Đề tài " Ergodic properties of rational mappings with large topological degree " pot

Annals of Mathematics


Ergodic properties of rational
mappings with large
topological degree


By Vincent Guedj


Annals of Mathematics, 161 (2005), 1589–1607
Ergodic properties of rational mappings
with large topological degree
By Vincent Guedj
Abstract
Let X be a projective manifold and f : X → X a rational mapping with
large topological degree, d
t
> λ
k−1
(f) := the (k − 1)
th
dynamical degree of f.
We give an elementary construction of a probability measure µ
f
such that
d
−n
t
(f

n
)
∗
Θ → µ
f
for every smooth probability measure Θ on X. We show
that every quasiplurisubharmonic function is µ
f
-integrable. In particular µ
f
does not charge either points of indeterminacy or pluripolar sets, hence µ
f
is
f-invariant with constant jacobian f
∗
µ
f
= d
t
µ
f
. We then establish the main
ergodic properties of µ
f
: it is mixing with positive Lyapunov exponents, preim-
ages of ”most” p oints as well as repelling periodic points are equidistributed
with respect to µ
f
. Moreover, when dim
C

X ≤ 3 or when X is complex homo-
geneous, µ
f
is the unique measure of maximal entropy.
Introduction
Let X be a projective algebraic manifold and ω a Hodge form on X nor-
malized so that

X
ω
k
= 1, k = dim
C
X. Let f : X → X be a rational
mapping. We shall always assume in the sequel that f is dominating; i.e., its
jacobian determinant does not vanish identically in any coordinate chart. We
let I
f
denote the indeterminacy locus of f (the points where f is not holomor-
phic): this is an algebraic subvariety of codimension ≥ 2. We let d
t
denote the
topological degree of f: this is the number of preimages of a generic point.
Define f
∗
ω
k
to be the trivial extension through I
f
of (f

|X\I
f
)
∗
ω ∧ · · · ∧
(f
|X\I
f
)
∗
ω. This is a Radon measure of total mass d
t
. When d
t
> λ
k−1
(f) (see
Section 1 below), we give an elementary construction of a probability measure
µ
f
such that d
−n
t
(f
n
)
∗
ω
k
→ µ

f
. We show that every quasiplurisubharmonic
function is µ
f
-integrable (Theorem 2.1). In particular µ
f
does not charge
pluripolar sets. This answers a question raised by Russakovskii and Shiffman
[RS 97] which was addressed by several authors (see [HP 99], [FG 01], [G 02],
[Do 01], [DS 02]). This also shows that µ
f
is an invariant measure with positive
entropy ≥ log d
t
> 0. Thus f has positive topological entropy.
1590 VINCENT GUEDJ
Building on the work of Briend and Duval [BD 01], we then establish the
main ergodic properties of µ
f
: it is mixing with p ositive Lyapunov exponents,
preimages of ”most” points as well as repelling periodic points are equidis-
tributed with respect to µ
f
(Theorem 3.1). Moreover, when dim
C
X ≤ 3 or
when the group of automorphisms Aut(X) acts transitively on X, µ
f
is the
unique measure of maximal entropy (Theorem 4.1).

Acknowledgements. We thank Jeffrey Diller, Julien Duval and Charles
Favre for several interesting conversations.
1. Numerical invariants
In this section we define and establish inequalities between several numer-
ical invariants. This involves some technicalities because our mappings are not
holomorphic and also, the psef/nef-cones are not well understood in dimension
≥ 4. What follows is quite simple when f is holomorphic (the only nontrivial
part, the link between entropy and dynamical degrees, goes back to Gromov
[Gr 77]). When X = P
k
, a clean treatment of the dynamical degrees is given
by Russakovskii and Shiffman in [RS 97]: the situation is simpler since P
k
is
a complex homogeneous manifold whose cohomology vector spaces H
l,l
are all
one-dimensional.
1.1. Dynamical degrees. Given a smooth form α of bidegree (l, l),
1 ≤ l ≤ k, we define the pull-back of α by f in the following way: let
Γ
f
⊂ X × X denote the graph of f and consider a desingularization
˜
Γ
f
of Γ
f
.
We have a commutative diagram

˜
Γ
f
π
1

π
2

X
f
−→ X
where π
1
, π
2
are holomorphic maps. We set f
∗
α := (π
1
)
∗
(π
∗
2
α) where we push
forward the smooth form π
∗
2
α by π

1
as a current. Note that f
∗
α is actually
a form with L
1
loc
-coefficients which coincides with the usual smooth pull-back
(f
|X\I
f
)
∗
α on X \ I
f
; thus the definition does not depend on the choice of
desingularization. In other words, f
∗
α is the trivial extension, as current, of
(f
|X\I
f
)
∗
α through I
f
.
This definition induces a linear action on the cohomology space H
l,l
(X, R)

which preserves H
l,l
a
(X, R), the subspace generated by complex subvarieties
of codimension l. We let H
l,l
psef
(X, R) denote the closed cone generated by
effective cycles.
RATIONAL MAPPINGS WITH LARGE TOPOLOGICAL DEGREE
1591
Definition 1.1. Set δ
l
(f) :=

X
f
∗
ω
l
∧ ω
k−l
. We define the l
th
-dynamical
degree of f to be
λ
l
(f) := lim inf
n→+∞

[δ
l
(f
n
)]
1/n
.
This definition clearly does not depend on the choice of the K¨ahler form ω.
Observe that for l = k, λ
k
(f) is the topological degree of f, i.e. the number
of preimages of a generic point, which we shall preferably denote by d
t
(f) (or
simply d
t
when no confusion can arise).
Proposition 1.2. i) The sequence l → λ
l
(f)/λ
l+1
(f) is nondecreasing,
0 ≤ l ≤ k − 1; i.e., log λ
l
is a concave function of l. In particular if d
t
=
λ
k
(f) > λ

k−1
(f), then d
t
> λ
k−1
(f) > · · · > λ
1
(f) > 1.
ii) There exists C > 0 such that for all dominating rational self-maps
f, g : X → X,
δ
1
(g ◦ f) ≤ Cδ
1
(f)δ
1
(g).
In particular δ
1
(f
n+m
) ≤ Cδ
1
(f
n
)δ
1
(f
m
) so that λ

1
(f) = lim[δ
1
(f
n
)]
1/n
. More-
over λ
1
(f) is invariant under birational conjugacy.
iii) Let r
1
(f) denote the spectral radius of the linear action induced by f
∗
on H
1,1
a
(X, R) and set λ

1
(f) = lim sup r
1
(f
n
)
1/n
. There exists C > 0 and for
every ε > 0 there exists C
ε

> 0, such that for all n,
0 ≤ r
1
(f
n
) ≤ Cδ
1
(f
n
) ≤ C
ε
[λ

1
(f) + ε]
n
.
In particular λ
1
(f) = λ

1
(f).
Proof. i) It is equivalent to prove that λ
l+1
(f)λ
l−1
(f) ≤ λ
l
(f)

2
for all
1 ≤ l ≤ k − 1. This is a consequence of δ
l+1
(f
n
)δ
l−1
(f
n
) ≤ δ
l
(f
n
)
2
, which
follows from Teissier-Hovanskii mixed inequalities: it suffices to apply Theorem
1.6.C
1
of [Gr 90] in the graph
˜
Γ
f
n
to the smooth semi-positive forms π
∗
1
ω
i

and
π
∗
2
ω
k−i
.
ii) Let f, g : X → X be dominating rational self-maps. It is possible to
define f
∗
T for any positive closed current T of bidegree (1, 1) (see [S 99]). In
particular, f
∗
(g
∗
ω) is a globally well defined positive closed current of bidegree
(1, 1) on X which coincides with (g◦f)
∗
ω in X \I
f
∪f
−1
(I
g
). Now (g◦f )
∗
ω is a
form with L
1
loc

coefficients, thus it does not charge the proper algebraic subset
I
f
∪ f
−1
(I
g
). Therefore we have an inequality between these two currents,
(g ◦ f)
∗
ω ≤ f
∗
(g
∗
ω)(†)
and the same inequality holds in H
1,1
psef
(X, R). Note that (†) does not hold in
general if we replace [ω] by the class of an effective divisor (see Remark 1.4
below).
Let N be a norm on H
l,l
(X, R). There exists C
1
> 0 such that for all class
α ∈ H
l,l
psef
(X, R), N (α) ≤ C

1

α ∧ ω
k−l
. We infer from (†) and the continuity
1592 VINCENT GUEDJ
of (α, β) →

α ∧ β that
δ
1
(g ◦ f) ≤

f
∗
(g
∗
ω) ∧ ω
k−1
=

g
∗
ω ∧ f
∗
ω
k−1
≤ Cδ
1
(g)δ

1
(f).
Note that we have used the fact that f
∗
[ω
k−1
] ∈ H
k−1,k−1
psef
(X, R) (see below for
the definition of f
∗
and related properties). We infer from the latter inequality
that the sequence (δ
1
(f
n
)) is quasisubmultiplicative, hence the lim inf can be
replaced by a lim (or an inf) in the definition of λ
1
(f). Moreover if g is
birational, we get
δ
1
(g ◦ f
n
◦ g
−1
) ≤ Cδ
1

(g)δ
1
(g
−1
)δ
1
(f
n
);
hence λ
1
(g ◦ f ◦ g
−1
) = λ
1
(f); i.e., λ
1
(f) is a birational invariant.
iii) Observe that H
1,1
psef
(X, R) is a closed convex cone with nonempty inte-
rior which is strict (i.e. H
1,1
psef
(X, R) ∩ −H
1,1
psef
(X, R) = {0}) and preserved by
f

∗
. Therefore there exists, for all n ∈ N, a class [θ
n
] ∈ H
1,1
psef
(X, R) such that
(f
n
)
∗
[θ
n
] = r
1
(f
n
)[θ
n
]. This can be thought of as a Perron-Frobenius-type
result (see Lemma 1.12 in [DF 01]).
Fix a basis [ω
1
] = [ω],[ω
2
], . . . , [ω
s
] of H
1,1
a

(X, R), where the ω

j
s are
smooth forms such that ω
j
≤ ω . We normalize θ
n
=

j
α
j,n
ω
j
so that
||[θ
n
]|| := max
j
|α
j,n
| = 1; thus θ
n
≤ sω. Observe that [θ] →

θ ∧ ω
k−1
is
a continuous form on H

1,1
a
(X, R) which is positive on H
1,1
psef
(X, R). Therefore
there exists C > 0 such that ||[θ]|| ≤ C

θ ∧ ω
k−1
, for all [θ] ∈ H
1,1
psef
(X, R).
This yields the first inequality:
r
1
(f
n
) = r
1
(f
n
)||[θ
n
]|| ≤ Cr
1
(f
n
)


θ
n
∧ ω
k−1
= C

(f
n
)
∗
θ
n
∧ ω
k−1
≤ Cs

(f
n
)
∗
ω ∧ ω
k−1
.
Conversely, fix ε > 0 and p > 1 such that r
1
(f
p
) ≤ (λ


1
(f) + ε/2)
p
.
Fix a norm N on H
1,1
a
(X, R). Since [θ] →

X
θ ∧ ω
k−1
defines a continuous
linear form on H
1,1
a
(X, R), there exists C
N
> 0 such that for all [θ], |

X
θ ∧
ω
k−1
| ≤ C
N
N([θ]). Set
˜
N(f) := sup
N([θ])=1

N(f
∗
[θ]). It follows from (†) that
N((f
n
)
∗
[ω]) ≤ N (f
∗
(. . . f
∗
[ω]) . . . ), hence
0 ≤

(f
n
)
∗
ω ∧ ω
k−1
≤ C
N
[
˜
N(f
p
)]
q
N([(f
r

)
∗
ω]),
where n = pq + r. Now for every ε > 0 one can find a norm N
ε
on H
1,1
a
(X, R)
such that r
1
(f
p
) ≤
˜
N
ε
(f
p
) ≤ r
1
(f
p
) + ε/2. This yields iii).
Remark 1.3. It is remarkable that the mixed inequalities λ
l+1
λ
l−1
≤ λ
2

l
contain all previously known inequalities, e.g. λ
l+l

(f) ≤ λ
l
(f)λ
l

(f) (which
are proved by Russakovskii and Shiffman [RS 97] when X = P
k
).
RATIONAL MAPPINGS WITH LARGE TOPOLOGICAL DEGREE
1593
Remark 1.4. One should be aware that simple inequalites like (†) are false
if we replace [ω] by the class of an effective divisor (in particular, Lemma 3
in [Fr 91] is wrong). Here is a simple 2-dimensional counterexample: consider
σ : Y → Y a biholomorphism of some projective surface Y with a nontrivial
2-cycle {p, σ(p)}. Let π : X → Y be the blow-up of Y at point p, E = π
−1
(p)
and q = π
−1
(σ(p)). Set f = π
−1
◦ σ ◦ π : X → X. This is a rational self-
map of X such that I
f
= {q}, f(q) = E, f(E) = q. Therefore f

∗
[E] = 0, so
f
∗
(f
∗
[E]) = 0 while (f ◦ f)
∗
[E] = [E] (contradicting Lemma 3 in [Fr 91]).
We define similarly the push-forward by f as f
∗
α := (π
2
)
∗
(π
∗
1
α). This
induces a linear action on the cohomology spaces H
l,l
(X, R) which is dual to
that of f
∗
on H
k−l,k− l
(X, R). The push-forward of any positive closed current
of bidegree (1, 1) is well defined and yields a positive closed current of bidegree
(1, 1) on X. Therefore H
1,1

psef
(X, R) is preserved by f
∗
(by duality, the dual
cone H
k−1,k−1
nef
(X, R) is preserved by f
∗
). We have a (†)

inequality
(g ◦ f)
∗
ω ≤ g
∗
(f
∗
ω).(†

)
This yields results on λ
k−1
(f) analogous to those obtained for λ
1
(f). We
summarize this in the following:
Proposition 1.5. The dynamical degree λ
k−1
(f) is invariant under bi-

rational conjugacy and satisfies
λ
k−1
(f) = lim[δ
k−1
(f
n
)]
1/n
= lim[r
k−1
(f
n
)]
1/n
,
where r
k−1
(f) denotes the spectral radius of the linear action induced by f
∗
on
H
k−1,k−1
a
(X, R).
Remark 1.6. When 2 ≤ l ≤ k − 2 (hence k = dim
C
X ≥ 4), it seems
unlikely that the cone H
l,l

psef
(X, R) (or its dual H
k−l,k− l
nef
(X, R)) is preserved by
f
∗
(or f
∗
), unless f is holomorphic. It follows however from previous proofs
that if H
l,l
psef
(X, R) is f
∗
-invariant and f
∗
[ω
l
] ≤ f
∗
(. . . f
∗
[ω
l
]) . . . ), then we
get similar information on λ
l
(f). These conditions are satisfied if e.g. X is a
complex homogeneous manifold.

1.2. Topological entropy. For p ∈ X, we define f (p) = π
2
π
−1
1
(p) and
f
−1
(p) = π
1
π
−1
2
(p): these are proper algebraic subsets of X. Note that I
f
=
{p ∈ X / dim f(p) > 0}. We set I
−
f
:= {p ∈ X / dim f
−1
(p) > 0} and let C
f
denote the critical set of f, i.e. the closure of the set of points in X \ I
f
where
Jf(p) = 0. Clearly I
−
f
⊂ f (C

f
) and I
−
f
n
⊂ f
n
(I
−
f
); thus
∪
n≥1
I
−
f
n
⊂ PC(f ) := ∪
n≥1
f
n
(C
f
) := postcritical set of f.
Observe that for a ∈ X \ ∪
n≥0
I
−
f
n

, we can define for all n ≥ 0 the probability
measures d
−n
t
(f
n
)
∗
ε
a
. Here ε
a
denotes the Dirac mass at point a. Therefore if
1594 VINCENT GUEDJ
ν is a probability measure on X which does not charge PC(f), we can define
ν
n
:=
1
d
n
t
(f
n
)
∗
ν =

1
d

n
t
(f
n
)
∗
ε
a
dν(a).
The latter are again probability measures which do not charge PC(f) since
f(PC(f)) ⊂ PC(f ). We will prove, when d
t
> λ
k−1
(f), that the ν

n
s converge
to an invariant measure µ
f
(Theorem 3.1).
We now give a definition of entropy which is suitable for our purpose (this
definition differs slightly from that of Friedland [Fr 91]). Observe that for all
n ≥ 0, I
f
n
⊂ f
−n
(I
f

). We set
Ω
f
:= X \ ∪
n∈
Z
f
n
(I
f
).
This is a totally invariant subset of X such that f
n
is holomorphic at ev-
ery point for all n ≥ 0. Following Bowen’s definition [Bo 73] we define the
topological entropy of f relative to Y ⊂ Ω
f
to be
h
top
(f
|Y
) := sup
ε>0
lim
1
n
log max{F / F (n, ε)-separated set in Y },
where F is said to be (n, ε)-separated if d
n

(x, y) ≥ ε whenever (x, y) ∈ F
2
,
x = y. Here d
n
(x, y) = max
0≤j≤n−1
d(f
j
(x), f
j
(y)) for some metric d on X.
We define h
top
(f) := h
top
(f
|Ω
f
). These definitions clearly do not depend on
the choice of the metric.
Given ν an ergodic probability measure such that ν(Ω
f
) = 1, we define
the metric entropy of ν following Brin-Katok [BK 83]: for almost every x ∈ Ω
f
,
h
ν
(f) := sup

ε>0
lim −
1
n
ν(B
n
(x, ε)),
where B
n
(x, ε) = {y ∈ Ω
f
/ d
n
(x, y) < ε}. One easily checks that the topolog-
ical entropy dominates any metric entropy:
h
top
(f) ≥ sup{h
ν
(f), ν ergodic with ν(Ω
f
) = 1}.
However it is not clear whether the reverse inequality holds, as it does for
nonsingular mappings. More generally if Y is a Borel subset of Ω
f
such that
ν(Y ) > 0, then h
ν
(f) ≤ h
top

(f
|Y
). This is what Briend and Duval call the
relative variational principle [BD 01].
Let Γ
n
= {(x, f(x), . . . , f
n−1
(x)), x ∈ Ω
f
} be the iterated graph of f and
set
lov(f) :=
lim
1
n
log(Vol(Γ
n
)) = lim
1
n
log


Γ
n
ω
k
n


,
where ω
n
=

n
i=1
π
∗
i
ω, π
i
being the projection X
n
→ X on the i
th
factor. A
well-known argument of Gromov [Gr 77] yields the estimate h
top
(f) ≤ lov(f).
When f is a holomorphic endomorphism (i.e. when I
f
= ∅), a simple coho-
RATIONAL MAPPINGS WITH LARGE TOPOLOGICAL DEGREE
1595
mological computation yields lov(f) = max
1≤j≤k
log λ
j
(f). Such computation

is more delicate for mappings which are merely meromorphic. The following
lemma will b e quite useful in our analysis.
Lemma 1.7. Assume dim
C
X ≤ 3 or X is a complex homogeneous mani-
fold. Fix ε > 0. Then there exists C
ε
> 0 such that
0 ≤

Ω
f
(f
n
1
)
∗
ω ∧ · · · ∧ (f
n
k−1
)
∗
ω ∧ ω ≤ C
ε
[ max
1≤j≤k−1
λ
j
(f) + ε]
max n

i
,
for all (n
1
, . . . , n
k−1
) ∈ N
k−1
.
Proof. We can assume n
1
≤ · · · ≤ n
k−1
without loss of generality.
When k = dim
C
X ≤ 2 everything is clear. Assume k = 3. Then

Ω
f
(f
n
1
)
∗
ω ∧ (f
n
2
)
∗

ω ∧ ω ≤

X
ω ∧ (f
n
2
−n
1
)
∗
ω ∧ (f
n
1
)
∗
ω. Here we use the
fact that (f
n
2
−n
1
)
∗
ω ∧ (f
n
1
)
∗
ω is a globally well defined positive closed current
of bidegree (2, 2) on X. This follows from the intersection theory of positive

currents (see [S 99]), since (f
n
2
−n
1
)
∗
ω and (f
n
1
)
∗
ω have continuous potentials
outside a set of codimension ≥ 2. Using Propositions 1.2 and 1.5, we thus get,
for ε > 0 fixed,
0 ≤

Ω
f
(f
n
1
)
∗
ω ∧ (f
n
2
)
∗
ω ∧ ω ≤ CN((f

n
2
−n
1
)
∗
[ω])N((f
n
1
)
∗
[ω])
≤ C
ε
[λ
1
(f) + ε]
n
2
−n
1
[λ
2
(f) + ε]
n
1
≤ C
ε
max
j=1,2

[λ
j
(f) + ε]
n
2
.
When dim
C
X ≥ 4, it becomes more difficult to define and control the
positivity of (f
i
1
)
∗
ω ∧ (f
i
2
)
∗
ω ∧ (f
i
3
)
∗
ω on X \ Ω
f
. However, when X is a
complex homogeneous manifold (i.e. when the group of automorphisms Aut(X)
acts transitively on X), one can regularize every positive closed current T
within the same cohomology class and get this way an approximation of T by

smooth positive closed forms T
ε
 T (see [Hu 94]). Proceeding as above and
replacing each singular term (f
n
)
∗
ω, (f
m
)
∗
ω by a smooth approximant, we
see that Fatou’s lemma yields the desired inequality (this argument is used in
[RS 97] to obtain related inequalities).
Corollary 1.8. Assume dim
C
X ≤ 3 or X is complex homogeneous.
Then
h
top
(f) ≤ lov(f) ≤ max
1≤j≤k
log λ
j
(f).
Proof. By definition Vol(Γ
n
) =

0≤i

1
, ,i
k
≤n−1

Ω
f
(f
i
1
)
∗
ω ∧ · · · ∧ (f
i
k
)
∗
ω.
Assume i
1
≤ · · · ≤ i
k
and fix ε > 0. Then

Ω
f
(f
i
1
)

∗
ω ∧ · · · ∧ (f
i
k
)
∗
ω = d
t
(f)
i
1

Ω
f
(f
i
2
−i
1
)
∗
ω ∧ · · · ∧ (f
i
k
−i
1
)
∗
ω ∧ ω
≤ C

ε
d
t
(f)
i
1
[ max
1≤j≤k−1
λ
j
(f) + ε]
i
k
−i
1
≤ C
ε
[ max
1≤j≤k
λ
j
(f) + ε]
n
.
1596 VINCENT GUEDJ
Therefore Vol(Γ
n
) ≤ C
ε
n

k
[max λ
j
(f) + ε]
n
, hence lov(f) ≤ log[max λ
j
(f) + ε].
When ε → 0 we have the desired inequality.
We will also need a relative version of this estimate.
Corollary 1.9. Assume dim
C
X ≤ 3 or X is complex homogeneous. Let
Y be a proper subset of Ω
f
. If Y is algebraic then
h
top
(f
|Y
) ≤ lov(f
|Y
) ≤ max
1≤j≤k−1
log λ
j
(f).
In the general case, we simply get
h
top

(f
|Y
) ≤ lim
1
n
log(Vol(Γ
n
|Y )
ε
),
where ε > 0 is fixed, Γ
n
|Y denotes the restriction of Γ
n
to Y and (Γ
n
|Y )
ε
is
the ε -neighborhood of Γ
n
|Y in Γ
n
.
2. A canonical invariant measure µ
f
Theorem 2.1. Let f : X → X be a rational mapping such that d
t
(f) >
λ

k−1
(f). Then there exists a probability measure µ
f
such that if Θ is any
smooth probability measure on X,
1
d
t
(f)
n
(f
n
)
∗
Θ −→ µ
f
,
where the convergence holds in the weak sense of measures. Moreover :
i) Every quasiplurisubharmonic function is in L
1
(µ
f
). In particular µ
f
does not charge pluripolar sets and log
+
||Df
±1
|| ∈ L
1

(µ
f
).
ii) f
∗
µ
f
= d
t
(f)µ
f
; hence µ
f
is invariant f
∗
µ
f
= µ
f
.
iii) h
top
(f) ≥ h
µ
f
(f) ≥ log d
t
(f) > 0. In particular µ
f
is a measure of

maximal entropy when dim
C
X ≤ 3 or when X is complex homogeneous.
Proof. Fix a a noncritical value of f and r > 0 such that f admits
d
t
= d
t
(f) well defined inverse branches on B(a, r). Fix Θ a smooth prob-
ability measure with compact support in B(a, r). Then d
−1
t
f
∗
Θ is a smooth
probability measure on X. Since X is K¨ahler, the dd
c
-lemma (see [GH 78,
p. 149]) yields
1
d
t
f
∗
Θ = Θ + dd
c
(S),
where S is a smooth form of bidegree (k −1, k − 1). Replacing S by S + Cω
k−1
if necessary, we can assume 0 ≤ S ≤ Cω

k−1
for some constant C > 0. We now
take the pull-back of the previous equation by f, as explained in Section 1.
RATIONAL MAPPINGS WITH LARGE TOPOLOGICAL DEGREE
1597
Recall that (f
n
)
∗
dd
c
S = dd
c
(f
n
)
∗
S for all n (because (π
1
)
∗
, π
∗
2
commute
with d, d
c
). We infer, by induction, that
1
d

n
t
(f
n
)
∗
Θ = Θ + dd
c
S
n
, S
n
=
n−1

j=0
1
d
j
t
(f
j
)
∗
S.
Indeed observe that (f
n+1
)
∗
Θ = (f

n
)
∗
(f
∗
Θ), since these are the pull-backs of
smooth forms; they are smooth and coincide in X \

I
f
n
∪ I
f
n+1

, hence they
coincide everywhere since they have L
1
loc
-coefficients. Therefore
1
d
n+1
t
(f
n+1
)
∗
Θ =
1

d
n
t
(f
n
)
∗

1
d
t
f
∗
Θ

=
1
d
n
t
(f
n
)
∗
(Θ + dd
c
S) = Θ + dd
c
S
n+1

.
The sequence of positive currents (S
n
) is increasing since (f
j
)
∗
S ≥ 0.
Setting ||S
n
|| :=

X
S
n
∧ ω, we get
0 ≤ ||S
n
|| ≤ C
n−1

j=0
1
d
j
t

Ω
f
(f

j
)
∗
ω
k−1
∧ ω ≤ C
ε

j≥0

λ
k−1
(f) + ε
d
t

j
< +∞,
using Proposition 1.5 with ε > 0 small enough. Therefore (S
n
) converges
towards some p ositive current S
∞
; hence
1
d
n
t
(f
n

)
∗
Θ −→ µ
f
:= Θ + dd
c
S
∞
.
Observe that if Θ

is another smooth probability measure, then Θ

= Θ+dd
c
R,
for some smooth form R of bidegree (k − 1, k − 1). Since ||(f
n
)
∗
R|| = o(d
n
t
),
we have again d
−n
t
(f
n
)

∗
Θ

→ µ
f
.
Let ϕ be a quasiplurisubharmonic (qpsh) function on X, i.e. an upper
semi-continuous function which is locally given as the sum of a plurisubhar-
monic function and a smooth function. Translating and rescaling ϕ if necessary,
we can assume ϕ ≤ 0 and dd
c
ϕ ≥ −ω. It follows from a regularization result
of Demailly (see [De 99]) that there exist C > 0 and ϕ
ε
≤ 0 a smooth sequence
of functions pointwise decreasing towards ϕ such that dd
c
ϕ
ε
≥ −Cω. Using
Stokes’ theorem we get
0 ≤

(−ϕ
ε
)dµ
f
=

(−ϕ

ε
)Θ +

S
∞
∧ (−dd
c
ϕ
ε
) ≤

(−ϕ
ε
)Θ + C

S
∞
∧ ω,
since S
∞
≥ 0. The monotone convergence theorem thus implies
0 ≤

X
(−ϕ)dµ
f
≤

X
(−ϕ)Θ + C


X
S
∞
∧ ω < +∞.
Since any pluripolar set is included in the −∞ locus of a qpsh function,
µ
f
does not charge pluripolar sets. In particular µ
f
(I
f
) = 0; hence f
∗
µ
f
= µ
f
;
i.e. µ
f
is an invariant probability measure. Similarly µ
f
(I
−
f
) = 0 so that
f
∗
µ

f
= d
t
µ
f
; i.e. µ
f
has constant jacobian d
t
.
1598 VINCENT GUEDJ
It follows from the Rohlin-Parry formula (see [P 69]) that h
µ
f
(f) ≥ log d
t
.
Since µ
f
(Ω
f
) = 1, we get in particular h
top
(f) ≥ log d
t
> 0. This is reminiscent
of the well-known result of Misiurewicz and Przytycki that the topological
entropy of a C
1
-smooth endomorphism of a compact manifold is minorated by

log d
t
(see [KH 95]). When dim
C
X ≤ 3 or when X is complex homogeneous,
we get
h
µ
f
(f) ≤ h
top
(f) ≤ max
1≤j≤k
log λ
j
(f) = log d
t
,
by Proposition 1.2 and Corollary 1.8; hence µ
f
is a measure of maximal entropy.
3. First ergodic properties of µ
f
In this section we adapt the work of Briend and Duval [BD 01] to establish
some ergodic properties of µ
f
.
Theorem 3.1. Let f, µ
f
be as in Theorem 2.1. Then the following hold :

i) If ν is a probability measure which does not charge the postcritical set
PC(f) := ∪
j≥1
f
j
(C
f
), then d
−n
t
(f
n
)
∗
ν → µ
f
.
ii) The measure µ
f
is mixing.
iii) If we let χ
k
≥ · · · ≥ χ
1
denote the Lyapunov exponents of µ
f
, then
χ
1
≥

1
2
log(d
t
/λ
k−1
(f)) > 0.
iv) Let RPer
n
(f) denote the set of repelling periodic points of order n.
They are equidistributed with respect to µ
f
if lim sup(RPer
n
(f)/d
n
t
) ≤ 1. The
latter holds when dim
C
X ≤ 3 or when X is complex homogeneous.
Remark 3.2. When X = P
k
and f is holomorphic (i.e. when I
f
= ∅), the
measure µ
f
was constructed by Hubbard and Papadopol [HP 94] and Fornæss
and Sibony [FS 94]. The latter also proved ii) and a weaker version i


) of i):
they showed the existence of an exceptional pluripolar set E
f
⊂ P
k
such that
d
−n
t
(f
n
)
∗
ε
a
→ µ
f
if a /∈ E
f
. The remaining assertions iii), iv) were established
by Briend and Duval [BD 99], [BD 01], who also proved that the exceptional
set E
f
is actually a totally invariant algebraic subset of PC(f).
When X = P
k
but f is merely meromorphic, the measure µ
f
was con-

structed by Russakovskii and Shiffman [RS 97] by proving i

). Following
[BD 01], we actually show that E
f
is a subset of PC(f). Note however, that
one can not expect E
f
to be algebraic in the meromorphic case.
This result heavily relies on the following lemma. We thank Julien Duval
for explaining to us the construction of inverse branches on balls.
RATIONAL MAPPINGS WITH LARGE TOPOLOGICAL DEGREE
1599
Lemma 3.3. Set V
l
= ∪
l
j=1
f
j
(C
f
), where C
f
denotes the critical set of f.
Fix ε > 0 and an embedding X ⊂ CP
N
. Fix 1 < δ < d
t
/λ

k−1
, δ arbitrarily
close to d
t
/λ
k−1
. Then there exists l  1 such that the following hold:
i) For every holomorphic disk
∆ ⊂ L ∩ X \ V
l
, where L is a generic
projective linear subspace of codimension dim
C
X−1 in CP
N
, there are (1−ε)d
n
t
inverse branches of f
n
(n ≥ l) whose images ∆
−n
i
satisfy
diam(∆
−n
i
) ≤ Cδ
−n/2
,

where C is independent of n.
ii) For every ball
B ⊂ X \ V
l
, there are (1 − ε)d
n
t
inverse branches of f
n
on B, n ≥ l, whose images B
−n
i
satisfy
diam(B
−n
i
) ≤ Cδ
−n/2
.
Proof. Fix ε > 0 small and δ = d
t
/(λ
k−1
(f) + ε).
i) Let V
1
= f(C
f
) denote the set of critical values of f. Let D be an
algebraic curve on X which is not included in PC(f). Then V

1
∩ f
−n
D is finite
(possibly empty) for all n ≥ 0. Let α be a closed smooth form of bidegree
(k − 1, k − 1) which is cohomologous to [D]. Then α ≤ C
D
ω
k−1
for some
constant C
D
> 0. Note that C
D
= C
1
can be chosen independent of D if
we restrict ourselve to curves D which are the trace on X of projective linear
subspaces of P
N
; in this case we can choose α = (ω
k−1
FS
)
|X
, where ω
FS
denotes
the Fubini-Study K¨ahler form on P
N

. We assume in the sequel that V
1
is a
hypersurface of X (in general V
1
may have codimension ≥ 2 in X; in this case
we simply replace V
1
by some hypersurface
˜
V
1
containing V
1
). Let β be a closed
smooth (1, 1)-form cohomologous to [V
1
], β ≤ C
2
ω. Then
V
1
∩ f
−n
D =

[V
1
] ∩ (f
n

)
∗
[D] ≤ C
1

[V
1
] ∧ (f
n
)
∗
ω
k−1
≤ C
1
C
2

ω ∧ (f
n
)
∗
ω
k−1
≤ C
ε
[λ
k−1
(f) + ε]
n

= C
ε
δ
−n
d
n
t
,
where the last inequality follows from Proposition 1.5.
Since
∆ ∩ V
l
= ∅, there are d
l
t
well defined inverse branches f
−l
i
of f
l
on
∆. Set ∆
−l
i
= f
−l
i
∆. We can further define d
t
inverse branches of f on ∆

−l
i
if ∆
−l
i
∩ V
1
= ∅. It follows from the computation above that at most C
ε
δ
−l
d
l
t
of the ∆
−l
i
’s may intersect V
1
. Therefore we can define d
l+1
t
(1 − C
ε
δ
−l
) inverse
branches of f
l+1
on ∆. A straightforward induction shows that we can define

d
n
t
(1 − C
ε
δ
−l

j≥0
δ
−j
) ≥ d
n
t
(1 − ε/2) inverse branches of f
n
on ∆, if we fix
l large enough so that C
ε
δ
−l
(1 − δ
−1
) < ε/2. Let I
n
ε
denote the set of indices
such that f
−n
i

is well defined on ∆. Now,

i∈I
n
ε
Area(∆
−n
i
) =

i∈I
n
ε

[f
−n
i
(∆)] ∧ ω ≤

(f
n
)
∗
[D] ∧ ω ≤ C
ε
[λ
k−1
(f) + ε]
n
.

1600 VINCENT GUEDJ
Therefore


i ∈ I
n
ε
/ Area(∆
−n
i
) >
2C
ε
ε
δ
−n

≤
ε
2
d
n
t
;
hence for (1− ε)d
n
t
inverse branches f
−n
i

, we get an upper bound Area(∆
−n
i
) ≤
C

ε
δ
−n
. It is now a standard fact that the area controls the diameter of slightly
smaller disks
˜
∆
−n
i
= f
−n
i
(
˜
∆),
diam(
˜
∆
−n
i
) ≤ C

ε
δ

−n/2
.
We refer the reader to the appendix in [BD 01] where this is proved using
the notion of extremal length. Note that when the ∆
−n
i
’s are included in
a relatively compact ball of some affine chart (i.e. if we already know that
diam(
˜
∆
−n
i
) is small enough), this follows from Cauchy’s formula.
ii) Let now B = B(p, 8r
ε
) be a ball such that B∩V
l
= ∅. We now construct
(1 − ε)d
n
t
inverse branches f
−n
i
of f
n
on B(p, 4r
ε
) such that

diam(f
−n
i
B(p, r
ε
)) ≤ C

ε
δ
−n/2
.
There are d
l
t
well defined inverse branches f
−l
i
of f
l
on B = B(p, 8r
ε
). Set
B
−l
i
= f
−l
i
B. For n ≥ l, we set r
n

= 1 − ρ
n
with ρ
n
=

n
j=l
j
−2
. We can
further define d
t
inverse branches of f on f
−l
i
(r
l+1
B) if f
−l
i
(r
l+1
B) ∩ V
1
= ∅.
Assume f
−l
i
(r

l+1
B) ∩V
1
= ∅; then f
l
(B
−l
i
∩ V
1
) ∩r
l+1
B = ∅. Let Z
l
denote the
analytic set f
l
(B
−l
i
∩V
1
) and pick x
l
a point on Z
l
such that B(x
l
, 8r
ε

l
−2
) ⊂ B.
Thus Z
l
∩B(x
l
, 8r
ε
l
−2
) is an analytic subset of B(x
l
, 8r
ε
l
−2
) without b oundary.
It follows from Jensen’s inequality that

[f
l
(B
−l
i
∩ V
1
)] ∧ ω
k−1
≥


B(x
l
,8r
ε
l
−2
)
[Z
l
] ∧ ω
k−1
≥ C
0
(8r
ε
l
−2
)
2(k−1)
,
for some uniform constant C
0
> 0. This is because Z
l
has Lelong number ≥ 1
at point x
l
. On the other hand,


i

[f
l
(B
−l
i
∩ V
1
)] ∧ ω
k−1
≤

(f
l
)
∗
[V
1
] ∧ ω
k−1
≤ C[λ
k−1
(f) + ε]
l
.
Therefore {i / f
−l
i
(r

l+1
B) ∩ V
1
= ∅} ≤ C

l
4(k−1)
δ
−l
d
l
t
. Continuing the induc-
tion, slightly shrinking the radius of the ball at each step as indicated above,
we construct d
n
:= d
n
t
(1 − C


n−1
j=l
l
4(k−1)
δ
−l
d
l

t
) inverse branches of f
n
on the
ball B
n
= B(p, 8r
ε
r
n
). Now r
n
≥ 1 −

j≥l
j
−2
so that B
n
⊃ B(p, 4r
ε
) and
d
n
≥ d
n
t
(1 − ε/2) for all n ≥ l, if l is chosen large enough.
Let ω


=

[L
θ
]dν(θ), where L
θ
denotes the trace of a projective line
through p and ν is the Fubini-Study volume form on the set of lines  P
N−1
,
so that ω

is a positive closed current of bidegree (k − 1, k −1) which is smooth
in X \ {p}. Thus
0 ≤

i

B(p,4r
ε
)
(f
−n
i
)
∗
ω

∧ ω ≤


(f
n
)
∗
ω

∧ ω ≤ C

[λ
k−1
(f) + ε]
n
.
RATIONAL MAPPINGS WITH LARGE TOPOLOGICAL DEGREE
1601
We infer that

B(p,4r
ε
)
(f
−n
i
)
∗
ω

∧ ω ≤
2C


ε
δ
−n
for at least (1 − ε)d
n
t
inverse
branches.
Let I
n
ε
denote the corresponding set of indices. Set ∆
θ
= L
θ
∩ B(p, 4r
ε
).
For i fixed in I
n
ε
, we get
Area(f
−n
i
∆
θ
) ≤
4C


ε
δ
−n
on a set of projective lines A
n
i
⊂ P
N−1
of measure ≥ 1/2. Therefore
diam(f
−n
i
1
2
∆
θ
) ≤ C
ε
δ
−n/2
for θ ∈ A
n
i
. Now the sets A
n
i
have projective ca-
pacity ≥ 1/2, so it follows from a result of Sibony and Wong [A 81] (see also
[DS 02], where this is used in a dynamical context ) that
diam


f
−n
i
1
4
∆
θ

≤ C
ε
δ
−n/2
for every line L
θ
. The desired bound on diam(f
−n
i
B(p, r
ε
)) follows.
Proof of Theorem 3.1. Let a, b ∈ X \ PC(f). We claim d
n
t
(f
n
)
∗
(ε
a

− ε
b
)
→ 0. Indeed let 0 ≤ χ ≤ 1 be a test function. Fix ε > 0 and l = l
ε
 1 as in
Lemma 3.3. Let ∆ be a holomorphic disk joining a to b such that
∆ ∩ V
l
= ∅.
Using Lemma 3.3, we construct (1 − ε)d
n
t
inverse branches f
−n
i
of f
n
on ∆
with small diameter. Thus





(f
n
)
∗
(ε

a
− ε
b
)
d
n
t
, χ





≤ 2ε sup |χ| +
(1−ε)d
n
t

i=1
|χ ◦ f
−n
i
(a) − χ ◦ f
−n
i
(b)|
d
n
t
< 3ε,

if n is large enough so that diam(f
−n
i
∆) is smaller than the modulus of conti-
nuity of χ with respect to ε. This proves the claim.
Now let a /∈ PC(f). Using the identities µ
f
=

ε
b
dµ
f
(b) and f
∗
µ
f
= d
t
µ
f
,
we get
µ
f
−
1
d
n
t

(f
n
)
∗
ε
a
=
1
d
n
t
(f
n
)
∗
(µ
f
− ε
a
) =

1
d
n
t
(f
n
)
∗
(ε

b
− ε
a
)dµ
f
(b) → 0,
by the dominated convergence theorem, by the fact that µ
f
(PC(f)) = 0.
Similarly, if ν is a probability measure such that ν(PC(f)) = 0, we get
d
−n
t
(f
n
)
∗
ν =

d
−n
t
(f
n
)
∗
ε
a
dν(a) → µ
f

. In particular let χ be a test func-
tion. Translating and rescaling, we can assume 0 ≤ χ and c
χ
:=

χdµ
f
= 1
so that χµ
f
is a probability measure. Since χµ
f
(PC(f)) = 0, we obtain
χ ◦ f
n
µ
f
=
1
d
n
t
(f
n
)
∗
(χµ
f
) → µ
f

= c
χ
µ
f
.
This says precisely that the measure µ
f
is mixing (see [KH 95]).
In particular µ
f
is ergodic. Moreover log
+
||Df
±1
|| ∈ L
1
(µ
f
) (by Theorem
2.1.i); hence µ
f
has well defined (finite) Lyapunov exp onents χ
k
≥ · · · ≥ χ
1
.
It follows from Birkhoff’s ergodic theorem that
χ
1
= lim

n→+∞
−
1
n

log ||(D
x
f
n
)
−1
||dµ
f
(x).
1602 VINCENT GUEDJ
Fix ε > 0, l = l
ε
 1 and x ∈ Supp µ
f
\ V
l
a generic point. Using Lemma 3.3,
we construct (1 − ε)d
−n
t
inverse branches f
−n
i
of f
n

on B = B(x, r
ε
) whose
images B
−n
i
have small diameter. Let x
−n
i
denote the preimages of x under
f
n
. Since Df
−n
i
(x) = (Df
n
(x
−n
i
))
−1
, it follows from Cauchy’s inequalities and
Lemma 3.3 that
||(D
x
−n
i
f
n

)
−1
|| ≤ Cδ
−n/2
,
where δ is arbitrarily close to d
t
/λ
k−1
, C is independent of n and 1 ≤ i ≤
(1 − ε)d
n
t
. Let

Ω
f
:= {x = (x
n
)
n∈
Z
∈ Ω
Z
f
: f(x
n
) = x
n+1
for all n ∈ Z}

be the natural extension of (f, Ω
f
). It is well-known that the dynamical system
(Ω
f
, f, µ
f
) lifts to (

Ω
f
,

f, µ
f
), where

f denotes the shift on

Ω
f
and µ
f
is the
unique invariant probability measure on

Ω
f
such that (π
n

)
∗
µ
f
= µ
f
, where π
n
denotes the projection onto the n
th
coordinate. Set

B := π
−1
0
B and

B
ε
:=
{x ∈

B : ∀n ≥ 1, x
−n
= f
−n
i
(x
0
) for some 1 ≤ i ≤ (1 − ε)d

n
t
}. Observe that

B
ε
=

n≥l


f
n

∪
(1−ε)d
n
t
i=1

B
−n
i

, where B
−n
i
= f
−n
i

B.
Therefore
µ
f
(

B
ε
) = lim µ
f

∪
(1−ε)d
n
t
i=1

B
−n
i

= lim
(1−ε)d
n
t

i=1
µ
f
(B

−n
i
) = (1 − ε)µ
f
(B) > 0,
when µ
f
is

f-invariant, µ
f
(

A) = µ
f
(A), and µ
f
(B
−n
i
) = d
−n
t
µ
f
(B) (because
(f
n
)
∗

µ
f
= d
n
t
µ
f
and f
n
is injective on B
−n
i
).
Set ϕ := − log ||(D
x
f)
−1
|| and ϕ = ϕ ◦ π
0
∈ L
1
(µ
f
). Then χ
1
=

ϕdµ
f
=


ϕdµ
f
. The measure µ
f
is mixing since µ
f
is; hence by Birkhoff’s theorem,
1
n
n−1

j=0
ϕ ◦

f
−j
(x) → χ
1
for almost every x.
Fix x a generic point in

B
ε
. Then
1
n
n−1

j=0

ϕ ◦

f
−j
(x) = −
1
n
n−1

j=0
log ||(D
x
−j
f)
−1
||
= −
1
n
log ||D
x
f
−n
i
|| ≥
log δ
2
−
log C
n

,
hence χ
1
≥
1
2
log δ. The desired lower b ound follows when δ → d
t
/λ
k−1
.
Set ν
n
:= [RPer
n
(f)]
−1

x∈RPer
n
(f)
ε
x
and let ν be any cluster point of
ν
n
. Fix ε > 0 and x ∈ Supp µ\PC(f). Using lemma 3.3, we construct (1−ε)d
n
t
RATIONAL MAPPINGS WITH LARGE TOPOLOGICAL DEGREE

1603
inverse branches f
−n
i
of f
n
on B = B(x, r
ε
) whose images have small diameter.
We now prove the following inequality:
(1 − ε)
3
µ
f
(B) ≤ ν(B).(††)
Clearly (††) implies µ
f
≤ ν. Indeed any Borel subset A can be approximated
by disjoint union of small balls satisfying (††); hence (1 − ε)
4
µ
f
(A) ≤ ν(A).
One can then let ε → 0. Finally since µ
f
and ν are probability measures, we
actually get µ
f
= ν; hence ν
n

→ µ
f
.
It remains to prove (††). We can assume µ
f
(B) > 0. Fix B

⊂⊂ B

⊂⊂ B
such that µ
f
(B

) ≥ (1 − ε)µ
f
(B). We consider as above

B
ε
the set of histories
of points in B given by the inverse branches f
−n
i
. Since µ
f
is mixing, we get
µ
f
(


f
−n
(

B
ε
) ∩

B

) → µ
f
(

B
ε
)µ
f
(

B

). Thus, for n large enough,
(1 − ε)
3
µ
f
(B)
2

≤ (1 − ε)µ
f
(

B
ε
)µ
f
(

B

)
≤ µ
f
(

f
−n
(

B
ε
) ∩

B

) ≤
(1−ε)d
n

t

i=1
µ
f
(B
−n
i
∩ B

).
Observe that either B
−n
i
∩ B

= ∅ or B
−n
i
⊂ B

⊂⊂ B since diam(B
−n
i
) → 0.
When B
−n
i
∩ B


= ∅, f
−n
i
is thus a contraction on B. Therefore it admits a
unique attracting fixed point which is henceforth a repelling periodic point of
order n for f. Using again that µ
f
(B
−n
i
) = d
−n
t
µ
f
(B), we infer
(1 − ε)
3
µ
f
(B)
2
≤
RPer
n
(f)
d
n
t
ν

n
(B)µ
f
(B).
Letting n
i
→ +∞ yields (††) if limRPer
n
(f)/d
n
t
≤ 1. Note that d
t
(f) >
max
1≤j≤k−1
λ
j
(f) by Proposition 1.2. When dim
C
X ≤ 3 or when X is complex
homogeneous, each dynamical degree λ
l
(f) equals the asymptotical growth
of the spectral radii r
l
(f
n
) of the linear action induced by f
∗

on H
l,l
a
(X, R)
(see Proposition 1.2 and Remark 1.3.ii). In these cases, the upper bound on
RPer
n
(f) follows from the Lefschetz fixed point formula if f has no curve of
periodic points. Note that f cannot have a curve of repelling periodic points.
The bound therefore follows from a perturbation argument.
4. Uniqueness of the measure of maximal entropy
Theorem 4.1. Assume dim
C
X ≤ 3 or X is complex homogeneous. Then
the measure µ
f
is the unique measure of maximal entropy.
Here again we follow Briend and Duval [BD 01] who proved this result for
holomorphic endomorphisms of CP
k
.
Proof. Let ν be an ergodic measure such that ν(PC(f)) > 0. Then
ν(f
j
(C
f
)) > 0 for some j ∈ N, so that it follows from the relative variational
1604 VINCENT GUEDJ
principle and Corollary 1.9 that
h

ν
(f) ≤ h
top
(f
|f
j
(C
f
)
) ≤ max
1≤j≤k−1
log λ
j
(f) < log d
t
(f).
Consider now an ergodic probability measure ν of entropy h
ν
(f) >
max
1≤j≤k−1
log λ
j
(f). Then ν does not charge PC(f); hence d
−n
t
(f
n
)
∗

(ν)
→ µ
f
. Assume ν = µ
f
. Then ν does not have constant jacobian, i.e. f
∗
ν = d
t
ν.
Therefore one can construct a simply connected domain U in X \ f(C
f
) with
ν(U) = Vol(U ) = 1 admitting U
1
, . . . , U
d
t
preimages on which f is one-to-one
and not equally well ν-distributed, say with ν(U
1
) > σ > d
−1
t
(see [BD 01]
for more details on this construction). We are going to show that this implies
h
ν
(f) < log d
t

(f).
Observe that ν(Ω
f
) = 1; otherwise h
ν
(f) ≤ max
1≤j≤k−1
log λ
j
(f) by
Corollary 1.9. Consider O a slightly smaller open subset of U
1
such that
O
ε
⊂ U
1
, where O
ε
denotes the ε -neighborhood of O, and ν(O) > σ. Set
Y = {a ∈ Ω
f
: {0 ≤ j ≤ n − 1, f
j
(a) ∈ O} ≥ nσ for n ≥ m}. It follows from
Birkhoff’s theorem that ν(Y ) > 0 for m large enough. The relative variational
principle yields
h
ν
(f) ≤ h

top
(f
|Y
) ≤ lim sup
1
n
Vol(Γ
n
|Y )
ε
,
where Γ
n
= {(a, . . . , f
n−1
(a)) : a ∈ Ω
f
} is the iterated graph of f (see Sec-
tion 1). Up to a zero volume set, we get
(Γ
n
|Y )
ε
⊂

α∈Σ
n
Γ
n
(α),

where Σ
n
= {α ∈ {1, . . . , d
t
} : {q, α
q
= 1} ≥ nσ} and Γ
n
(α) = Γ
n
∩
(U
α
1
× · ·· × U
α
n
). Indeed the U

j
s form a partition of X (up to a zero vol-
ume set) and {Γ
n
(α)} is the induced partition on Γ
n
. Therefore
Vol(Γ
n
|Y )
ε

≤

α∈Σ
n

Γ
n
(α)
ω
k
n
≤

i∈{0, ,n−1}
k

α∈Σ
n

π(Γ
n
(α))
(f
i
1
)
∗
ω ∧ · · · ∧ (f
i
k

)
∗
ω,
where π denotes the projection of X
n
on the first factor. Fix ε > 0 so small
that β + ε < d
t
, where β := max
1≤j≤k−1
λ
j
(f). Fix γ < 1 to be chosen later
and define, following a trick of Briend and Duval,
I = {i ∈ {0, . . . , n − 1}
k
: i
1
, . . . i
k
≥ γn} and II = {0, . . . , n − 1}
2
\ I.
Fix i ∈ II and assume i
1
≤ · · · ≤ i
k
(hence i
1
≤ γn). Since the π(Γ

n
(α)) form
a partition of Ω
f
, we get
RATIONAL MAPPINGS WITH LARGE TOPOLOGICAL DEGREE
1605

α∈Σ
n

π(Γ
n
(α))
(f
i
1
)
∗
ω ∧ · · · ∧ (f
i
k
)
∗
ω ≤

Ω
f
(f
i

1
)
∗
ω ∧ · · · ∧ (f
i
k
)
∗
ω
= d
i
1
t

Ω
f
ω ∧ (f
i
2
−i
1
)
∗
ω ∧ · · · ∧ (f
i
k
−i
1
)
∗

ω
≤ C
ε
d
i
1
t
[β + ε]
i
k
−i
1
≤ C
ε
d
γn
t
[β + ε]
n(1−γ)
,
where the existence of C
ε
is as in Lemma 1.7. Therefore

i∈II

α∈Σ
n

π(Γ

n
(α))
(f
i
1
)
∗
ω ∧ · · · ∧ (f
i
k
)
∗
ω ≤ C
ε
n
k
d
γn
t
[β + ε]
n(1−γ)
.
Now fix i ∈ I, α ∈ Σ
n
and set q = [γn]. Since f
q
is injective on π(Γ
n
(α)),
assuming i

1
≤ · · · ≤ i
k
, we get

π(Γ
n
(α))
(f
i
1
)
∗
ω ∧ · · · ∧ (f
i
k
)
∗
ω
=

π(Γ
n
(α))
(f
q
)
∗

(f

i
1
−q
)
∗
ω ∧ · · · ∧ (f
i
k
−q
)
∗
ω

≤

Ω
f
(f
i
1
−q
)
∗
ω ∧ · · · ∧ (f
i
k
−q
)
∗
ω

≤ C
ε
d
i
1
−q
t
[β + ε]
i
k
−i
1
=

d
t
β + ε

i
1
−q
[β + ε]
i
k
−q
≤ C
ε
d
n−1−q
t

≤ C
ε
d
n(1−γ)
t
.
By Lemma 7.2 in [L 83] there exists ρ < 1 such that Σ
n
≤ d
nρ
t
. Therefore

i∈I

α∈Σ
n

π(Γ
n
(α))
(f
i
1
)
∗
ω ∧ · · · ∧ (f
i
k
)

∗
ω ≤ C
ε
n
k
d
ρn
t
d
n(1−γ)
t
.
Altogether this yields
h
ν
(f) ≤ max([1 + ρ − γ] log d
t
(f), γ log d
t
(f) + [1 − γ] log(β + ε)),
so that h
ν
(f) < log d
t
(f) if we choose ρ < γ < 1.
Laboratoire Emile Picard, Universit
´
e Paul Sabatier, Toulouse, France
E-mail address:
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(Received February 4, 2003)