Table Of ContentLyapunov exponents, bifurcation currents and laminations in
bifurcation loci
8
Giovanni Bassanelli and Franc¸ois Berteloot
0
0
2
n Abstract
a
J Bifurcation loci in the moduli space of degree d rational maps are shaped by the
7 hypersurfaces defined by the existence of a cycle of period n and multiplier 0 or eiθ.
1
Using potential-theoretic arguments, we establish two equidistribution properties for
these hypersurfaces with respect to the bifurcation current. To this purpose we first
]
V establish approximation formulas for the Lyapunov function. In degree d = 2, this
C allows us to build holomorphic motions and show that the bifurcation locus has a
. lamination structure in the regions where an attracting basin of fixed period exists.
h
t
a
m Giovanni Bassanelli, Dipartimento di Matematica, Universit`a di Parma, Parco Area delle Scienze,
[ Viale Usberti 53/A I–43100 Parma, Italia. Email: [email protected]
Fran¸cois Berteloot, Universit´e Paul Sabatier MIG. Institut de Math´ematiques de Toulouse. 31062
1
v Toulouse Cedex 9, France. Email: [email protected]
0
Math Subject Class: 37F45; 37F10
9
5
2
. 1 Introduction
1
0
8 A bifurcation is usually said to occur when the dynamical behaviour of a map changes
0
drastically under perturbation. For rational maps on the Riemann sphere, the study of
:
v bifurcations started with the seminal paper of Man˜´e, Sad and Sullivan [16] and, since,
i
X constitutes a large field of research. Let us recall that, in particular, this work character-
r izes the bifurcation locus as the subset of the parameter space where the Julia set does
a
not move continuously.
In any parameter space M of some holomorphic family (f ) or in the moduli space
λ λ∈M
M of degree d rational maps, the subsets Per (w) of rational maps possessing a cycle of
d n
exact period n and multiplier w ∈ C\{1} turn out to be hypersurfaces. As the Man˜´e-
Sad-Sullivan theory shows, the bifurcation locus is shaped by the hypersurfaces Per (0)
n
and Per (eiθ); it actually coincides with the closure of the union over n and θ of the hy-
n
persurfaces Per (eiθ). Moreover, the work of Milnor [19] reveals that the global structure
n
of the bifucation locus is intimately related to the behaviour of these hypersurfaces in a
convenient compactification of M . We aim, in the present paper, to combine the ap-
d
proach of Milnor with the point of view of bifurcation currents. As we shall now explain,
these currents allow to both exploit the properties of the Lyapunov function and general
measure or potential-theoretic methods.
1
The bifurcation locus of any holomorphic family (f ) , as well as the bifurcation
λ λ∈M
locus of a moduli space M , carries a closed, positive (1,1)-current. This current, which
d
has been introduced by DeMarco [6], is called the bifurcation current and denoted T .
bif
Both Tbif and its powers Tbkif (k ≤ dimCM) areextremely convenient tools for thestudyof
measurableorcomplexanalyticpropertiesofthebifurcationlocusandhavebeenexploited
in several recent works ([1],[20],[10],[9]). A crucial fact is that the bifurcation current
admits both the Lyapunov function L(λ) and the sum of values of the Green function on
the critical points as potentials (see [7] or [1]):
T = ddcL(λ) = ddc G (c ).
bif λ λ
X
We recall that L(λ) denotes the Lyapunov exponent of f with respect to its maximal
λ
entropy measure. The above formula actually enlights the double nature of the bifurca-
tion current: T may either detect the instability of repelling cycles or the instability
bif
of critical orbits. This reflects the essence of the classical Man˜´e-Sad-Sullivan theory on
bifurcations (see [1], Section 5). Ourpoint of view, initiated in [1], is based on the identity
T = ddcL(λ).
bif
Letusnowdescribethecontentofthepaper. Wefirstinvestigatehowthehypersurfaces
Per (0) and Per (eiθ) accumulate the bifurcation current and establish the following
n n
equidistribution formulas in a quite general context (see theorem 3.5 and, for similar
results regarding Tk , theorem 3.9):
bif
T = limd−n[Per (0)]
bif n
n
d−n 2π
T = lim [Per (eiθ)]dθ.
bif n 2π Z n
0
It is important to stress here that these formulas are deduced from the identity T =
bif
ddcL(λ) and the following property of the Lyapunov exponent (RP(f ,n) is the set of
λ
repulsiv points of f of exact period n):
λ
L(λ) =limd−n log|f′|.
λ
n X
RP(fλ,n)
A similar fact has been proved for holomorphic endomorphisms of Pk ([2]). Although the
proof is definitely easier when k = 1, the above formula seems to benew even in that case.
In the remaining of the paper we work in the moduli space M of quadratic rational
2
maps. As it has been shown by Milnor [19], we may identify M with C2. Our goal is
2
to use the above formulas to see how the distribution of the hypersurfaces Per (0) deter-
n
mines the shape of the bifurcation locus.
Besides the equidistribution formulas, we introduce two other tools which may be of in-
dependant interest. The first one is a parametrization “`a la Douady-Hubbard” of certain
hyperbolic components of M (see theorem 5.2). This parametrization yields a holo-
2
morphic motion of the hyperbolic components of any curve Per (0). The second is an
n0
extension principle for special kind of holomorphic motions in P2. This principle is com-
parable with the classical λ-lemma for holomorphic motions in the Riemann sphere. It
2
allows us to obtain the motion of the bifurcation locus as a limit of motions of hyperbolic
components (see theorem 4.4).
Our main result describes the bifurcation locus Bifn0 of the open region Un0 ⊂ M2
consisting of “parameters” which do have an attracting cycle of period n . We compare
0
Bifn0 with the bifurcation locus Bifnc0 of the “central curve” Pern0(0) (see theorem 6.4 for
a precise statement). We show that Bifn0 is a lamination whose transverse measure is the
bifurcation measure µc on the central curve Per (0). We actually construct, in a very
n0 n0
natural way, a holomorphic motion σ : Bifnc0 ×∆ −→ Bifn0 such that:
T | = [σ(λ,∆)] µc .
bif Un0 ZBifc n0
n0
When n = 1, our result is sharper and also easier to visualize. The open region U is the
0 1
subset of M which consists of parameters having an attracting fixed point. The central
2
curve Per (0) is a line in C2 = M . This line is the family of quadratic polynomials on
1 2
which the bifurcation locus is the boundary of the Mandelbrot set M . The bifurcation
2
measure is the harmonic measure of M (see corollary 3.8). Our result says, in particular,
2
that the bifurcation locus in U is obtained by moving holomorphically the boundary of
1
the Mandelbrot set. Using Slodkowski’s theorem, we extend our holomorphic motion to
the full line Per (0). This shows that every stable component moves holomorphically.
1
Consequently, we observe that if some non-hyperbolic stable component would exist on
the line of quadratic polynomials, such a component would also exist within quadratic
rational maps (see theorem 7.1).
We now end this intoduction with a few words on related works. In their paper ([10],
Theorems 1 and 4.2) Dujardin and Favre have studied the distribution of hypersurfaces of
M defined by the pre-periodicity of a critical point. They have shown that the bifurcation
current is the limit of suitably normalized currents of integration on these hypersurfaces.
The first assertion of our theorem 3.5 corresponds to the case of a periodic critical point.
It is interesting to note that the approaches are based on the two different interpretations
of the bifurcation current which we have previously described. Dujardin and Favre use
T = ddc G (c ) and investigate the activity of critical points while we work with
bif λ λ
T = ddcLP(λ) and investigate the instability of cycles.
bif
Using Branner-Hubbard holomorphic motion and quasi-conformal surgery, Uhre [25] has
builtamotion similar toour intheregion U . Thelaminarity propertiesof thebifurcation
1
currentin the family of cubicpolynomials have beenstudied by Dujardin [9], his approach
differs from our.
The paper is organized as follows. The section 2 describes the general concepts and
results usedin thepaper, wediscussin particular theMilnor compactification of M . The
2
equidistribution formulas for T and Tk are proved in section 3 in the context of holo-
bif bif
morphic families, these results remain valid for moduli spaces. In section 4, we introduce
the notion of guided holomorphic motion in C2 and establish, for these motions, an exten-
sion principle. The section 5 is devoted to the uniformization of hyperbolic components
in M . Finally, the laminarity properties of the bifurcation current in M are studied in
2 2
sections 6 and 7.
3
2 The framework
2.1 Bifurcation current and Lyapunov function
Every rational map of degree d ≥ 2 on the Riemann sphere admits a maximal entropy
measure µ . This measure satisfies π∗µ = ddcG where π is the canonical projection
f f F
from C2 \ {0} onto P1 and G := limd−nlnkFnk is the Green function of any lift F
F
of f to C2. The Lyapunov exponent of f with respect to the measure µ is defined by
f
L(f)= ln|f′|µ (see the book [22] for a general exposition in any dimension).
P1 f
R
When f : M × P1 → P1 is a holomorphic family of degree d rational maps, the
Lyapunov function L on the parameter space M is defined by:
L(λ)= ln|f′|µ
Z λ λ
P1
where µ is the maximal entropy measure of f . It turns out that the function L is p.s.h
λ λ
and H¨older continuous on M (see [1] Corollary 3.4). The bifurcation current T of the
bif
family is a closed, positive (1,1)-current on M which may be defined by
T := ddcL(λ).
bif
As it has been shown by DeMarco [7], the support of T concides with the bifurcation
bif
locus of the family in the classical sense of Man˜´e-Sad-Sullivan (see also [1], Theorem 5.2).
All these constructions also make sense on the moduli spaces M (see [1], Section 6).
d
2.2 The moduli spaces M and the hypersurfaces Per (w)
d n
The space Rat of all rational maps of degree d on P1 may be viewed as an open subset of
d
P2d+1 on which the group of M¨obius transformations, which is isomorphic to PSL(2,C),
acts by conjugation. The moduli space M is, by definition, the quotient resulting from
d
this action. We shall denote as follows the canonical projection:
Rat −→ M
d d
f 7−→ f¯
Although the action of PSL(2,C) is not free, it may be proven that M is a normal
d
quasi-projective variety [23]. For simplifying we shall sometimes commit the abuse of
language which consists in considering a parameter λ ∈ M as a rational map. For
d
instance, ”λ ∈ M has a n-cycle” means that every f ∈ Rat such that f¯= λ posseses
d d
such a cycle.
We shall now recall why the subsets of M whose elements are the λ which have a
d
n-cycle of multiplier w are (at least when w 6= 1) hypersurfaces of M .
d
Let n ∈ IN∗. We first consider the hypersurfaces Q∗ of Rat ×P1 defined by
n d
Q∗ := {(f,z) ∈ Rat ×P1 / fn(z) = z}
n d
4
and then only retain the components whose generic points (f,z) has the property that z
is exactly of period n:
Q := Q∗ \ Q∗.
n n k
[
k<n,k|n
We may now define a holomorphic function w on Q by setting
n n
w : Q −→ C
n n
(f,z) 7−→ (fn)′(z)
and consider its graph Γ ⊂ Q ×C ⊂ Rat ×P1×C. We are lead to the following key
n n d
point:
Definition-Proposition 2.1 Let Per be the image of Γ under the canonical projection
n n
Rat ×P1 ×C → M ×C. Then, Per is an algebraic subvariety of M ×C and the
d d n d
projection Per → M is an analytic covering map. By construction, (f¯,w) ∈ Per if
n d n
and only if f there exists z ∈ P1 such that (f,z) ∈ Q and w (f,z) = w.
n n
We denote by ν (n) the number of distinct n-periodic points for a generic f ∈ Rat .
d d
Then, N(n) := νd(n) is the cardinality of the generic fibers of the cover Per → M and
n n d
taking multiplicity into account, the fiber of any λ ∈ M may be written
d
w (λ),...,w (λ).
n,1 n,N(n)
We are now ready to give a precise description of the hypersurfaces Per (w)
n
Definition-Proposition 2.2 The hypersurface Per ⊂ M ×C is defined by the alge-
n d
braic equation p (λ,w) = 0 where
n
N(n)
p (λ,w) := Π (w−w (λ)).
n j=1 n,j
For any w ∈ C, the hypersurface Per (w) of M is defined by:
n d
Per (w) := {λ ∈ M /p (λ,w) = 0}.
n d n
If w 6= 1: λ ∈Per (w) ⇔ λ posseses a cycle of multiplier w and exact period n.
n
The last assertion essentially follows from the fact that (f,z) is a smooth point of Q
n
if f¯∈ Per (w) when w 6= 1. This is a consequence of the implicit function theorem since,
n
in that case, ∂ (fn(z)−z) = w (f,z)−1 6= 0. Although much more delicate, a specific
∂z n
description of Per (1) is possible (see [19] and [11]). Let us stress that, in the present
n
work, we will not have to use the hypersurfaces Per (1).
n
2.3 Milnor’s compactification of M
2
We recall here some fundamental geometric properties of the moduli space of quadratic
rational maps. Our main references for this subsection are the paper of Milnor [19] and
the fourth chapter of the book of Silverman [24].
5
A generic f ∈ Rat has 3 fixed points with multipliers µ ,µ ,µ . The symmetric
2 1 2 3
functions
σ := µ +µ +µ , σ := µ µ +µ µ +µ µ , σ := µ µ µ
1 1 2 3 2 1 2 1 3 2 3 3 1 2 3
are clearly well defined on M and it follows from the holomorphic index formula that
2
σ −σ +2 = 0. Milnor has actually shown that (σ ,σ ) induces a good parametrization
3 1 1 2
of M ([19]).
2
Theorem 2.3 (Milnor) The map M → C2 defined by f¯ 7→ (σ ,σ ) is a biholomor-
2 1 2
phism. Using this identification, p (λ,w) is a polynomial on C2×C. Moreover, for every
n
fixed w ∈ C, the degree of p (·,w) is equal to ν2(n) which is the number of hyperbolic
n 2
components of period n in the Mandelbrot set.
This gives a projective compactification
M ∋ f¯7−→ (σ :σ : 1) ∈P2
2 1 2
whose corresponding line at infinity will be denoted by L
L := {(σ : σ :0); (σ ,σ ) ∈ C2\{0}}.
1 2 1 2
AnyPer (w)maybeseenasacurveinP2. Itisimportanttostressthatthiscompact-
n
ification is actually natural and that the ”behaviour near L” captures a lot of dynamically
meaningful informations. We shall use the following facts, also due to Milnor ([19]).
Proposition 2.4 (Milnor)
1) For all w ∈ C the curve Per (w) is a line whose equation in C2 is (w2 +1)λ −
1 1
wλ −(w3 +2) = 0 and whose point at infinity is (w : w2 +1 : 0). In particular,
2
Per (0) = {λ = 2} is the line of quadratic polynomials, its point at infinity is
1 1
(0 : 1 :0).
2) For n > 1 and w ∈ C the points at infinity of the curves Per (w) are of the form
n
(u :u2+1 :0) with uq = 1 and q ≤ n.
Working with the same compactification, Epstein [11] has proved the boundedness of
certain hyperbolic components of M :
2
Theorem 2.5 (Epstein) Let H be a hyperbolic component of M whose elements are
2
possessing two distinct attracting cycles. If neither attractor is a fixed point then H is
relatively compact in M
2
3 Approximation formulas
In all the section we consider a fixed holomorphic family f : M ×P1 → P1 of degree d
rational maps whose parameter space M is an m-dimensional complex manifold.
As for the modulispace M (see section 2), oneassociates to such afamily a collection
d
of polynomials and hypersurfaces of M:
6
Definition-Proposition 3.1 Let f : M×P1 → P1 be a family of degree d rational maps
on the Riemann sphere. For any n ∈ IN∗ there exists a monic polynomial p ∈ O(M)[X]
n
such that for any w ∈ C\{1}:
p (λ,w) = 0 ⇔ f has a cycle of multiplier w and exact period n.
n λ
The hypersurfaces Per (w) are defined by Per (w) := {p (·,w) = 0}.
n n n
We then define the currents [Per (w)] by setting:
n
[Per (w)] := ddcln|p (·,w)|.
n n
Up to multiplicity, [Per (w)] is the current of integration on the hypersurface Per (w).
n n
The main goal of this section is to show that the bifurcation current T of the family
bif
(f ) is a limit of laminar currents whose dynamical interpretation is clear:
λ λ∈M
d−n 2π
T = limd−n[Per (0)] = lim [Per (eiθ)]dθ.
bif n n n 2π Z n
0
Our proof essentially relies on an approximation formula for L in terms of potentials of
the form ln|p (·,w)| (see theorem 3.4) and on standard compacity properties for positive
n
currents.
3.1 Approximation formulas for the Lyapunov function
Definition 3.2 We use the notations introduced in the definition 3.1. For every integer
n we define a pair of functions L and L0 on the parameter space M by setting:
n n
L0 = d−nlog|p (λ,0)|
n n
d−n 2π
L = log|p (λ,eiθ)|dθ.
n n
2π Z
0
These functions are actually good approximations of the Lyapunov function L. As we
shall see, this basically follows from the fact that the set of attracting cycles is finite and
the following property of the Lyapunov function L:
Theorem 3.3 Let f be a rational map of degree d and RP(f,n) be the set of repelling
n-periodic points of f. Then:
d−n
L(f)= limd−n log|f′| = lim log|(fn)′|.
n X n n X
RP(f,n) RP(f,n)
AsimilarpropertyactuallyholdsforholomorphicendomorphismsofPk hasithasbeen
proved in [2]. The proof is more direct when k = 1; basically one simply has to combine
Lyubich’s construction of repulsiv n-periodic points [15] with Koebe theorem to control
(fn)′ on them. More precisely, while in the proof given in [2] one uses a specific theorem
(the main result of that paper) to get the estimate (a ) that is to compare 1 ln|Jacfn(z)|
2 n
with L(f), one may simply use Cauchy estimates and the Koebe distorsion theorem.
Let us now state the approximation result on which will be based our study.
7
Theorem 3.4 We use the notations introduced in the above definition.
1) The functions L0 are p.s.h on M. The sequence (L0) is locally uniformly bounded
n n n
from above and converges pointwise and in L1 to L.
loc
2) The functions L are positive, continuous and p.s.h on M. The sequence (L ) is
n n n
locally uniformly bounded and converges pointwise (and therefore in L1 ) to L.
loc
Proof. 1) The function L0 is clearly p.s.h on M. Let us denote by w (λ) the roots of the
n n,j
polynomial p (λ,·):
n
N(n)
p (λ,w) = Π (w−w (λ)).
n j=1 n,j
Using this decomposition one gets:
N(n) N(n)
L0(λ) = d−n log|w (λ)| ≤ d−n log+|w (λ)| =
n n,j n,j
X X
j=1 j=1
d−n
= log|(fn)′| = d−n log|f′|. (3.1)
n λ λ
X X
RP(fλ,n) RP(fλ,n)
As fn has at most dn +1 fixed points, this shows that the sequence (L0) is locally
λ n n
uniformly bounded from above. Moreover, since f has only a finite number of attracting
λ
cycles and w (λ) is the multiplier of such a cycle of period n when |w (λ)| < 1, the
n,j n,j
inequality in 3.1 is an equality for all but a finite number of values of n. Then, for n big
enough we have:
N(n)
d−n
L0(λ) = d−n log+|w (λ)| = log|(fn)′|. (3.2)
n n,j n λ
X X
j=1 RP(fλ,n)
Thus, (L0) is a sequence of p.s.h functions which is locally uniformly bounded from
n n
above and, according to theorem 3.3, converges pointwise to L. By Slutsky’s lemma, L is
therefore the only limit value of (L0) in L1 . Then, by a classical compacity theorem for
n n loc
families of p.s.h functions (see [22] appendix), (L0) converges to L in L1 .
n n loc
2) Using the same decomposition than above for p (λ,w) and the classical formula
n
log+|a| = 1 2πlog|eiθ −a|dθ, one gets:
2π 0
R
N(n)
L (λ) = d−n log+|w (λ)|. (3.3)
n n,j
X
j=1
As p (λ,w) is holomorphic in λ, this shows that the function L is positive, continuous
n n
and p.s.h on M. The sequence (L ) is, as before, locally uniformly bounded from above.
n n
It now suffices to observe that L0(λ) = L (λ) for n big enough, as it follows from 3.3 and
n n
3.2. ⊔⊓
8
3.2 The bifurcation current as a limit of uniformly laminar currents
Since the bifurcation current T is given by T = ddcL, our approximation formulas for
bif bif
L lead to the following result:
Theorem 3.5 We use the notations introduced in the definition 3.1. The bifurcation cur-
rent T of a holomorphic family of rational maps coincides with the limit of the following
bif
sequences of laminar currents:
1) T = lim d−n[Per (0)]
bif n n
2) T = lim d−n 2π[Per (eiθ)]dθ.
bif n 2π 0 n
R
Theprooffollowsimmediatelyfromtheorem3.4bytakingddc. Forthesecondassertion
however, one first has to observe that the following identity occurs:
Lemma 3.6 ddcL = d−n 2π[Per (eiθ)]dθ.
n 2π 0 n
R
Proof. Let φ be a (m−1,m−1) test form on M where m is the complex dimension of M.
One has to check that:
d−n 2π
hddcL ,φi = h[Per (eiθ)],φidθ. (3.4)
n n
2π Z
0
Let dV be a volume form on M and ϕ be a smooth function such that ddcφ = ϕdV. Let
us temporarily assume that the function ϕ(λ)log|p (λ,eiθ)| is integrable on M ×[0,2π].
n
Then (3.4) easily follows from Fubini’s theorem:
d−n 2π
hddcL ,φi = hL ,ddcφi = log|p (λ,eiθ)|dθ ddcφ
n n n
2π Z Z
M (cid:0) 0 (cid:1)
d−n 2π
= ϕ(λ)log|p (λ,eiθ)|dθ dV
n
2π Z Z
M (cid:0) 0 (cid:1)
d−n 2π
= ϕ(λ)log|p (λ,eiθ)|dV dθ
n
2π Z Z
0 (cid:0) M (cid:1)
d−n 2π
= hddclog|p (λ,eiθ)|,φidθ.
n
2π Z
0
It remains to see that log|p (λ,eiθ)| is integrable on Supp φ × [0,2π]. Let c be
n n
an upper bound for log|p (λ,eiθ)| on Supp φ × [0,2π].(cid:0) Then,(cid:1)the negative function
n
log|p (λ,eiθ)| − c is indeed integrable(cid:0) on Su(cid:1)pp φ × [0,2π]. This follows from the
n n
positivity of L (see theorem 3.4: (cid:0) (cid:1)
n
2π
log|p (λ,eiθ)|−c dθ dV = 2πdn L dV −2πc dV
n n n n
Z Z Z Z
Suppφ(cid:0) 0 (cid:0) (cid:1) (cid:1) Suppφ Suppφ
≥ −2πc dV.
n
Z
Suppφ
⊔⊓
Let us formulate some immediate consequences of the first assertion of theorem 3.5.
9
Corollary 3.7 A point λ is in the bifurcation locus of the family f : M × P1 → P1
0
if and only if λ = lim λ where f has a super-attracting cycle of period k for some
0 n n λn n
increasing sequence of integers (k ) .
n n
Proof. If λ belongs to the bifurcation locus then the existence of the sequence (λ ) fol-
0 n n
lows immediately from theorem 3.5. Conversely, the existence of such a sequence (λ )
n n
implies that λ is not stable since, otherwise, f would have an infinite number of at-
0 λ0
tracting cycles. ⊔⊓
Let us also recall that for the polynomial family f := zd+λ (λ ∈C), the bifurcation
λ
locus is the boundary of the Mandelbrot set M and the bifurcation current is a measure
d
which turns out to be the harmonic measure of M . In this setting the first assertion of
d
theorem 3.5 yields the following corollary which was first proved by Levin [14]:
Corollary 3.8 let M be the set of complex numbers λ for which the Julia set of f :=
d λ
zd+λ is connected. Then
limd−n δ = µ
λ
n X
fn(0)=0
λ
where µ is the harmonic measure of M .
d
3.3 Approximation of the bifurcation measure
k
The higher degree bifurcation currents T (k ≤ m) of holomorphic families have been
bif
studied in [1]. As one may expect, they(cid:0) can(cid:1) be obtained as limits of intersections of the
laminar currents introducedintheformer subsection. For thesake of brevity weshallonly
consider here the case of the bifurcation measure µ = (ddcL)m.
bif
Our aim is to approximate µ by measures of the following type:
bif
[Per (eiθ1)]∧···∧[Per (eiθm)] dθ ···dθ . (3.5)
Z n1 nm 1 m
[0,2π]m
Let us observe that, for any fixed variety Per (eiθp), the set of θ ∈ [0,2π] for which
p
Per (eiθp) shares a non trivial component with Per (eiθ) for some m ∈ IN∗ is at most
p m
countable. This follows from Fatou’s theorem on the finiteness of the set of non-repelling
cycles. In particular, the wedge products [Pern1(eiθ1)]∧···∧[Pernm(eiθm)] make sense for
almost every (θ ,···,θ ) ∈ [0,2π]m and the measures given in 3.5 are well defined. Our
1 m
result may be stated as follows:
Theorem 3.9 We use the notations introduced in the definition 3.1. Let µ be the bifur-
bif
cation measure of a holomorphic family (f ) of rational maps. There exists increasing
λ λ∈M
sequences of integers k (n),...,k (n) such that:
2 m
d−(n+k2(n)+···+km(n)) m
µ = lim [Per (eiθ1)]∧ [Per (eiθj)] dθ ···dθ .
bif n (2π)m Z[0,2π]m n j^=2 kj(n) 1 m
Proof. The problem is local and we may therefore replace M by Cm. We shall use the two
following lemmas:
10
