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Bifurcation measure and
postcritically finite rational
maps
Xavier Buff and Adam L. Epstein
Bassanelli and Berteloot [BB07] have defined a bifurcation measure µbif on the
moduli space Md of rational maps of degree d ≥ 2. They have proved that it is
a positive measure of finite mass and that its support is contained in the closure
of the set Zd of conjugacy classes of rational maps of degree d having 2d − 2
indifferent cycles.
Denote by Xd the set of conjugacy classes of strictly postcritically finite rational maps of degree d which are not flexible Lattès examples. We prove that
Supp(µbif ) = Xd = Zd . Our proof is based on a transversality result due to the
second author.
A similar result was obtained with different techniques by Dujardin and Favre
[DF] for the bifurcation measure on moduli spaces of polynomials of degree d ≥ 2.
Introduction
A result of Lyubich [Lyu83] asserts that for each rational map f : P1 → P1
of degree d ≥ 2, there is a unique probability measure µf of maximal
entropy log d. It is ergodic, satisfies f ∗ µf = d · µf and is carried outside
the exceptional set of f . This measure is called the equilibrium measure of
f.
The Lyapunov exponent of f with respect to the measure µf can be
defined by
Z
logDf dµf ,
L(f ):=
P1
where k · k is any smooth metric on P1 . Since µf is ergodic, the quantity
eL(f ) records the average rate of expansion of f along a typical orbit with
respect to µf .
Denote by Ratd the set of rational maps of degree d. It is a smooth
complex manifold of dimension 2d + 1. The function L : Ratd → R is
continuous [Mañ88] and plurisubharmonic [DeM03] on Ratd . The positive
(1, 1)-current
Tbif :=ddc L
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1. Bifurcation measure and Misiurewicz maps
is called the bifurcation current in Ratd .
The bifurcation locus in Ratd is the closure of the set of discontinuity
of the map f ∈ Ratd 7→ Jf , where Jf stands for the Julia set of f and
the continuity is for the Hausdorff topology for compact subsets of P1 .
DeMarco [DeM03] proved that the support of the bifurcation current Tbif
is equal to the bifurcation locus.
Given f ∈ Ratd , denote by O(f ) the set of rational maps which are
conjugate to f by a Möbius transformation. It is a smooth submanifold
of Ratd of complex dimension 3. It is isomorphic to Aut(P1 )/Γ, where
Aut(P1 ) ≃ PSL(2, C) is the group of Möbius transformations, and where
Γ ⊂ Aut(P1 ) is the finite subgroup of Möbius transformations which commute with f .
The moduli space Md is the quotient Ratd /Aut(P1 ), where Aut(P1 )
acts on Ratd by conjugation. It is an orbifold of complex dimension 2d − 2.
It is a normal, quasiprojective variety. We denote by Π : Ratd → Md the
canonical projection.
The Lyapunov exponent is invariant under holomorphic conjugacy, thus
is constant on the orbits O(f ). The map L : Ratd → R descends to a map
L̂ : Md → R which is continuous, bounded from below and plurisubharmonic on Md (see [BB07] proposition 6.2). The measure
µbif :=(ddc L̂)∧(2d−2)
is called the bifurcation measure on Md . By construction, we have
Π∗ µbif = Tbif ∧(2d−2) .
The critical set C(f ) of a rational map f is the set of critical points of
f . The postcritical set is the set:
[ [
P(f ) =
f ◦n (ω).
ω∈C(f ) n≥1
A rational map f is postcritically finite if P(f ) is finite. It is strictly postcritically finite if P(f ) is finite and if f does not have any superattracting
cycle. It is a Lattès map if it is obtained as the quotient of an affine map
A : z 7→ az +b on a complex torus C/Λ: there is a finite-to-one holomorphic
map Θ : C/Λ → P1 such that the following diagram commutes:
C/Λ
A
/ C/Λ
Θ
Θ
P1
f
/ P1 .
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A Lattès example is strictly postcritically finite (see [Mil06]). It is a flexible
Lattès example if we can choose Θ with degree 2 and A(z) = az + b with
a > 1 an integer.
Let us introduce the following notation:
• Xd ⊂ Md for the set of conjugacy classes of strictly postcritically
finite rational maps of degree d which are not flexible Lattès examples;
• Xd∗ ⊂ Xd for the subset of conjugacy classes of strictly postcritically
finite rational maps of degree d which have simple critical points and
satisfy C(f ) ∩ P(f ) = ∅;
• Zd ⊂ Md for the set of conjugacy classes of maps having 2d − 2
indifferent cycles (we do not count multiplicities).
Bassanelli and Berteloot [BB07] proved that
• the bifurcation measure does not vanish identically on Md and has
finite mass,
• the conjugacy class of any non-flexible Lattès map lies in the support
of µbif ,
• the support of µbif is contained in the closure of Zd .
Our main result is the following.
Main Theorem The support of the bifurcation measure in Md is:
Supp(µbif ) = Xd∗ = Xd = Zd .
Remark We shall prove in section 1 below that
Xd∗ = Xd = Zd .
(0.1)
Then, thanks to the results in [BB07], we will only have to prove the
inclusion
Xd∗ ⊂ Supp(µbif ).
Remark We do not know how to prove that the conjugacy classes of flexible
Lattès examples lie in the support of µbif . It would be enough to prove that
every flexible Lattès example can be approximated by strictly postcritically
finite rational maps which are not Lattès examples.
Remark The underlying idea of the proof is a potential-theoretic interpre-
tation of a result of Tan Lei [Tan90] concerning the similarities between
the Mandelbrot set and Julia sets.
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1. Bifurcation measure and Misiurewicz maps
Our proof relies on a transversality result that we will now present.
Assume f0 ∈ Ratd is a strictly postcritically finite rational map which
has 2d − 2 distinct critical points ωj (f0 ), j ∈ {1, . . . , 2d − 2}, and satisfies
C(f0 ) ∩ P(f0 ) = ∅. There are integers ℓj ≥ 1 such that
◦ℓj
αj (f0 ):=f0
ωj (f0 )
are periodic points of f0 . The critical points are simple and the periodic
points are repelling. By the Implicit Function Theorem, there are
• analytic germs ωj : (Ratd , f0 ) → P1 following the critical points of f
as f ranges in a neighborhood of f0 in Ratd and
• analytic germs αj : (Ratd , f0 ) → P1 following periodic points of f as
f ranges in a neighborhood of f0 in Ratd .
Let F : (Ratd , f0 ) → (P1 )2d−2 and G : (Ratd , f0 ) → (P1 )2d−2 be defined
by :


F := 
F1
..
.
F2d−2



with Fj (f ):=f ◦ℓj
ωj (f )


and G:= 
α1
..
.
α2d−2


.
Denote by Df0 F and Df0 G the differentials of F and G at f0 .
The transversality result we are interested in is the following.
Proposition 1 If f0 is not a flexible Lattès example, then the linear map
Df0 F − Df0 G : Tf0 Ratd → Tα1 (f0 ) P1 × · · · × Tα2d−2 (f0 ) P1
has rank 2d − 2. The kernel of Df0 F − Df0 G is the tangent space to O(f0 )
at f0 .
There are several proofs of this result. Here, we include a proof due to
the second author. A different proof was obtained by van Strien [vS00].
His proof covers a more general setting where the critical orbits are allowed
to be preperiodic to a hyperbolic set. Our proof covers the case of maps
commuting with a non trivial group of Möbius transformations (a slight
modification of van Strien’s proof probably also covers this case).
Remark A similar transversality theorem holds in the space of polynomials
of degree d and the arguments developed in this article lead to an alternative proof of the result of Dujardin and Favre.
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1. The sets Xd∗ , Xd and Zd
5
Acknowledgements
We would like to thank our colleagues in Toulouse who brought this problem to our attention, in particular François Berteloot, Julien Duval and
Vincent Guedj. We would like to thank John Hubbard for his constant
support.
1 The sets Xd∗, Xd and Zd
In this section, we prove equality (0.1): Xd∗ = Xd = Zd . Since Xd∗ ⊂ Xd , it
is enough to prove that
Xd ⊂ Zd
and Zd ⊂ Xd∗ .
1.1 Tools
Our proof relies on the following three results.
The first result is an immediate consequence of the Fatou-Shishikura
inequality on the number of non repelling cycles of a rational map (see
[Shi87] and/or [Eps99]). For f ∈ Ratd , denote by
• Natt (f ) the number of attracting cycles of f ,
• Nind (f ) the number of distinct indifferent cycles of f and
• Ncrit (f ) the number of critical points of f , counting multiplicities,
whose orbits are strictly preperiodic to repelling cycles.
Theorem 1 For any rational map f ∈ Ratd , we have:
Ncrit (f ) + Nind (f ) + Natt (f ) ≤ 2d − 2.
The second result is a characterization of stability due to Mañe, Sad
and Sullivan [MSS83] (compare with [McM94] section 4.1). Let Λ be a
complex manifold. A family of rational maps {fλ }λ∈Λ is an analytic family
if the map (λ, z) ∈ Λ × P1 7→ fλ (z) ∈ P1 is analytic. The family is stable
at λ0 ∈ Λ if the number of attracting cycles of fλ is locally constant at λ0 .
Theorem 2 Let {fλ }λ∈Λ be an analytic family of rational maps parametrized
by a complex manifold Λ. The following assertions are equivalent.
• The family is stable at λ0 .
• For all µ ∈ S 1 and p ≥ 1, the number of cycles of fλ having multiplier
µ and period p is locally constant at λ0 .
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1. Bifurcation measure and Misiurewicz maps
• For all ℓ ≥ 1 and p ≥ 1, the number of critical points ω of fλ such
that f ◦ℓ (ω) is a repelling periodic point of period p is locally constant
at λ0 .
• The maximum period of an indifferent cycle of fλ is locally bounded
at λ0 .
• The maximum period of a repelling cycle of fλ contained in the postcritical set P(fλ ) is locally bounded at λ0 .
The third result is due to McMullen [McM87]. Assume Λ is an irreducible quasiprojective complex variety. A family of rational maps {fλ }λ∈Λ
is an algebraic family if the map (λ, z) ∈ Λ × P1 7→ fλ (z) ∈ P1 is a rational
mapping. The family is trivial if all its members are conjugate by Möbius
transformations. The family is stable if it is stable at every λ ∈ Λ.
Theorem 3 A stable algebraic family of rational maps is either trivial or it
is a family of flexible Lattès examples.
1.2 Spaces of rational maps with marked critical points and marked
periodic points
crit,per
n
In the sequel, it will be convenient to consider the set Ratd
of rational
maps of degree d with marked critical points and marked periodic points
of period dividing n. This can be defined as follows.
First, the unordered sets of m points in P1 can be identified with Pm .
This yields a rational map σm : (P1 )m → Pm which might be defined as
follows:
σm [x1 : y1 ], . . . [xm : ym ] = [a0 : · · · : am ]
with
m
X
aj xj y m−j =
j=0
m
Y
(xyi − yxi ).
i=1
Second, to a rational map f ∈ Ratd , we associate the unordered set
{ω1 , . . . , ω2d−2 } of its 2d−2 critical points, listed with repetitions according
2d−2
to their multiplicities. This induces a rational map
Crit : Ratd → P
which might be defined as follows: if f [x : y] = P (x, y) : Q(x, y) with
P and Q homogeneous polynomials of degree d, then
Crit(f ) = [a0 : · · · : a2d−2 ] with
2d−2
X
j=0
aj xj y 2d−2−j =
∂P ∂Q ∂P ∂Q
−
.
∂x ∂y
∂y ∂x
Third, given an integer n ≥ 1, to a rational map f ∈ Ratd , we associate
the unordered set {α1 , . . . , αdn +1 } of the fixed points of f ◦n , listed with
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1. The sets Xd∗ , Xd and Zd
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repetitions according to their multiplicities. This induces a rational map
2n +1
Pern : Ratd →
which might be defined as follows:
P
◦n
if f
[x : y] = Pn (x, y) : Qn (x, y) with Pn and Qn homogeneous polynomials of degree dn , then
Pern (f ) = [a0 : · · · : adn +1 ]
with
n
dX
+1
aj xj y d
n
+1−j
= yPn (x, y) − xQn (x, y).
j=0
The space of rational maps of degree d with marked critical points and
marked periodic points of period dividing n is the set:
n
(f, ω, α) ∈ Ratd × (P1 )2d−2 × (P1 )d +1 such that
crit,pern
.
Ratd
:=
Crit(f ) = σ2d−2 (ω) and Pern (f ) = σdn +1 (α)
crit,per
n
n
Then, Ratd
is an algebraic subset1 of Ratd × (P1 )2d−2 × (P1 )d +1 .
Since there are 2d − 2 + dn + 1 equations, the dimension of any irreducible
crit,pern
component of Ratd
is at least that of Ratd , i.e. 2d + 1. Since the
crit,pern
fibers of the projection Ratd
→ Ratd are finite (a rational map has
finitely many critical points and finitely many periodic points of period
dividing n), the dimension of any component is exactly 2d + 1.
1.3 The inclusion Xd ⊂ Zd
Assume f0 ∈ Ratd is a strictly postcritically finite rational map but not
a flexible Lattès example. We must show that any neighborhood of f0 in
Ratd contains a rational map with 2d − 2 indifferent cycles. This is an
immediate consequence of the following lemma.
Lemma 1 Assume r ≥ 1 and s ≥ 0 are integers such that r + s = 2d − 2.
Assume f0 ∈ Ratd is not a flexible Lattès example, Ncrit (f0 ) = r and
Nind (f0 ) = s. Then, arbitrarily close to f0 , we can find a rational map f1
such that Ncrit (f1 ) = r − 1 and Nind (f1 ) = s + 1.
Proof: Let p1 , . . . , ps be the periods of the indifferent cycles of f0 . Denote
crit,perM
by M their least common multiple. A point λ ∈ Ratd
is of the form
gλ , ω1 (λ), . . . , ω2d−2 (λ), α1 (λ), . . . αdM +1 (λ) .
crit,perM
We choose λ0 ∈ Ratd
so that
1 By algebraic we mean a quasiprojective variety in some projective space. It is
classical that any product of projective spaces embeds in some PN . Thus, it makes
sense to speak of algebraic subsets of products of projective spaces
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1. Bifurcation measure and Misiurewicz maps
• gλ0 = f0 ,
• ω1 (λ0 ), . . . ωr (λ0 ) are preperiodic to repelling cycles of f0 and
• α1 (λ0 ),. . . , αs (λ0 ) are indifferent periodic points of f0 belonging to
distinct cycles.
For i ∈ [1, r], let ki ≥ 1 and pi ≥ 1 be the least integers such that
gλ◦k0 i ωi (λ0 ) = gλ◦k0 i +pi ωi (λ0 ) .
For j ∈ [1, s], let µj be the multiplier of gλ◦M
at αj (λ0 ). In particular,
0
µj ∈ S 1 .
Consider the algebraic subset
∀i ∈ [1, r − 1], gλ◦ki ωi (λ) = gλ◦ki +pi ωi (λ) and
crit,perM
λ ∈ Ratd
;
∀j ∈ [1, s], the multiplier of gλ◦M at αj (λ) is µj
and let Λ be an irreducible component containing λ0 . Note that there are
r − 1 + s = 2d − 3 equations. It follows that the dimension of Λ is at least
(2d + 1) − (2d − 3) = 4. In particular, the image of Λ by the projection
crit,perM
Ratd
→ Ratd cannot be contained in O(f0 ).
Embedding Λ in some PN (with N sufficiently large) and slicing with an
appropriate projective subspace, we deduce that there is an algebraic curve
crit,perM
Γ ⊂ Λ containing λ0 , whose image by the projection Ratd
→ Ratd
is not contained in O(f0 ). Desingularizing the algebraic curve, we obtain a
smooth quasiprojective curve Σ (i.e. a Riemann surface of finite type) and
an algebraic map Σ → Γ which is surjective and generically one-to-one.
We let σ0 ∈ Σ be a point which is mapped to λ0 ∈ Γ. With an abuse
of notation,
we write gσ , ωi (σ) and αj (σ) in place of gλ(σ) , ωi λ(σ) and
αj λ(σ) . Then, we have an algebraic family of rational maps {gσ }σ∈Σ
parametrized by a smooth quasiprojective curve Σ, coming with marked
critical points ωi (σ) and marked periodic points αj (σ).
The family is not trivial and gσ0 is not a flexible Lattès example. Assume that the family were stable at σ0 . Then, according to theorem 2, the
number of critical points which are preperiodic to repelling cycles would
be locally constant at σ0 . Thus, the critical orbit relation
gσ◦kr (ωr σ) = gσ◦kr +pr (ωr σ)
would hold in a neighborhood of σ0 in Σ, thus for all σ ∈ Σ by analytic continuation. As a consequence, for all σ ∈ Σ, we would have
Ncrit (gσ ) + Nind (gσ ) = 2d − 2. According to theorem 1, this would imply that Natt (gσ ) = 0 for all σ ∈ Σ. The family would be stable which
would contradict theorem 3.
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1. The sets Xd∗ , Xd and Zd
9
Thus, the family {gσ }σ∈Σ is not stable at σ0 . According to theorem 2,
the maximum period of an indifferent cycle of gσ is not locally bounded at
σ0 . Thus, we can find a parameter σ1 ∈ Σ arbitrarily close to σ0 such that
gσ1 has at least s + 1 indifferent cycles.
1.4 The inclusion Zd ⊂ Xd∗
The proof follows essentially the same lines as the previous one. We begin
with a rational map f0 ∈ Ratd having 2d − 2 indifferent cycles. We must
show that arbitrarily close to f0 , we can find a postcritically finite rational
map f1 ∈ Ratd which has simple critical points, satisfies C(f1 ) ∩ P(f1 ) = ∅
and is not a flexible Lattès example.
The following lemma implies (by induction) that arbitrarily close to f0 ,
we can find a rational map f1 ∈ Ratd whose postcritical set contains 2d − 2
distinct repelling cycles. Automatically, the 2d − 2 critical points of f1
have to be simple, strictly preperiodic, with disjoint orbits. In particular,
C(f1 )∩P(f1 ) = ∅. In addition, Lattès examples lie in a Zariski closed subset
of Ratd . Since f0 has indifferent cycles, it cannot be a Lattès examples.
So, if f1 is sufficiently close to f0 , then f1 is not a Lattès example.
Lemma 2 Assume r ≥ 0 and s ≥ 1 are integers such that r + s = 2d − 2.
Assume f0 ∈ Ratd satisfies Ncrit (f0 ) = r, Nind (f0 ) = s, the postcritical
set of f0 containing r distinct repelling cycles. Then, arbitrarily close to
f0 , we can find a rational map f1 which satisfies Ncrit (f1 ) = r + 1 and
Nind (f1 ) = s − 1, the postcritical set of f1 containing r + 1 distinct repelling
cycles.
Proof: The proof is similar to the proof of lemma 1; we do not give all the
details. First , note that since s ≥ 1, f0 has an indifferent cycle, and thus,
is not a Lattès example. Second, we can find a non trivial algebraic family
of rational maps {gσ }σ∈Σ parametrized by a smooth quasiprojective curve
Σ with marked critical points ω1 (σ), . . . , ω2d−2 (σ) and marked periodic
points α1 (σ), . . . , αdM +1 (σ) such that
• gσ0 = f0 ,
• for j ∈ [1, s], the periodic points αj (σ0 ) belong to distinct indifferent
cycles of f0 ,
• for all σ ∈ Σ and all i ∈ [1, r], the critical point ωi (σ) is preperiodic
to a periodic cycle of gσ and
• for all σ ∈ Σ and all j ∈ [1, s − 1], the periodic points αj (σ) are
indifferent periodic points of gσ .
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1. Bifurcation measure and Misiurewicz maps
If the family were stable at σ0 , the indifferent periodic point αs (σ0 ) would
be persistently indifferent in a neighborhood of σ0 in Σ, thus in all Σ by
analytic continuation. The relation Ncrit (gσ ) + Nind (gσ ) = 2d − 2 would
hold throughout Σ. Thus, for all σ ∈ Σ, we would have Natt (gσ ) = 0 and
the family would be stable. This is not possible since gσ0 is not a Lattès
example and since the family {gσ }σ∈Σ is not trivial.
It follows that the period of a repelling cycle contained in the postcritical
set of gσ is not locally bounded at σ0 . In addition, for i ∈ [1, r], the critical
points ωi (σ0 ) of gσ0 are preperiodic to distinct repelling cycles. Thus, we
can find a rational map gσ1 , with σ1 arbitrarily close to σ0 , such that gσ1
has r + 1 critical points preperiodic to distinct repelling cycles.
2 Transversality
In this section, we prove proposition 1. Assume f0 ∈ Ratd is a strictly
postcritically finite rational map which has 2d − 2 distinct critical points
and satisfies C(f0 ) ∩ P(f0 ) = ∅. Assume the analytic germs
F : (Ratd , f0 ) → (P1 )2d−2
and G : (Ratd , f0 ) → (P1 )2d−2
are defined as in the introduction. We want to show that the rank of
Df0 F − Df0 G is 2d − 2.
Since Tf0 Ratd has complex dimension 2d + 1, the rank of Df0 F − Df0 G
is 2d − 2 if and only if the kernel has dimension 3.
Note that on O(f0 ), we have F ≡ G. Thus, every tangent vector to
O(f0 ) at f0 is in the kernel of Df0 F − Df0 G. Since O(f0 ) has complex
dimension 3, we must therefore show that every vector in the kernel of
Df0 F − Df0 G is tangent to O(f0 ).
Thus, it is enough to show that if {ft } is a holomorphic family of rational
maps of degree d ≥ 2, parametrized by a neighborhood of t = 0 in C and if
ξ :=
dft dt t=0
belongs to Ker(Df0 F − Df0 G), then ξ is a tangent vector to O(f0 ).
2.1 The tangent vectors to Ratd .
Assume {ft } is a holomorphic family of rational maps of degree d ≥ 2,
dft parametrized by a neighborhood of t = 0 in C, and assume ξ :=
.
dt t=0
1
Then, for all z ∈ P , the vector ξ(z) is tangent to the curve t 7→ ft (z) at
t = 0. This vector belongs to Tf0 (z) P1 .
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2. Transversality
11
In other words, if f ∈ Ratd and if ξ ∈ Tf Ratd , then for all z ∈ P1 ,
ξ(z) ∈ Tf (z) P1 . If z is not a critical point of f , then
Dz f : Tz P1 → Tf (z) P1
is an isomorphism and we set:
ηξ (z) := Dz f
−1
ξ(z) .
We obtain a vector field ηξ defined and holomorphic on P1 \ C(f ).
Remark Assume {ft } is a holomorphic family of rational maps, parametrized
dft . If z0 is not a critical
dt t=0
point of f0 , the equation ft (z) = f0 (z0 ) defines implicitly a map z(t),
holomorphic in a neighborhood of t = 0 with z(0) = z0 . The vector
ηξ (z0 ) ∈ Tz0 P1 is tangent to the curve t 7→ z(t) at t = 0.
by a neighborhood of t = 0 in C, with ξ =
Remark If f has simple critical points, then the vector field ηξ has simple
poles or removable singularities along C(f ). This can be seen by working
in coordinates, using the fact that f ′ has simple zeroes at points of C(f ).
Proposition 2 Assume f ∈ Ratd and assume {φt } is an analytic family of
Möbius transformations parametrized by a neighborhood of t = 0 in C, with
dφt φ0 = Id. Let θ be the holomorphic vector field on P1 defined by θ:=
.
dt t=0
Then,
d(φt ◦ f ◦ φt−1 ) ⇐⇒ ηξ = f ∗ θ − θ.
ξ=
dt
t=0
Remark The vector field f ∗ θ is meromorphic on P1 and holomorphic on
P1 \ C(f ). If z is not a critical point of f , then
−1
(f ∗ θ)(z):= Dz f
θ(f (z)) .
Proof: Set ft :=φt ◦ f ◦ φ−1
t . Note that f0 = f . Differentiating the equality
φt ◦ f (z) = ft ◦ φt (z) with respect to t and evaluating at t = 0, we see that
dft ξ=
if and only if for all z ∈ P1 , we have
dt t=0
θ f (z) = ξ(z) + Dz f θ(z) .
(2.2)
If equality (2.2) holds, then applying (Dz f )−1 to both sides, we obtain the
following equality of vector fields:
f ∗ θ = ηξ + θ.
Conversely, if f ∗ θ = ηξ + θ, then for all z ∈ P1 \ C(f ), we have the equality
of vectors f ∗ θ(z) = ηξ (z) + θ(z). Applying Dz f to both sides shows that
equality (2.2) hold on P1 \ C(f ), thus on P1 by analytic continuation. i
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1. Bifurcation measure and Misiurewicz maps
Corollary 1 The vector ξ ∈ Tf Ratd is tangent to O(f ) if and only if there
is a vector field θ, holomorphic on P1 , such that:
ηξ = f ∗ θ − θ.
Proof: If ξ ∈ Tf O(f ), there is a holomorphic family of Möbius transforma
d(φt ◦ f ◦ φ−1
t )
tions φt such that φ0 = Id and ξ =
. The vector field
dt
t=0
dφt 1
is holomorphic on P , and according to the previous proposiθ:=
dt t=0 ∗
tion, ηξ = f θ − θ.
Conversely, ηξ = f ∗ θ − θ with θ holomorphic on P1 , we can find a
dφt = θ.
family of Möbius transformations φt such that φ0 = Id and
dt t=0
−1
According to the previous proposition, the family {ft :=φt ◦f ◦φt } satisfies
dft ξ=
, and since ft ∈ O(f ), we have ξ ∈ Tf O(f ).
dt t=0
2.2 Following critical points and critical values
In the following proposition, {ft } is a holomorphic family of rational maps,
dft and let
parametrized by a neighborhood of t = 0 in C. We set ξ:= dt t=0
1
ηξ be the corresponding meromorphic vector field on P . We assume that
ω0 is a simple critical point of f0 and that v0 is the corresponding critical
value. Finally, we let
ω : (C, 0) → (P1 , ω0 ) and v : (C, 0) → (P1 , v0 )
be holomorphic germs following respectively a critical point of ft and the
corresponding critical value of ft (in view of the Implicit Function Theorem,
since ω0 is a simple critical point of f0 , those germs exist and are unique).
Proposition 3 Assume θ is a vector field, defined and holomorphic in a
neighborhood of v0 . Then, f0∗ θ − ηξ is holomorphic in a neighborhood of ω0
if and only if
dv(t) θ(v0 ) =
.
dt t=0
In that case,
dω(t) (f0∗ θ − ηξ )(ω0 ) =
.
dt t=0
Proof: Let us first show that there exists a vector field θ0 , defined and
holomorphic in a neighborhood of v0 , such that f0∗ θ0 − ηξ is holomorphic
in a neighborhood of ω0 .
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2. Transversality
13
Choose a holomorphic family of local biholomorphisms φt sending v0
to v(t), with φ0 = Id. Note that there is a holomorphic family of local
biholomorphisms ψt sending ω0 to ω(t), with ψ0 = Id and
φt ◦ f0 = ft ◦ ψt .
Note that
dφt dt t=0
is defined and holomorphic in a neighborhood of v0 and
dψt ζ0 :=
dt t=0
θ0 :=
is defined and holomorphic in a neighborhood of ω0 . Differentiating the
equality φt ◦ f0 (z) = ft ◦ ψt (z) with respect to t and evaluating at t = 0
yields the following equality of vectors:
θ0 f0 (z) = ξ(z) + Dz f0 ζ0 (z) .
Applying (Dz f0 )−1 to both sides of the equation yields the equality of
vector fields:
f0∗ θ0 = ηξ + ζ0 .
As required, f0∗ θ0 − ηξ = ζ0 is holomorphic in a neighborhood of ω0 . In
addition,
dv(t) θ0 (v0 ) =
dt t=0
and
dω(t) (f0∗ θ0 − ηξ )(ω0 ) = ζ0 (ω0 ) =
.
dt t=0
Let us now assume that θ is a vector field, defined and holomorphic in a
neighborhood of v0 . If θ agrees with θ0 at v0 , then θ−θ0 . Consequently, the
pullback f0∗ (θ − θ0 ) is holomorphic vanishes at ω0 . It follows that f0∗ θ − ηξ
is holomorphic and agrees with f0∗ θ0 − ηξ at ω0 .
Finally, if θ does not agree with θ0 at v0 , then θ − θ0 is not vanishing at
v0 . Thus, the vector field f0∗ (θ − θ0 ) has a pole at ω0 . As a consequence,
f0∗ θ − ηξ has a pole at ω0 .
2.3 Proof of proposition 1
Let us recall the hypotheses. The rational map f0 ∈ Ratd is postcritically
finite. It has 2d − 2 distinct critical points ωj (f0 ) and C(f0 ) ∩ P(f0 ) = ∅.
In particular, there are integers ℓj ≥ 1 such that
◦ℓ
αj (f0 ):=f0 j ωj (f0 )
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14
1. Bifurcation measure and Misiurewicz maps
are periodic points of f0 . The critical points are simple and the periodic
points are repelling. By the Implicit Function Theorem, there are
• analytic germs ωj : (Ratd , f0 ) → P1 following the critical points of f
and
• analytic germs αj : (Ratd , f0 ) → P1 following periodic points of f .
The analytic germs
F : (Ratd , f0 ) → (P1 )2d−2
are defined by


F1


..
F := 

.
F2d−2
and G : (Ratd , f0 ) → (P1 )2d−2
with Fj (f ):=f ◦ℓj ωj (f )


and G:= 
α1
..
.
α2d−2


.
Assume {ft } is a holomorphic family of rational maps of degree d ≥ 2,
parametrized by a neighborhood of t = 0 in C. Suppose
ξ :=
dft ∈ Ker(Df0 F − Df0 G).
dt t=0
Let ηξ be the associated holomorphic vector field on P1 \ C(f0 ). We want
to show that ξ ∈ Tf O(f ), and for this purpose, we will show that there is
a globally holomorphic vector field θ such that
ηξ = f0∗ θ − θ.
First, we define θ on Pf0 as follows. If z0 ∈ Pf0 , then
• z0 is not a precritical point of f0 and
• z0 is preperiodic to a repelling periodic point of f0 .
It follows from the Implicit Function Theorem that there is a holomorphic
germ z : (C, 0) → (P1 , z0 ) such that z(t) is a preperiodic point of ft and
this germ is unique. Indeed, there are integers ℓ ≥ 0 and p ≥ 1 such that
f0◦ℓ (z0 ) = α0
with f0◦p (α0 ) = α0 .
Since α0 is a repelling periodic point of f0 , (f0◦p )′ (α0 ) 6= 1 and since z0 is
not precritical, (f0◦ℓ )′ (z0 ) 6= 0 whence
(f0◦ℓ+p )′ (z0 ) = (f0◦p )′ (α0 ) · (f0◦ℓ )′ (z0 ) 6= (f0◦ℓ )′ (z0 ).
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2. Transversality
15
We set
dz(t) .
dt t=0
Second, we have the following equality, valid on the postcritical set
P(f0 ):
f0∗ θ = θ + ηξ .
θ(z0 ):=
Indeed, assume that z0 ∈ P(f0 ) and w0 = f0 (z0 ) ∈ P(f0 ). Then,
dz(t) dw(t) θ(z0 ) =
and θ(w0 ) =
with w(t) = ft z(t) .
dt t=0
dt t=0
Evaluating the derivative of the equation w(t) = ft z(t) at t = 0 yields:
θ(w0 ) = Dz0 f0 θ(z0 ) + ξ(z0 ).
Applying (Dz0 f0 )−1 to both sides yields the required equation.
Third, we may reformulate the condition that ξ belongs to the kernel
of Df0 F − Df0 G as follows. Let vj : (Ratd , f0 ) → P1 be the analytic germs
following the critical values of f :
vj (f ):=f ωj (f ) .
Lemma 3 If ξ belongs to the kernel of Df0 F − Df0 G, then for all integer
j ∈ {1, . . . , 2d − 2}, we have
dvj (ft ) θ vj (f0 ) =
.
dt
t=0
Proof: To lighten notation, let us leave aside the index j. Choose a critical
point ω(f0 ) and let v(f0 ) be the corresponding critical value. We have
f0◦ℓ ω(f0 ) = α(f0 ) for some integer ℓ ≥ 1 and if ξ belongs to the kernel of
Df0 F − Df0 G, then
dft◦ℓ ω(ft ) dα(ft ) =
.
dt
dt t=0
t=0
In addition, we have a point zt depending holomorphically on t, such that
z0 = v(f0 ) and ft◦ℓ−1 (zt ) = α(ft ).
Using
we easily derive
dft◦ℓ−1 v(ft ) df ◦ℓ−1 (zt ) = t
,
dt
dt
t=0
t=0
Dv(f0 ) f0◦ℓ−1
dv(ft ) dzt = Dz0 f0◦ℓ−1
.
dt t=0
dt t=0
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1. Bifurcation measure and Misiurewicz maps
Since v(f0 ) = z0 and since there are no critical points in the forward orbit
of v(f0 ), the map Dv(f0 ) f0◦ℓ−1 = Dz0 f0◦ℓ−1 is invertible. So:
dzt dv(ft ) =
= θ v(f0 ) .
dt t=0
dt t=0
We now come to the heart of the argument.
Lemma 4 The vector field θ is the restriction to P(f0 ) of a vector field which
is holomorphic on P1 .
Proof: A rational map whose postcritical set contains only two points is
conjugate to z 7→ z ±d. It follows that P(f0 ) which contains at three points.
Choose three distinct points z1 , z2 , z3 in P(f0 ) and let θ̃ be the unique
holomorphic vector field on P1 which coincides with θ at z1 , z2 and z3 .
We must show that θ̃ coincides with θ at any other point of P(f0 ). If
there are only three points in P(f0 ), we are done. Otherwise, let z4 be a
point in P(f0 ) \ {z1 , z2 , z3 }. Consider a quadratic differential q which is
holomorphic on P1 \ {z1 , z2 , z3 , z4 } and has simple poles at z1 , z2 , z3 and
z4 . Then, the 1-form q ⊗ θ̃ is holomorphic on P1 and has a most simple
poles on {z1 , z2 , z3 , z4 }. The sum of residues of a meromorphic 1-form on
P1 is 0. It follows that θ and θ̃ coincide at z4 if and only if
X
X
Res(q ⊗ θ, z) = 0.2
Res(q ⊗ θ̃, z) =
0=
z∈{z1 ,z2 ,z3 ,z4 }
z∈{z1 ,z2 ,z3 ,z4 }
Denote by Q(f0 ) the set of quadratic differentials which are holomorphic
on P1 \ P(f0 ) and have at most simple poles on P(f0 ). By the above
discussion, in order to prove the lemma, it is enough to show that for all
q ∈ Q(f0 ), we have:
X
Res(q ⊗ θ, z) = 0.
hq, θi :=2iπ
z∈P(f0 )
For this purpose, choose a vector field which is C ∞ on P1 , holomorphic in
a neighborhood of P(f0 ) and which coincides with θ on P(f0 ). To lighten
notation, let us still denote it by θ. Without loss of generality, we may
assume that U is a finite union of disjoint disks. Then, according to the
residue theorem and to Stokes’ theorem, for all q ∈ Q(f0 ), we have:
Z
Z
Z
¯ =−
¯
hq, θi =
q⊗θ =−
q ⊗ ∂θ
q ⊗ ∂θ.
∂U
P1 \U
P1
2 To define the residue, we choose any holomorphic extension of θ to a neighborhood
of P(f0 ) (compare with [HS94] for example).
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3. The bifurcation measure
17
For the last equality, we used ∂θ = 0 in U . If q ∈ Q(f0 ), we therefore have:
Z
Z
Z
¯ =−
¯ =−
q ⊗ ∂¯ f0∗ θ − ηξ
q ⊗ f0∗ ∂θ
(f0 )∗ q ⊗ ∂θ
(f0 )∗ q, θ = −
P1
P1
P1
¯ ξ ≡ 0. However according to proposition 3, f ∗ θ − ηξ is a C ∞ vector
since ∂η
0
field on P1 . Since f0∗ θ − ηξ coincides with θ on P(f0 ), we see that:
∀q ∈ Q(f0 ),
(f0 )∗ q, θ = hq, θi .
Consider the linear map ∇ : Q(f0 ) → Q(f0 ) defined by ∇q = (f0 )∗ q −q.
We have just seen that for all q ∈ Q(f0 ), we have h∇q, θi = 0.
We will now use the fact that f0 is not a flexible Lattès example. Under
this hypothesis ∇ : Q(f0 ) → Q(f0 )is injective (see [DH93]), thus surjective.
It follows that for all q ∈ ∇ Q(f0 ) = Q(f0 ), we have hq, θi = 0.
We still denote by θ the holomorphic extension to P1 . Note that the
vector field f0∗ θ is defined and holomorphic on P1 \ C(f0 ). According to
lemma 4, for j ∈ {1, . . . , 2d − 2}, we have:
dft ωj (t) θ vj (f0 ) =
.
dt
t=0
According to proposition 3, the vector field f0∗ θ − ηξ is holomorphic in a
neighborhood of C(f0 ), thus on P1 . It coincides with θ on P(f0 ) which
contains at least three points. Two holomorphic vector fields on P1 which
coincide at three points are equal: indeed, in any affine coordinate, their
expressions are polynomials of degree ≤ 2. As a consequence, we have the
required equality:
f0∗ θ − ηξ = θ.
3 The bifurcation measure
We will now prove that Xd∗ ⊂ Supp(µbif ). This will complete the proof of
our main theorem.
The idea is the following. Assume f0 ∈ Ratd is a strictly postcritically
finite rational map which is not a flexible Lattès example, has simple critical
points and satisfies C(f0 ) ∩ P(f0 ) = ∅. It is enough to show that there is
a 2d − 2-dimensional complex manifold Σ transverse to O(f0 ) at f0 , and a
basis of neighborhoods Σn of f0 in Σ, such that
Z
(Tbif )∧(2d−2) > 0.
Σn
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1. Bifurcation measure and Misiurewicz maps
Let ωj : (Ratd , f0 ) → P1 be analytic germs following the critical points
◦ℓ
of f . There are integers ℓj ≥ 1 such that αj (f0 ):=f0 j ωj (f0 ) are repelling periodic points of f0 of periods pj . Let Vj be simply connected
◦p
neighborhoods of αj (f0 ) such that V j ⊂ Wj :=f0 j (Vj ) and such that
◦pj
f0 : Vj → Wj are isomorphisms.
Denote by M the least common multiple of the periods pj and define
Σn ⊂ Σ as the connected component containing f0 of the set
f ∈ Σ ; ∀j ∈ {1, . . . , 2d − 2}, f ◦ℓj +nM ωj (f ) ∈ Vj .
Our proof essentially consists in proving that
Z
Σn
(Tbif )∧(2d−2)
∼
n→+∞
(2d − 2)!
dnM(2d−2)+
P
ℓj
·
2d−2
Y
µf0 (Vj )
j=1
where µf0 is the equilibrium measure of f0 . Since the disks Vj intersect the
Julia set of f0 , we have µf0 (Vj ) > 0 for all j, and so, for n large enough,
Z
(Tbif )∧(2d−2) > 0.
Σn
3.1 Another definition of the bifurcation current
We will use a second definition of the bifurcation current Tbif due to DeMarco [DeM01] (see [DeM03] or [BB07] for the equivalence of the two
definitions). The current Tbif can be defined locally by considering the
behavior of the critical orbits as follows.
We still denote by π : C2 \ {0} → P1 the canonical projection. Assume
f0 ∈ Ratd has only simple critical points and let U be a sufficiently small
neighborhood of f0 so that there are:
• holomorphic functions ωj : U → P1 j∈{1,...,2d−2} following the critical points of f ,
• holomorphic functions ω̃j : U → C2 \ {0} j∈{1,...,2d−2} such that
π ◦ ω̃j = ωj and
• a family {Pf }f ∈U of non-degenerate homogeneous polynomials of degree d depending analytically on f ∈ U such that π ◦ Pf = f ◦ π.
The map G : U × C2 → R defined by
G(f, z):= lim
n→+∞
is plurisubharmonic on U × C2 .
1
logPf◦n (z)
dn
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3. The bifurcation measure
19
For j ∈ {1, . . . , 2d − 2}, define Gj : U → R by:
Gj (f ):=G f, ω̃j (f ) .
DeMarco [DeM03] proved that
• if Pf is chosen so that Res(Pf ) = 1, 3 and
• if ω̃j are chosen so that det(Dz Pf ) =
2d−2
Y
j=1
then
2d−2
X
z ∧ ω̃j (f ) ,
Gj (f ) = L(f ) + log d.
j=1
As a consequence,
2d−2
X
ddc Gj .
Tbif U =
j=1
3.2 A parametrized family of rational maps
From now on, we assume that f0 ∈ Ratd is a strictly postcritically finite
rational map which is not a flexible Lattès example, has simple critical
points and satisfies C(f0 ) ∩ P(f0 ) = ∅.
Let ωj : (Ratd , f0 ) → P1 be analytic germs following the critical points
of f and for n ≥ 1, let vjn : (Ratd , f0 ) → P1 be the analytic germs defined
by
vjn (f ) = f ◦n ωj (f ) .
There are integers ℓj ≥ 1 and pj ≥ 1 such that
ℓ
αj (f0 ):=vj j (f0 )
are repelling periodic points of f0 of periods pj . Let αj : (Ratd , f0 ) → P1
be analytic germs following those periodic points and let
φj,f : P1 , αj (f ) → (C, 0)
be linearizing maps depending analytically on f , i.e.
φj,f ◦ f ◦pj = µj (f ) · φj,f
with
3 If
z “= (z1 , z2 ) and w = (w
= z1 w2 − z2 w1 and if
” 1 , w2 ), then z ∧ w
Q
Q
Q
,
then
Res(F
)
=
z
∧
a
,
z
∧
b
i
j
i,j ai ∧ bj .
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F (z) =
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1. Bifurcation measure and Misiurewicz maps
• φj,f αj (f ) = 0,
• φj,f is locally injective at αj (f ) and
• µj (f ) is the multiplier of f ◦pj at αj (f ).
Consider the analytic map τ : (Ratd , f0 ) → (C2d−2 , 0) defined by
τ :=(τ1 , . . . , τ2d−2 ) with
ℓ
τj (f ):=φj,f ◦ vj j (f ).
Lemma 5 The rank of Df0 τ : Tf0 Ratd → C2d−2 is 2d − 2. The kernel is
Tf0 O(f0 ).
Proof: Differentiating the equality φj,f αj (f ) = 0 with respect to f , we
have
dφj,f + Dαj (f0 ) φj,f0 ◦ Df0 αj = 0.
df
f0 ,αj (f0 )
As a consequence, we have
Df0 τj =
dφj,f df
f0 ,αj (f0 )
+ Dαj (f0 ) φj,f0 ◦ Df0 vjℓj
ℓ
= Dαj (f0 ) φj,f0 ◦ Df0 vj j − Df0 αj .
Thus, ξ ∈ Tf0 Ratd belongs to the kernel of Df0 τ if and only if it belongs to
ℓ
the kernel of Df0 vj j − Df0 αj for all j ∈ {1, . . . , 2d − 2}, i.e. if and only if
it belongs to the kernel of Df0 F − Df0 G. According to proposition 1, such
a ξ belongs to Tf0 O(f0 ). Thus, the kernel Df0 τ has dimension 3 and the
image has dimension 2d − 2 (the dimension of Tf0 Ratd is 2d + 1).
Corollary 2 If r > 0 is sufficiently small and if ∆:=D(0, r), then there is a
section
χ : ∆2d−2 → Ratd
with
χ(0) = f0
and
τ ◦ χ = Id.
From now on, we assume that such a section χ : ∆2d−2 → Ratd has
been chosen and we let Σ ⊂ Ratd be the complex manifold of dimension
2d − 2 defined by
Σ:=χ(∆2d−2 ).
Then, Σ is an analytic familyof rational maps parametrized by the positions of the points f ◦ℓj ωj (f ) in the linearizing coordinates φj,f .
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3. The bifurcation measure
21
3.3 The proof
Let us now assume that Vj ⊂ P1 is a sufficiently small neighborhood of
αj (f0 ) so that
• φj,f0 is defined and univalent on a neighborhood of V j and
• π : C2 \ {0} → P1 has an analytic section σj : Vj → C2 \ {0} defined
on Vj .
Decreasing r if necessary, we may assume that ∆ is relatively compact in
φj,f0 (Vj ) for all j ∈ {1, . . . , 2d − 2}.
Let us now denote by M the least common multiple of the periods pj
and set mj = M/pj . For n ≥ 0, we define χn : ∆2d−2 → Σ by
!
x1
x2d−2
nm1 , . . . ,
nm2d−2
χn (x1 , . . . , x2d−2 ):=χ
µ1 (f0 )
µ2d−2 (f0 )
and set
Σn :=χn (∆2d−2 ) ⊂ Σ.
Lemma 6 As n → +∞, the sequences of functions
µj ◦ χn
µj (f0 )
n
: ∆2d−2 → C
converge uniformly to 1.
Proof: Since the map χ : ∆2d−2 → Ratd and the germs µj : (Ratd , f0 ) → C
are analytic, there is a constant C > 0 such that
µj ◦ χn
n
µj (f0 ) − 1 ≤ Cλ
∞
with
λ:=
min
j∈{1,...,2d−2}
It follows easily that
(
1
µj (f0 )mj
)
µj ◦ χn n
−
1
−→ 0.
µj (f0 )
n→+∞
∞
.
Corollary 3 For n sufficiently large and all j ∈ {1, . . . , 2d − 2}, we have
ℓ +nM
vj j
(Σn ) ⊂ Vj .
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1. Bifurcation measure and Misiurewicz maps
Proof: Since ∆ = D(0, r) is relatively compact in φj,f0 (Vj ), there is a neighborhood U of f0 in Ratd and a κ > 1 such that for all f ∈ U and all
j ∈ {1, . . . , 2d − 2}, D(0, κr) ⊂ φj,f (Vj ). Let us choose n sufficiently large
so that χn (∆2d−2 ) ⊂ U and so that
2d−2
∀f ∈ χn (∆
µj (f ) nmj
< κ.
µj (f0 ) ), ∀j ∈ {1, . . . , 2d − 2},
If f ∈ Σn , then f = χn (x1 , . . . , x2d−2 ) with (x1 , . . . , x2d−2 ) ∈ ∆2d−2 ,
ℓ
φj,f vj j (f ) =
xj
nmj
µj (f0 )
nmj
·
µj (f )
and
In addition, φj,f ◦ f ◦pj = µj (f ) · φj,f . Thus,
ℓ +nM
vj j
(f )
=
ℓ +nmj pj
(f )
vj j
=
φ−1
j,f
xj
nmj ∈ φj,f (Vj ).
µj (f0 )
nmj
·
µj (f )
xj
nmj
µj (f0 )
!
∈ Vj .
From now on, we assume that n is sufficiently large so that for all integer
j ∈ {1, . . . , 2d − 2}, we have
ℓ +nM
vj j
(Σn ) ⊂ Vj .
Define plurisubharmonic functions Gnj : ∆2d−2 → R by
ℓ +nM
Gnj (x):=G χn (x), σj ◦ vj j
Lemma 7 For j ∈ {1, . . . , 2d − 2}, we have:
◦ χn (x) .
ddc Gnj = dℓj +nM · χ∗n (ddc Gj ).
Proof: Assume f ∈ Σn and j ∈ {1, . . . , 2d − 2}. Note that
◦ℓj +nM
π ◦ Pf
Thus,
ℓ +nM
ℓ +nM
(f ) = π ◦ σj ◦ vj j
(f ).
ω̃j (f ) = f ◦ℓj +nM ωj (f ) = vj j
ℓ +nM
Pf◦n ω̃j (f ) = λnj (f ) · σ ◦ vj j
(f )
for some holomorphic functions λnj : Σn → C∗ . Note that
G f, Pf (z) = d · G(f, z) and ∀λ ∈ C∗ ,
G(f, λz) = G(f, z) + log |λ|.
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3. The bifurcation measure
23
Thus, if f = χn (x) with x ∈ ∆2d−2 , then
◦ℓj +nM
Gnj (x) = G f, Pf
ω̃j (f ) − logλnj (f )
= dℓj +nM G f, ω̃j (f ) − logλnj (f )
= dℓj +nM Gj (f ) − logλnj (f )
= dℓj +nM Gj ◦ χn (x) − logλnj ◦ χn (x).
Since log |λnj ◦ χn | is pluriharmonic on ∆2d−2 , we have
ddc Gnj = dℓj +nM · χ∗n (ddc Gj ).
It follows that
χ∗n (Tbif ) =
2d−2
X
χ∗n (ddc Gj ) =
j=1
1
dnM
2d−2
X
j=1
1
ddc Gnj .
dℓj
Lemma 8 We have
lim
n→+∞
Z
∆2d−2


2d−2
X
j=1
∧(2d−2)
2d−2
(2d − 2)! Y
1
−1
c n
P
(∆)
.
φ
µ
·
dd
G
=
f
0
j
j,f
0
dℓj
d ℓj
j=1
Proof: If f = χn (x1 , . . . , x2d−2 ), then
ℓ +nM
vj j
(f )
=
φ−1
j,f
µj (f )
µj (f0 )
nmj
· xj .
ℓ +nM
Thus, as n → +∞, the sequence of functions vj j
converge uniformly to
◦ χn : ∆2d−2 → Vj
(x1 , . . . , x2d−2 ) 7→ φ−1
j,f0 (xj ).
As a consequence, as n → +∞, the sequences of functions Gnj : ∆2d−2 → R
converge uniformly to
−1
G∞
j : (x1 , . . . , x2d−2 ) 7→ G f0 , σj ◦ φj,f0 (xj ) .
Thanks to the uniform convergence of the potentials, we can write:
lim
n→+∞
Z
∆2d−2

2d−2
X

j=1

∧(2d−2)
∧(2d−2)
Z
2d−2
X 1
1


=
.
ddc Gnj 
ddc G∞
j
ℓj
dℓj
d
∆2d−2
j=1
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24
1. Bifurcation measure and Misiurewicz maps
Note that G∞
j only depends on the j-th coordinate. It follows that


2d−2
X
j=1
∧(2d−2)
2d−2
1
(2d − 2)! ^
c ∞
P
ddc G∞
·
dd
G
=
j .
j
dℓj
d ℓj
j=1
−1
g : Vj → R the
In addition, G∞
j (x1 , . . . , x2d−2 ) = gj ◦ φj,f0 (xj ) with
j
subharmonic function defined by gj (z) = G f0 , σj (z) . We have
ddc gj = µf0 Vj .
Therefore, according to Fubini’s theorem,


Z
2d−2
2d−2
Y Z
^
−1
c
c ∞

dd gj ◦ φj,f0
=
dd Gj
∆2d−2
j=1
j=1
=
2d−2
Y
j=1
=
2d−2
Y
j=1
∆
Z
c
dd gj
φ−1
j,f (∆)
0
µf0 φ−1
j,f0 (∆) .
!
We can now complete the proof. Since the points αj (f0 ) are repelling
periodic points of f0 , they are in the support of the equilibrium measure
µf0 . Thus,
µf0 φ−1
j,f0 (∆) > 0.
As a consequence,
lim dnM(2d−2) ·
n→+∞
Z
Σn
Tbif ∧(2d−2)
=
2d−2
(2d − 2)! Y
P
(∆) > 0.
µf0 φ−1
·
j,f
0
ℓ
d j
j=1
Thus, f0 is in the support of Tbif ∧(2d−2) and the conjugacy class of f0 is in
the support of µbif .
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Bibliography
[BB07]
Giovanni Bassanelli and François Berteloot, Bifurcation currents
in holomorphic dynamics on Pk , J. Reine Angew. Math. 608
(2007), 201–235.
[DeM01] Laura DeMarco, Dynamics of rational maps: a current on the
bifurcation locus, Math. Res. Lett. 8 (2001), no. 1-2, 57–66.
[DeM03] Laura DeMarco, Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity, Math. Ann. 326 (2003), no. 1,
43–73.
[DF]
Romain Dujardin and Charles Favre, Distribution of rational
maps wih a preperiodic critical point, Amer. Journ. Math., to
appear.
[DH93]
Adrien Douady and John H. Hubbard, A proof of Thurston’s
topological characterization of rational functions, Acta Math. 171
(1993), no. 2, 263–297.
[Eps99]
Adam L. Epstein, Infinitesimal Thurston rigidity and the FatouShishikura inequality, Stony Brook IMS Preprint (1999).
[HS94]
John H. Hubbard and Dierk Schleicher, The spider algorithm,
Complex dynamical systems (Cincinnati, OH, 1994), Proc. Sympos. Appl. Math., vol. 49, Amer. Math. Soc., Providence, RI,
1994, pp. 155–180.
[Lyu83]
M. Ju. Lyubich, Entropy properties of rational endomorphisms of
the Riemann sphere, Ergodic Theory Dynam. Systems 3 (1983),
no. 3, 351–385.
[Mañ88] Ricardo Mañé, The Hausdorff dimension of invariant probabilities of rational maps, Dynamical systems, Valparaiso 1986, Lecture Notes in Math., vol. 1331, Springer, Berlin, 1988, pp. 86–117.
25
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i
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i
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i
i
i
26
BIBLIOGRAPHY
[McM87] Curt McMullen, Families of rational maps and iterative rootfinding algorithms, Ann. of Math. (2) 125 (1987), no. 3, 467–493.
[McM94] Curtis T. McMullen, Complex dynamics and renormalization,
Annals of Mathematics Studies, vol. 135, Princeton University
Press, Princeton, NJ, 1994.
[Mil06]
John Milnor, On Lattès maps, Dynamics on the Riemann sphere,
Eur. Math. Soc., Zürich, 2006, pp. 9–43.
[MSS83] R. Mañé, P. Sad, and D. Sullivan, On the dynamics of rational
maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193–217.
[Shi87]
Mitsuhiro Shishikura, On the quasiconformal surgery of rational
functions, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 1, 1–29.
[Tan90]
Lei Tan, Similarity between the Mandelbrot set and Julia sets,
Comm. Math. Phys. 134 (1990), no. 3, 587–617.
[vS00]
Sebastian van Strien, Misiurewicz maps unfold generically (even
if they are critically non-finite), Fund. Math. 163 (2000), no. 1,
39–54.
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