphic family of rational maps equipped with 2d − 2 marked critical points c

1

, . . . , c

2d−2

: Λ → P

¹

.

Definition 2.1. — A critical point c is said to be marked if there exists a holomorphic function c : Λ −→ P

¹

satisfying f

_λ⁰

(c(λ)) = 0 for every λ ∈ Λ.

We say that the critical point c is active at λ

0

∈ Λ if (f

_λⁿ

(c(λ)))

n≥0

is not a normal family in any neighborhood of λ

₀

. Otherwise we say that c is passive at λ

₀

. The activity locus of c is the set of parameters λ ∈ Λ at which c is active.

Let us construct the bifurcation current of the critical point c

_i

. One can define a fibered dynamical system ˆ f acting on Λ × P

¹

f ˆ : Λ × P

¹

−→ Λ × P

¹

(λ, z) 7−→ (λ, f

_λ

(z)) .

We denote by p

_Λ

: Λ × P

¹

→ Λ and p

_P¹

: Λ × P

¹

→ Λ the respective natural projections and by ˆ

ω := (p

_P¹

)

^∗

ω

FS

, where ω

FS

is the Fubini-Study form on P

¹

normalized so that R

P¹

ω

FS

= 1. We say that a function ψ is ˆ ω-psh if it can be locally written as the sum of a smooth function and a plurisubharmonic function (psh for short) and dd

^c

ψ + ˆ ω ≥ 0 in the sense of currents. Then, there exists a ˆ ω-psh function g such that

( ˆ f)

^∗

(ˆ ω + dd

^c

g) = d · (ˆ ω + dd

^c

g) . (1)

Indeed, since d

⁻¹

( ˆ f )

^∗

ω ˆ and ˆ ω are in the same cohomology class, there exists a smooth ˆ ω-psh function u such that d

⁻¹

( ˆ f )

^∗

ω ˆ = ˆ ω + dd

^c

u. Taking

u

n

:=

n−1

X

j=0

u ◦ ( ˆ f )

^j

d

^j

,

we defined g := lim

n

u

n

. The function g is continuous and ˆ ω-plurisubharmonic on Λ × P

¹

, since kg − u

_n

k

_∞

= O(d

⁻ⁿ

). The function g is the Green function of ˆ f and is unique up to an additive constant. We shall use the following notation in the sequel

T b := ˆ ω + dd

^c

g . One can give the following definition.

Definition 2.2. — The bifurcation current of the critical point c

_i

in (f

_λ

)

λ∈Λ

is T

i

:= (p

Λ

)

∗

(ˆ ω + dd

^c

g) ∧ [ ˆ V

i

]

,

where V ˆ

i

= {(λ, v

_i

(λ)) : λ ∈ Λ} is the graph of the map v

i

(λ) := f

λ

(c

i

(λ)).

The holomorphic family (f

_λ

)

λ∈Λ

admits local lifts, i.e. for any small enough V ⊂ Λ, there exists a holomorphic family (F

_λ

)

λ∈V

of non-degenerate homogeneous degree d polynomial endo- morphims of C

²

, and the Green function of the lift is then

G(λ, x, y) := lim

n→∞

d

⁻ⁿ

log kF

_λⁿ

(x, y)k , (x, y) ∈ C

²

\ {0} .

It actually is a continuous and psh function on V × ( C

²

\ {0}). According to [B, Section 3.2.2], one can prove that for any local holomorphic section σ : U ⊂ P

¹

→ C

²

\ {0} of the canonical projection C

²

\ {0} → P

¹

, then

ˆ

ω + dd

^c

g = dd

^c

G(λ, σ(z)) , on V × U.

We thus can locally write T

i

= dd

^c

G(λ, σ ◦ v

i

(λ)) = d · dd

^c

G(λ, σ ◦ c

i

(λ)) on U.

Remark. — The definition we give here is not the classical one. The usual definition of the bifurcation currents is to take locally ˜ T

i

:= dd

^c

G(λ, σ ◦ c

i

(λ)) = d

⁻¹

· T

i

, which does not change the support of the current.

The important information concerning the current T

_i

is the following (see [DF]).

Proposition 2.3 (Dujardin-Favre). — The support of T

i

is the activity locus of c

i

.

Another way to characterize the bifurcation current of c

_i

is the following. Let ξ

_nⁱ

: Λ → P

¹

be the map given by ξ

_nⁱ

(λ) := f

_λⁿ

(v

i

(λ)), for n ≥ 0 and 1 ≤ i ≤ 2d − 2. The sequence of forms d

⁻ⁿ

(ξ

_nⁱ

)

^∗

ω

_FS

converge to the current T

_i

in the sense of currents (see e.g. [Du]).

2.2. Bifurcation currents of a holomorphic family

Recall that f ∈ Rat

_d

admits a unique maximal entropy measure µ

_f

. The Lyapounov exponent of f with respect to the measure µ

f

is the real number L(f) := R

P¹

log |f

⁰

|µ

_f

. For a holomorphic family (f

_λ

)

λ∈Λ

of degree d rational maps, we denote by L(λ):=L(f

_λ

). Then, the function λ 7−→

L(λ) is called the Lyapounov function of the family (f

λ

)

λ∈Λ

. It is a psh and continuous function on Λ (see [BB] Corollary 3.4). The Margulis-Ruelle inequality implies that L(f ) ≥

^log₂^d

.

When (f

_λ

)

λ∈Λ

is with 2d −2 distinct marked critical points c

₁

, . . . , c

2d−2

, the bifurcation locus in the sense of Ma˜ n´ e-Sad-Sullivan and Lyubich (see [Ly, MSS]) coincides with the union of the activity loci of the c

_i

’s. According to DeMarco [De], we have the following.

Theorem 2.4 (DeMarco). — Let (f

λ

)

λ∈Λ

be a holomorphic family of degree d rational maps with 2d − 2 distinct marked critical points. Then the current T

_bif

:= dd

^c

L is exactly supported by the bifurcation locus. Moreover, T

_bif

= d

⁻¹

P

2d−2

i=1

T

_i

.

The current T

_bif

is the bifurcation current of of the family (f

_λ

)

λ∈Λ

. The self-intersections of the current T

_bif

have been first studied by Bassanelli and Berteloot [BB].

Definition 2.5. — We define the m

^th

-bifurcation current of the family (f

_λ

)

λ∈Λ

by setting T

_bif^m

:=

m

^

i=1

T

_bif

.

It is known that for all 1 ≤ i ≤ 2d − 2, we have T

_i

∧ T

_i

= 0 (see [G1, Theorem 6.1]) and one can show that

T

_bif^m

= m! X

i1<...<im

T

_i1

∧ · · · ∧ T

_im

.

(2)

2.3. A formula for higher bifurcation currents

We want here to give a similar expression as the one given in Definition 2.2 for the higher bifurcation current associated to a m-tuple of critical points. Let us introduce some notations.

Let (f

λ

)

λ∈Λ

be a holomorphic family of degree d rational maps with m marked critical points, c

₁

, . . . , c

_m

: Λ → P

¹

, with 1 ≤ m ≤ min(2d − 2, dim Λ). As above, we define v

_j

: Λ → P

¹

for λ ∈ Λ by v

j

(λ) := f

λ

(c

j

(λ)). This time, we let

V

_j

:= {(λ, z) ∈ Λ × ( P

¹

)

^m

: z

j

= v

j

(λ)}

We finally let π

_Λ

: Λ × ( P

¹

)

^m

−→ Λ and, for 1 ≤ j ≤ m, we let

π

j

: Λ × ( P

¹

)

^m

−→ Λ × P

¹

(λ, z) 7−→ (λ, z

_j

)

be the respective natural projection. Let T

_i

be the bifurcation current of c

_i

in (f

_λ

)

λ∈Λ

.

Lemma 2.6. — With the above notations, we have

m

^

j=1

T

j

= (π

Λ

)

∗





m

^

j=1

(π

j

)

^∗

T b

∧ [V

_j

]



 .

Proof. — It is a local problem, so we can assume that Λ = B is a ball of C

^N

for some N ≥ 1.

Recall that, up to reducing the ball, one can also write T b = dd

^c_λ,z

G

_λ

(σ(z)), where G

_λ

is the Green function of a holomorphic family of non-degenerate homogeneous polynomial lift F

_λ

of f

λ

, i.e.

G

_λ

(z

1

, z

2

) := lim

n→∞

d

⁻ⁿ

log kF

_λⁿ

(z

1

, z

2

)k, (z

1

, z

2

) ∈ C

²

\ {0} ,

and σ is any local section of the natural projection C

²

\{0} → P

¹

. Let σ

j

be such a section which up to reducing the ball contains the image of the map v

j

: B → P

¹

, then dd

^c

G

λ

(σ

j

◦ v

j

(λ)) = T

j

(see e.g. [B, Section 3.2.2]).

Let now φ be a smooth test (N − m, N − m)−form on B and let V := T

j

V

_j

. Let p(λ) :=

(λ, v

₁

(λ), . . . , v

_m

(λ)), so that π

_Λ

◦ p = id

_B

and p( B ) = V. Then

* (π

_Λ

)

∗





m

^

j=1

(π

_j

)

^∗

T b

∧ [V

_j

]



 , φ +

= Z

^m

^

j=1

(π

_j

)

^∗

T b

∧ [V

_j

] ∧ (π

_Λ

)

^∗

φ

= Z

V m

^

j=1

(π

_j

)

^∗

T b

∧ (π

_Λ

)

^∗

φ

= Z

V m

^

j=1

(π

_j

)

^∗

dd

^c_λ,z

(G

_λ

◦ σ

_j

) ∧ (π

_Λ

)

^∗

φ

= Z

p(B) m

^

j=1

dd

^c_λ,z

G

λ

(σ

j

(z

j

)) ∧ (π

Λ

)

^∗

φ

= Z

B m

^

j=1

dd

^c

G

_λ

(σ

j

◦ v

j

(λ)) ∧ ((p

^∗

(π

Λ

)

^∗

φ))

=

*

_m

^

j=1

T

_j

, φ +

.

3. Generalized large scale condition and the bifurcation currents

For the whole section, we let (f

_λ

)

λ∈Λ

be a holomorphic family of rational maps with m ≥ 1 marked critical points c

₁

, . . . , c

_m

: Λ −→ P

¹

. As above, we use the notation v

_j

(λ) := f

_λ

(c

_j

(λ)).

We set c := (c

1

, . . . , c

m

) and we also use the following notation: For any m-tuple of positive integers n = (n

₁

, . . . , n

_m

), any 1 ≤ j ≤ m and any λ ∈ Λ, we let

ξ

_n^j_j

(λ) := f

_λ^nj

(v

_j

(λ)) , and V

^c_n

(λ) := ξ

¹_n1

(λ), . . . , ξ

_n^m_m

(λ) .

This way, we define a holomorphic map V

^c_n

: Λ −→ ( P

¹

)

^m

. We denote by V

_n

the graph of V

^c_n

. We now can define the (generalized) large scale condition.

Definition 3.1 (Generalized large scale condition). — We say that a parameter λ

₀

∈ Λ satisfies the generalized large scale condition at f

λ0

for the m-tuple (c

1

, . . . , c

m

) in Λ if there exist a sequence n

_k

= (n

_k,1

, . . . , n

_k,m

) of m-tuples with n

_k,j

→ +∞ and a basis of neighbor- hood {Ω

_k

}

_k≥1

of f in Λ and δ > 0 such that for any k, the connected component of V

_n

k

∩ Ω

k

×

D (ξ

_n¹_k,1

(λ

0

), δ) × · · · × D (ξ

_n^m_k,m

(λ

0

), δ)

containing (λ

0

, V

^c_n

k

(λ

0

)) is contained in Ω

⁰_k

×

D(ξ

_n¹_k,1

(λ

0

), δ) × · · · × D(ξ

^m_nk,m

(λ

0

), δ)

for some Ω

⁰_k

b Ω

k

.

We prove in this section the following result which we view as a general sufficient condition for a parameter to belong to the support of a (higher) bifurcation current.

Theorem 3.2. — Let 1 ≤ m ≤ min(2d − 2, dim Λ) be an integer and let λ

₀

∈ Λ. Assume

that λ

0

satisfies the generalized large scale condition for c := (c

1

, . . . , c

m

) in a local submanifold

S 3 λ

₀

of Λ with dim S = m. Assume in addition that ω(c

_i

(λ

₀

)) ⊂ J

_λ0

for all 1 ≤ i ≤ m. Then

λ

₀

∈ supp(T

₁

∧ · · · ∧ T

_m

).

Proof of Theorem B. — By definition of µ

_bif

, we have [f] ∈ supp(µ

_bif

) if and only if f ∈ supp(T

_bif^2d−2

). On the other hand, since f has simple critical points, there exist a neighborhood U ⊂ Rat

d

of f and holomorphic maps c

1

, . . . , c

2d−2

: U → P

¹

with C(g) = {c

₁

(g), . . . , c

2d−2

(g)}

for all g ∈ U . We then can apply the above Theorem 3.2 to c = (c

₁

, . . . , c

2d−2

). Since, by (2), T

_bif^2d−2

= (2d − 2)!T

1

∧ · · · ∧ T

2d−2

on U , the result follows.

Remark. — Note that the assumption ω(c

_j

(λ

₀

)) ⊂ J

_λ0

is satisfied not only when c

_j

(λ

₀

) ∈ J

_λ0

, but also when c

j

(λ

0

) belongs to a parabolic basin.

First, note that it is sufficient to treat the case when dim Λ = m and λ

₀

satisfies the large scale condition for c := (c

1

, . . . , c

m

) in a Λ. Indeed, by [G1, Lemma 6.3], if λ

0

∈ supp((T

₁

∧ · · · ∧ T

_m

) |

_S

), then λ

₀

∈ supp (T

₁

∧ · · · ∧ T

_m

). We hence may assume S = Λ has dimension m and let

µ := T

1

∧ · · · ∧ T

m

. It defines a positive measure on Λ.

Let (λ

₁

, . . . , λ

_m

) be a local system of holomorphic coordinates centered at λ

₀

. We let D

^m_δ

be the polydisk of radius δ > 0 of Λ centered at λ

0

in those coordinates.

3.1. The case m = 1: a toy-model for the general case

We give here the proof of Theorem 3.2 in the case m = 1. We let c : Λ → P

¹

be the marked critical point satisfying the large scale condition at λ

₀

. As above, for n ≥ 0 and λ ∈ Λ, write

ξ

_n

(λ) := f

_λⁿ⁺¹

(c(λ)) = f

_λⁿ

(ξ

₀

(λ)) .

Of course, in that case, it is easy to see that the large scale condition implies the non-normality of the family (f

_λⁿ

(c(λ)))

n

but we provide here a proof that can be adapted to work with higher degree bifurcation currents. For that, recall that V

_n

is the graph of ξ

n

: Λ → P

¹

. Recall that d

⁻¹

( ˆ f )

^∗

T b = T b . Choose any > 0. We shall prove that µ(D(λ

0

, )) > 0 for all > 0 small enough. For any n ≥ 1

µ (D(λ

0

, )) = Z

D(λ0,)×P¹

T b ∧ [V

₀

]

= d

⁻ⁿ

Z

D(λ0,)×P¹

( ˆ f

ⁿ

)

^∗

T b

∧ [V

₀

] . As a consequence,

I

n

:= d

ⁿ

µ (D(λ

0

, )) = Z

D(λ0,)×P¹

( ˆ f

ⁿ

)

^∗

T b

∧ [V

₀

]

= Z

D(λ0,)×P¹

T b ∧ ( ˆ f

ⁿ

)

∗

[V

₀

] = Z

D(λ0,)×P¹

T b ∧ [V

_n

], since on one hand ˆ f

⁻¹

(D(λ

0

, ) × P

¹

) = D(λ

0

, ) × P

¹

= ˆ f D(λ

0

, ) × P

¹

and, on the other hand, ( ˆ f

ⁿ

)

∗

[V

₀

] = [V

_n

]. Let now δ > 0, (n

k

) and (Ω

k

) be given by the large scale condition. Up to extraction, we can assume ξ

_nk

:= ξ

_nk

(λ

₀

) converges to some point x ∈ J

_λ0

. As a consequence, there exists k

₀

≥ 1 such that Ω

_k

⊂ D (λ

₀

, ) and

ξ

_nk

(Ω

_k

) ⊃ D (x, δ/2) ,

for any k ≥ k

₀

. We now let S

_k

be the connected component of V

_nk

∩ Ω

_k

× D (x, δ/2) containing (0, ξ

n_k

). Then S

_k

⊂ Ω

_k

× D (x, δ/2), [S

_k

]/k[S

_k

]k has mass 1 and

[V

_nk

] ≥ 1

_{D(λ0,)×D(x,δ/2)}

[Γ

_nk

] ≥ [S

_k

] .

for all k ≥ k

₀

. Let now S be any weak limit of the sequence [S

_k

]/k[S

_k

]k. Then supp(S) = {λ

₀

} × D (x, δ/2) and S has mass 1 by the large scale condition. By extremality of the current [{λ

₀

} × D (x, δ/2)], we deduce S = M · [{λ

₀

} × D (x, δ/2)], where M

⁻¹

> 0 is the Fubini-Study area of D (x, δ/2). As T b has continuous potential, T b ∧ [S

_k

]/k[S

_k

]k → T b ∧ S as k → ∞ in the sense of measures so:

lim inf

k→∞

k[S

_k

]k

⁻¹

· I

_nk

≥ lim inf

k→∞

Z

T b ∧ [S

_k

] k[S

_k

]k ≥

Z

T b ∧ S = M · Z

T b ∧ [{λ

₀

} × D (x, δ/2)]

= M · µ

_λ0

( D (x, δ/2)) as T b |

_λ=0

= µ

_λ0

. Since x ∈ J

_f

= supp(µ

_λ0

) then µ

_λ0

( D (x, δ/2)) > 0, we get

lim inf

k→∞

k[S

_k

]k

⁻¹

· I

_nk

> 0 ,

which means that I

nk

> 0 for k large enough so µ(D(λ

0

, )) > 0. Since this holds for all > 0, this ends the proof.

3.2. First step: pulling-back by a fibered dynamical system

Let us define a family of fibered dynamical systems acting on Λ × (P

¹

)

^m

as follows: for any m-tuple n := (n

1

, . . . , n

m

) ∈ ( N

^∗

)

^m

, we let

F

_n

: Λ × ( P

¹

)

^m

−→ Λ × ( P

¹

)

^m

(λ, z

₁

, . . . , z

_m

) 7−→ (λ, f

_λⁿ¹

(z

₁

), . . . , f

_λ^nm

(z

_m

)) . For a m-tuple n = (n

1

, . . . , n

m

) of positive integers, we also set

|n| := n

₁

+ · · · + n

_m

.

Let us first partially rewrite the mass of µ on any open set in terms of iterated pull-back by one the F

_n

’s.

Lemma 3.3. — For any m-tuple n = (n

1

, . . . , n

m

) of positive integers, we let V

_n

be the graph in Λ × ( P

¹

)

^m

of V

_n

. Then, for any Borel set Ω ⊂ Λ, we have

µ(Ω) = d

^−|n|

Z

Ω×(P¹)^m





m

^

j=1

(π

_j

)

^∗

T b



 ∧ V

_n

.

Proof. — Recall that we defined in Section 2.1 a dynamical system ˆ f acting on Λ × P

¹

by f ˆ : Λ × P

¹

−→ Λ × P

¹

(λ, z) 7−→ (λ, f

_λ

(z)) . By definition of F

n

and ˆ f , for all j, the following diagram commutes

Λ × ( P

¹

)

^{m Fn}

^//

πj

Λ × ( P

¹

)

^m

πj

Λ × P

¹ _ˆ

f^nj

// Λ × P

¹

.

In particular, by (1), we get d

^−nj

(F

n

)

^∗

(π

j

)

^∗

T b

= d

^−nj

(π

j

◦ F

n

)

^∗

T b

= d

^−nj

( ˆ f

^nj

◦ π

j

)

^∗

T b

= d

^−nj

(π

_j

)

^∗

( ˆ f

^nj

)

^∗

T b

= (π

_j

)

^∗

T b

. According to Lemma 2.6, the change of variable formula gives

µ(Ω) = Z

Ω

(π

Λ

)

∗





m

^

j=1

(π

j

)

^∗

T b

∧ [V

_j

]





= Z

π⁻¹_Λ (Ω) m

^

j=1

(π

j

)

^∗

T b

∧ [V

_j

]

= d

^−|n|

Z

Ω×(P¹)^m

(F

_n

)

^∗





m

^

j=1

(π

_j

)

^∗

T b



 ∧





m

\

j=1

V

_j





where the last equality comes from π

_Λ⁻¹

(Ω) = Ω × ( P

¹

)

^m

. Whence µ(Ω) = d

^−|n|

Z

Ω×(P¹)^m





m

^

j=1

(π

_j

)

^∗

T b



 ∧ (F

_n

)

∗





m

\

j=1

V

_j





where we used (F

_n

)

∗

V

j

[V

_j

] = [V

_n

].

3.3. Second step: a phase-parameter transfer phenomenon For the sake of simplicity, we let in the sequel V

_k

:= V

^c_n

k

, where n

_k

is given by the large scale condition. Up to extracting a subsequence, we may assume that f

_λ^nk

0

(c

_j

(λ

₀

)) → x

_j

∈ J

_λ0

for all 1 ≤ j ≤ m. We also let x := (x

₁

, . . . , x

_m

) ∈ ( P

¹

)

^m

.

We want to reduce the problem to a purely dynamical datum of f . Building on the large scale condition, one actually gets the following.

Proposition 3.4. — There exist k

0

≥ 1 and δ, α > 0 such that for any k ≥ k

0

, we have Z

Ωk×(P¹)^m





m

^

j=1

(π

j

)

^∗

T b



 ∧ V

_n

k

> 0 . Proof of Proposition 3.4. — Set

I

_k

:=

Z

Ωk×(P¹)^m





m

^

j=1

(π

j

)

^∗

T b



 ∧ V

_n

k

,

and let δ be given by the large scale condition. Let S

_k

be the connected component of V

_n

k

∩ Ω

k

× D

^m_δ

(x) containing (0, V

_n

k

(0)). Up to replacing δ with δ/2, for any k ≥ k

1

, the current [S

_k

]/k[S

_k

]k is of a vertical current of mass 1 in Λ × D

^m_δ

(v

∞

) and

supp([S

_k

]) = S

_k

⊂ Ω

_k

× D

^m_δ

(x) .

As in the case m = 1, let S be any weak limit of the sequence [S

k

]/k[S

_k

]k. Then S is a closed

positive (m, m)-current of mass 1 in B × D

^m_δ

(x) with supp(S) = {λ

₀

} × D

^m_δ

(x) by the large

scale condition. Hence, by extremality of [{λ

₀

} × D

^m_δ

(x)], we have that S = M · [{λ

₀

} × D

^m_δ

(x)],

where M

⁻¹

> 0 is the volume of D

^m_δ

(x) for the volume form V

j

ω

_j

, where ω

_j

= (p

_j

)

^∗

ω

_FS

and p

j

: ( P

¹

)

^m

→ P

¹

is the projection on the j-th coordinate.

As a consequence, [S

_k

]/k[S

_k

]k weakly converges to S as k → ∞. Since V

m

j=1

(π

_j

)

^∗

T b

is the wedge product of (1, 1) current with continuous potentials, we have

m

^

j=1

(π

_j

)

^∗

T b

∧ [S

_k

] k[S

_k

]k →

m

^

j=1

(π

_j

)

^∗

T b

∧ S

so

lim inf

k→∞

k[S

_k

]k

⁻¹

· I

k

≥ lim inf

k→∞

Z

^m

^

j=1

(π

j

)

^∗

T b

∧ [S

k

] k[S

_k

]k ≥

Z

^m

^

j=1

(π

j

)

^∗

T b

∧ S.

By the above, this gives lim inf

k→∞

k[S

_k

]k

⁻¹

· I

_k

≥ M · Z

{λ₀}×D^mδ (x) m

^

j=1

(π

j

)

^∗

T b

≥ Z

^m

^

j=1

(π

_j

)

^∗

T b

∧ [{λ

₀

} × D

^mδ

(x)] . The proof of the proposition directly follows from the following lemma.

Lemma 3.5. — For any δ > 0, and any x = (x

1

, . . . , x

m

) ∈ (J

_f

)

^m

, we have Z

m

^

j=1

(π

j

)

^∗

T b

∧ [{λ

₀

} × D

^mδ

(x)] =

m

Y

j=1

µ

_λ0

( D (x

j

, δ)) > 0.

Proof. — Let us set ω

_j

:= ((π

_j

)

^∗

ω) ˆ |

_λ=λ0

. First, we can remark that g|

_λ=λ0

= g

_λ0

is the Green function of the rational map f

λ0

. We denote by p

j

: (P

¹

)

^k

→ P

¹

the canonical projection onto the j

^th

coordinate. A classical slicing argument gives

((π

_j

)

^∗

dd

^c

g) |

_λ=λ0

= dd

^c_z

(g

_λ0

◦ p

_j

) = (p

_j

)

^∗

dd

^c

g

_λ0

. In particular, since ω

_j

= (p

_j

)

^∗

ω

_FS

, we have

(π

_j

)

^∗

( T b )

λ=λ0

= ((π

_j

)

^∗

(ω

_FS

+ dd

^c

g))|

_λ=λ

0

= (p

_j

)

^∗

(µ

_λ0

) , where µ

λ0

is the maximal entropy measure of f

λ0

, hence

I :=

Z

^m

^

j=1

(π

_j

)

^∗

T b

∧ [{λ

₀

} × D

^m_δ

(x)] = Z

{λ0}×D^m_δ (x) m

^

j=1

(π

_j

)

^∗

T b

= Z

D^m_δ(x) m

^

j=1

(π

j

)

^∗

T b

λ=λ0

= Z

D^m_δ(x) m

^

j=1

(p

j

)

^∗

(µ

λ0

) . Since supp(µ

λ0

) = J

_λ0

and x

1

, . . . , x

m

∈ J

_λ0

, Fubini Theorem yields

I =

m

Y

j=1

Z

D(xj,δ)

µ

_λ0

!

=

m

Y

j=1

µ

_λ0

( D (x

j

, δ)) > 0 ,

which ends the proof of Lemma 3.5.

The proof of Theorem 3.2 directly follows from Lemma 3.3 and Proposition 3.4.

4. Misiurewicz maps and the generalized large scale condition

Fix d ≥ 2 and pick 1 ≤ k ≤ 2d − 2. A rational map f ∈ Rat

d

is k-Misiurewicz if the following properties hold:

• f has no parabolic periodic points,

• f has k critical points in its Julia set, counted with multiplicity,

• for any c ∈ C(f ) ∩ J

_f

, we have ω(c) ∩ C(f ) = ∅.

In this section, we want to emphasize that the large scale condition is the good condition for proving that specific parameters lie in the support of the bifurcation measure. Our motivation here is also to provide a simpler and more intrinsic proof of Theorem 1.4 of [G1]. Recall that for a critical point c

j

, we denote by v

j

the critical value f (c

j

). According to Theorem 3.2, it is sufficient to prove the existence of a (local) submanifold in which f satisfies the (generalized) large scale condition. More precisely, we prove the following.

Theorem 4.1. — Let f ∈ Rat

_d

and 1 ≤ k ≤ 2d − 2. Assume that f is k-Misiurewicz and that C(f ) ∩ J

_f

= {c

₁

, . . . , c

k

}. Assume that for all n ∈ N and all i 6= j ≤ k, f

ⁿ

(v

j

) 6= v

i

. If f is not a flexible Latt` es map, then there exists a k-dimensional local submanifold Λ

_f

⊂ Rat

_d

such that f satisfies the generalized large scale condition for (c

₁

, . . . , c

_k

) in Λ

_f

.

This approach allows us to exhibit the key expansion and distortion arguments, without using more elaborate tools that will necessarily be missing in a more general situation, such as linearizing coordinates along repelling orbits.

Observe also that the condition ”for all n ∈ N and all i 6= j ≤ k, f

ⁿ

(v

_j

) 6= v

_i

” is not an issue: any k-Misiurewicz map that does not satisfy this condition can be approximated by k- Misiurewicz maps that do satisfy it using Montel theorem. In particular, it is in the support of the bifurcation current T

1

∧ · · · ∧ T

k

.

4.1. Hyperbolic sets and holomorphic motions

We recall some classical definitions and facts. Let f ∈ Rat

_d

and E ⊂ P

¹

be a non-empty compact f -invariant set, i.e. such that f(E) ⊂ E. We say that E is f -hyperbolic if one of the following equivalent conditions is satisfied:

1. there exist C > 0 and α > 1 such that |(f

ⁿ

)

⁰

(z)| ≥ Cα

ⁿ

for all z ∈ E and all n ≥ 0, 2. for some appropriate metric on P

¹

, there exists K > 1 such that |f

⁰

(z)| ≥ K for all z ∈ E.

Such a constant K is called a hyperbolicity constant for f .

Recall also that a holomorphic motion of a set X ⊂ P

¹

over a complex manifold Λ centered at λ

0

∈ Λ is a map h : Λ × X → P

¹

such that:

• h

λ0

:= h(λ

0

, ·) : X → P

¹

is the identity map,

• h

_λ

:= h(λ, ·) : X → P

¹

is injective for all λ ∈ Λ,

• λ 7→ h(λ, x) is holomorphic on Λ for all x ∈ X.

First, notice that the classical λ-lemma of Ma˜ n´ e-Sad-Sullivan [MSS] says that any holomor-

phic motion of X extends continuously to a holomorphic motion of the closure of X. It is known

that a hyperbolic set E admits a natural holomorphic motion (see, e.g., [S, Property (1.2) page

229] or [J, Theorem C]).

Theorem 4.2. — Let (f

_λ

)

_λ∈B(0,r)

be a holomorphic family of degree d rational maps parametrized by a ball B (0, r) ⊂ C

^m

. Let E

0

⊂ P

¹

be a compact f

0

-hyperbolic set. Then there exist 0 < ρ ≤ r and a unique holomorphic motion

h : B(0, ρ) × E

0

−→ P

¹

(λ, z) 7−→ h

_λ

(z) centered at 0 and such that f

_λ

◦ h

_λ

(z) = h

_λ

◦ f

₀

(z) for all z ∈ E

₀

.

The proof of Theorem 4.2 relies on the compactness of E

₀

and on the next lemma, which is an immediate corollary of the Implicit Function Theorem. Let K > 1 be a hyperbolicity constant for E

₀

for a suitable metric α. In what follows, we denote by |.|

_α

, D

α

the distance and disk with respect to that metric. Up to reducing K and r, we can find a δ-neighborhood N

_δ

of E

0

in P

¹

such that

(3) |f

_λ⁰

(z)|

_α

≥ K > 1 for all (z, λ) ∈ N

_δ

× B(0, r) .

Lemma 4.3. — Under the assumption of Theorem 4.2, there exist ε > 0 and 0 < ρ ≤ r, such that for all z

₀

∈ E

₀

, there exists a map f

_z⁻¹

0,λ

(w) which depends holomorphically on (λ, w) ∈ B (0, ρ) × D

α

(f

₀

(z

₀

), ε) and taking values in D

α

(z

₀

, ε) which satisfies

1. f

_z⁻¹_0,0

f

₀

(z

₀

)

= z

₀

, 2. f

_λ

f

_z⁻¹

0,λ

(w)

= w for all (λ, w) ∈ B (0, ρ) × D

α

(f

₀

(z

₀

), ε), and 3.

f

_z⁻¹

0,λ

0

(w)

_α

≤

_K¹

for all (λ, w) ∈ B (0, ρ) × D

α

(f

₀

(z

₀

), ε).

4.2. Transversality for Misiurewicz maps

It is well known that if f is a k-Misiurewicz degree d rational map but not a flexible Latt` es map, then all periodic Fatou components of f are attracting basins and f does not carry any invariant line field on its Julia set. Moreover, there exists a positive integer n

₀

≥ 1 such that the set

E

_f

:= {f

ⁿ

(c) ; n ≥ n

₀

and c ∈ C(f) ∩ J

_f

} is a compact hyperbolic f-invariant set (see e.g. [Asp1, G1]).

If f ∈ Rat

d

is k-Misiurewicz, C(f ) ∩ J

_f

= {c

₁

, . . . , c

k

} and, for all n ∈ N and all i 6= j ≤ k, f

ⁿ

(v

j

) 6= v

i

, where v

i

= f (c

i

), we let h : B(f, r) × E

f

→ P

¹

be the dynamical holomorphic motion of E

_f

. Since all the critical points in the Julia set of f are simple (v

_i

6= v

_j

), we may follow them holomorphically in the neighborhood of f , as well as the critical values: denote by v

i

(λ), 1 ≤ i ≤ k, the corresponding critical values of f

_λ

.

Given a holomorphic curve λ 7→ f

_λ

∈ Rat

_d

passing through f = f

_λ0

, it will be convenient to denote by η the meromorphic vector field on P

¹

defined by

η(z) :=

d

dλ|λ=λ0

f

_λ

(z) f

⁰

(z) .

Recall that the pullback of η by f

ⁿ

is given in coordinates by (f

ⁿ

)

^∗

η(z) =

^η◦f_(fn)^n0(z)(z)

. As before, let ξ

_nⁱ

(λ) := f

_λⁿ

(v

i

(λ)).

Lemma 4.4. — Let (f

_λ

)

λ∈Λ

be a holomorphic curve of rational maps with a marked critical

value v

_i

(λ). Assume that the orbit of v

_i

:= v

_i

(λ

₀

) does not meet the critical set of f := f

_λ0

. We

have, for all n ≥ 1: : 1 (f

ⁿ

)

⁰

(v

i

)

d dλ

|λ=λ0

ξ

_nⁱ

(λ) = d dλ

|λ=λ0

v

i

(λ) +

n−1

X

k=0

(f

^k

)

^∗

η(v

i

).

Moreover, if v

i

lies in a hyperbolic set E

f

and h

λ

(v

i

) denote its holomorphic motion, then d

dλ

|λ=λ0

h

_λ

(v

_i

) = −

+∞

X

n=0

(f

ⁿ

)

^∗

η(v

_i

).

Proof. — To lighten the notations, we will denote by ˙ v

i

, ˙ f (z), etc. the derivatives

_dλ|λ=λ^d

0

v

i

(λ),

d

dλ|λ=λ₀

f

_λ

(z), etc.

We have: ξ

_nⁱ

(λ) = f

_λⁿ

(v

i

(λ)) = f

_λ

◦ f

_λⁿ⁻¹

(v

i

(λ)), so d

dλ

|λ=λ0

ξ

ⁱ_n

(λ) = ˙ f ◦ f

ⁿ⁻¹

(v

i

) + f

⁰

◦ f

ⁿ⁻¹

(v

i

) · d dλ

|λ=λ0

ξ

_n−1ⁱ

(λ).

Therefore

d

dλ|λ=λ0

ξ

_nⁱ

(λ)

(f

ⁿ

)

⁰

(v

_i

) = η ◦ f

ⁿ⁻¹

(v

i

) (f

ⁿ⁻¹

)

⁰

(v

_i

) +

d

dλ|λ=λ0

ξ

_n−1ⁱ

(λ)

(f

ⁿ⁻¹

)

⁰

(v

_i

) = (f

ⁿ⁻¹

)

^∗

η(v

i

) +

d

dλ|λ=λ0

ξ

_n−1ⁱ

(λ) (f

ⁿ⁻¹

)

⁰

(v

_i

) . The first statement then follows by induction on n.

Let us now prove the second statement. By Theorem 4.2, there exists a holomorphic family of homeomorphisms h

λ

: E

f

→ P

¹

, with h

λ0

= Id and such that

h

λ

◦ f = f

λ

◦ h

λ

. Differentiating with respect to λ at λ = λ

0

, we get:

h ˙ ◦ f = ˙ f + f

⁰

· h, ˙

which we may rewrite as f

^∗

h ˙ − h ˙ = η. Since E

_f

is hyperbolic, the operator f

^∗

is strictly contracting on the space of vector fields on E

_f

, and therefore

h(v ˙

i

) = −

+∞

X

n=0

(f

ⁿ

)

^∗

η(v

i

), ending the proof.

We can deduce the following from Theorem B of [Ast] (see the discussion after Theorem B in [Ast]):

Proposition 4.5. — Let f ∈ Rat

_d

be k-Misiurewicz, C(f )∩J

_f

= {c

₁

, . . . , c

_k

} and, for all n ∈ N and all i 6= j ≤ k, f

ⁿ

(v

j

) 6= v

i

. Then, there exists a local complex submanifold Λ

f

⊂ Rat

d

which contains f , such that the holomorphic map

λ ∈ Λ

_f

7−→ (v

₁

(λ) − h

_λ

(v

₁

), . . . , v

_k

(λ) − h

_λ

(v

_k

)) ∈ C

^k

defines local coordinates at f

0

= f in Λ

f

. Furthermore, the limits

τ

i

:= lim

n→∞

Dξ

_nⁱ

(0) (f

ⁿ

)

⁰

(v

i

)

exist as linear maps on T

₀

Λ

_f

, for 1 ≤ i ≤ k, and τ

₁

, . . . , τ

_k

are linearly independent.

4.3. The generalized large scale condition for k-Misiurewicz maps We now turn to the proof of Theorem 4.1.

Proof of Theorem 4.1. — Consider the set Ω

n

:=

λ ∈ Λ

f

, ∀i ≤ k, |h

_λ

(v

i

) − v

i

(λ)| ≤ 1

|(f

ⁿ

)

⁰

(v

i

)|

,

then, by Proposition 4.5, it can written as L

⁻¹_n

(Ω) where L

_n

(t

₁

, . . . , t

_k

) =

t

₁

|(f

ⁿ

)

⁰

(v

₁

)| , . . . , t

_k

|(f

ⁿ

)

⁰

(v

_k

)|

and Ω is some polydisk around f in Λ

_f

. For all 1 ≤ i ≤ k, let φ

ⁱ_n

: Ω → C be defined by:

φ

ⁱ_n

(λ) := ξ

_nⁱ

λ

|(f

ⁿ

)

⁰

(v

_i

)|

− ξ

_nⁱ

(0).

Let φ

_n

: Ω → C

^k

be defined by φ

_n

:= (φ

ⁱ_n

)

_1≤i≤k

.

We start by proving that the sequence (φ

n

)

n∈N

is bounded, hence normal, on some small polydisk ˜ ρ · Ω. Let i ≤ k, let us bound |ξ

ⁱ_n

(λ) − ξ

_nⁱ

(0)|. We have:

|ξ

ⁱ_n

(λ) − ξ

ⁱ_n

(0)| = |f

_λⁿ

(v

i

(λ)) − f

ⁿ

(v

i

)|

≤ |f

_λⁿ

(v

_i

(λ)) − f

_λⁿ

(h

_λ

(v

_i

))| + |f

_λⁿ

(h

_λ

(v

_i

)) − f

ⁿ

(v)|.

(4)

By chaining Lemma 4.3, there exists > 0 such that for all λ small enough, for all n ∈ N , there is an inverse branch g

_n,λ

of f

_λⁿ

that is well-defined on D

α

(f

_λⁿ

(h

_λ

(v

_i

)), ε), and that maps f

_λⁿ

(h

t

(v

i

)) to h

λ

(v

i

). As the standard metric and the metric α are equivalent, we have that the map g

_n,λ

is univalent on some disk D (f

_λⁿ

(h

_λ

(v

_i

)), C · ε) for some constant C that does not depend on n nor λ. By Koebe’s distortion theorem, the distortion of g

_n,λ

is uniformly bounded on D (f

_λⁿ

(h

_λ

(v

i

)), C · ε/2).

We prove by induction on n that there is a constant 0 < ρ < ˜ 1 depending only on f such that if |h

_λ

(v

_i

) − v

_i

(λ)| ≤

_|(fn)^ρ˜0(vi)|

, then |f

_λⁿ

(v

_i

(λ)) − f

_λⁿ

(h

_λ

(v

_i

))| ≤

^Cε₄

. Assume it is true for n − 1.

Observe first that by Lemma 6.6 of [G1], there exists C

₁

> 0 depending only on f and on the metric, such that we have:

(5) max |(f

_λⁿ⁻¹

)

⁰

(h

_λ

(v

_i

))|

|(f

ⁿ⁻¹

)

⁰

(v

_i

)| , |(f

ⁿ⁻¹

)

⁰

(v

_i

)|

|(f

_λⁿ⁻¹

)

⁰

(h

_λ

(v

i

))|

!

≤ e

^{(n−1)C1kλk}

≤ 2,

by our choice of λ. We can thus apply the induction hypothesis and the bound on the distortion of g

n−1,λ

, and then (5) to find:

|f

_λⁿ⁻¹

(v

_i

(λ)) − f

_λⁿ⁻¹

(h

_λ

(v

_i

))| ≤ 2|(f

_λⁿ⁻¹

)

⁰

(h

_λ

(v

_i

))| · |v

_i

(λ) − h

_λ

(v

_i

)|

≤ 4|(f

ⁿ⁻¹

)

⁰

(v

i

)| · |v

_i

(λ) − h

_λ

(v

i

)|.