VALUE DISTRIBUTION FOR SEQUENCES OF RATIONAL MAPPINGS AND COMPLEX DYNAMICS

ALEXANDER RUSSAKOVSKII AND BERNARD SHIFFMAN

Abstract. We study pre-images under the iterates P^{k} of a rational (not necessarily holomorphic) mapping P of ℙ^{n}. We show, assuming a condition on the topological degree λ of P, that there is a probability measure μ on ℙ^{n} and a pluripolar set E ⊂ ℙ^{n} such that λ^{−k}P^{k∗}ν → μ for all probability measures ν on ℙ^{n} \ E. We also obtain results on the asymptotic equidistribution of the pre-images of linear subspaces for sequences of rational mappings between projective spaces.

1. Introduction

In this paper, we investigate the asymptotic equidistribution of pre-images under the iterates of a rational mapping of ℙ^{n} and also under an arbitrary sequence of rational mappings with rapidly increasing degrees. The study of value distribution for sequences of mappings may be regarded as an analogue of Nevanlinna theory; instead of studying the asymptotic behavior of the area of a pre-image of an analytic set in a ball when its radius tends to infinity, one investigates the asymptotics of the pre-images of an analytic set under a sequence of mappings. The prototype for our investigation is the following theorem on the convergence of the iterated pull-backs of a measure by a rational function, proved independently by Lyubich [Ly] and by Freire-Lopes-Mañé [FLM] in 1983 (see also [ES, HP]):

Let R(z) be a rational function of degree λ ≥ 2, let R^{k} denote its k-th iterate, and let E be the set of points of ℙ^{1} = ℂ ∪ {∞} that have only finitely many iterated pre-image points. Then there is a probability measure μ on ℙ^{1}, such that for all probability measures ν on ℙ^{1}\E,

\frac{1}{λ^{k}} R^{k∗}ν → μ .

This result was proved by Brolin [Br] in 1965 for iterates of polynomials. The limit measure μ is the unique probability measure on ℙ^{1} \ E that is invariant in the sense that R^{∗}μ = λμ. (The support of μ is the Julia set of R.) Recall that E contains at most 2 points; in fact, E is empty unless R is conjugate to a polynomial or to the map z ↦ z^{−λ}. In particular, if we let ν be the delta measure at any point of ℙ^{1} \ E, we obtain the asymptotic equidistribution of the pre-images of points (with at most 2 exceptional values).

Several recent papers on holomorphic dynamics in several complex variables (see, for example, [BS1, BS2, BS3, BLS1, BLS2, FS1]) have dealt with various extensions of the Lyubich-Freire-Lopes-Mañé Theorem to polynomial biholomorphisms of ℂ^{2}. For holomorphic mappings P of complex projective n-space ℙ^{n} of degree δ ≥ 2, it was shown by Hubbard and Papadopol [HP] and by Fornaess and Sibony [FS2, FS3, FS4] that δ^{−k} log |P^{k}| (where P^{k} is the k-th iterate of P) converges uniformly to a Green function G on ℂ^{n+1}; see also Ueda [Ue]. Since dd^{c} log |P^{k}|^{2} = P^{k∗}ω where ω is the Kähler form of the Fubini-Study metric on ℙ^{n} (normalized so that ∫_{ℙ^{n}} ω^{n} = 1), it follows by Bedford-Taylor [BT] that

Research of the second author partially supported by National Science Foundation Grant DMS-9500491.

\frac{1}{δ^{nk}} P^{k∗}ω^{n} → (2dd^{c}G)^{n} = μ ,

where μ is a probability measure on ℙ^{n} which is invariant in the sense that δ^{−n}P^{∗}μ = μ. It then easily follows from the equidistribution theorem in [RS] (see below) that there is a pluripolar set E ⊂ ℙ^{n} such that for all probability measures ν on ℙ^{n} \ E,

\frac{1}{δ^{nk}} P^{k∗}ν → μ .

Fornaess and Sibony [FS4] also showed that for certain meromorphic maps P : ℙ^{2}−→ℙ^{2} of degree δ, the sequence {δ^{−k}P^{k∗}ω} converges to an invariant positive (1, 1)-current.

However, a different approach is needed to obtain convergence of iterated pull-back measures for rational, non-holomorphic maps. Let P be a rational map of ℙ^{n} given by polynomials of degree δ, and let λ denote its topological degree. One easily sees that the total mass of the measure P^{k∗}ω^{n} equals λ^{k}. If P is not holomorphic, then λ < δ^{n} (see Lemma 4.4) so that δ^{−nk}P^{k∗}ω^{n} → 0, and it follows that if the Green function G exists, then (dd^{c}G)^{n} vanishes on any domain where the iterates P^{k} are holomorphic (see Examples 2 and 4 in Section 2). Thus the methods of [HP, FS4, RS] do not yield a useful limit measure in the meromorphic case. Our main result (Theorem 1.1) gives the existence of a limit measure μ and a pluripolar set E such that λ^{−k}P^{k∗}ν → μ for all probability measures ν on ℙ^{n} \ E, provided that λ > δ^{n−1}. A new ingredient in our proof of Theorem 1.1 is a second derivative bound (Lemma 5.1) on a “proximity function” given as a fiber integral of a pull-back of the Levine form. In fact, Theorem 1.1 has a slightly weaker hypothesis on λ expressed in terms of an “intermediate dynamic degree” of P, which we now describe.

Let P : ℙ^{n}−→ℙ^{m} be a rational map. For 1 ≤ ℓ ≤ min(n, m), we let δ_{ℓ}(P) denote the degree of P^{−1}(W) for a generic projective (m − ℓ)-plane W ⊂ ℙ^{m}. (For generic W, the algebraic variety P^{−1}(W) has pure codimension ℓ and its degree is the number of points in its intersection with a generic ℓ-plane in ℙ^{n}.) We also set δ_{0}(P) = 1. One easily sees that δ_{1}(P) is the degree of the polynomials in a representation of P (using homogeneous coordinates). We now suppose that n = m; then δ_{n}(P) is the topological degree of P; i.e., the number of pre-images of a generic point. If P is holomorphic (i.e., regular), then δ_{ℓ}(P^{k}) = δ_{1}(P)^{ℓk} for 0 ≤ ℓ ≤ n, k ≥ 1. In Section 4, we give an analytic description of the “intermediate degrees” δ_{ℓ}(P) and we show that δ_{ℓ}(P^{j+k}) ≤ δ_{ℓ}(P^{j})δ_{ℓ}(P^{k}) and δ_{k+ℓ}(P) ≤ δ_{k}(P)δ_{ℓ}(P). Thus we can define the “intermediate dynamic degrees”

λ_{ℓ}(P) = lim_{k→∞} δ_{ℓ}(P^{k})^{1/k} = inf_{k≥1} δ_{ℓ}(P^{k})^{1/k} ≤ δ_{ℓ}(P) ≤ δ_{1}(P)^{ℓ} .

Intermediate degrees were also considered by Friedland [Fr] in the context of their relationship with topological entropy.

The following is our main result on limit measures and equidistribution for the iterates of a rational mapping of projective space:

Theorem 1.1. Let P : ℙ^{n}−→ℙ^{n} be a rational map, let λ = δ_{n}(P) denote the topological degree of P, and let P^{k} denote the k-th iterate of P. If λ > λ_{n−1}(P), then there exist a probability measure μ on ℙ^{n} and a pluripolar subset E ⊂ ℙ^{n} such that, for all probability measures ν on ℙ^{n} with ν(E) = 0,

\frac{1}{λ^{k}} P^{k∗}ν → μ

weakly (in the space of measures) as k → ∞.

A subset S of ℙ^{n} is said to be pluripolar if it it is contained in the −∞ locus of a quasiplurisubharmonic function on ℙ^{n} (or equivalently, if the cone over S is contained in the −∞ locus of a plurisubharmonic function on ℂ^{n+1}.) The pull-back measures P^{k∗}ν of Theorem 1.1 are defined as follows. For a rational map Q : ℙ^{n}−→ℙ^{n}, we write

E_{Q} = {W ∈ ℙ^{n} : dim Q^{−1}(W) > 0} ,

where Q^{−1}(W) denotes the set of Z ∈ ℙ^{n} such that (Z, W) is in the graph of Q. If ν is a finite measure on ℙ^{n} with ν(E_{Q}) = 0, there is a unique finite measure Q^{∗}ν on ℙ^{n} given by

(Q^{∗}ν, φ) = ∫_{ℙ^{n}\E_{Q}} Q_{∗}φdν , φ ∈ C(ℙ^{n}) .

The function Q_{∗}φ ∈ C(ℙ^{n} \ E_{Q}) is given by Q_{∗}φ(W) = ∑_{j} φ(Z_{j}), where Z_{1}, …, Z_{δ_{n}(Q)} are the points of Q^{−1}(W) repeated according to their multiplicities. (Alternately, Q_{∗}φ can be given by first lifting φ to the graph of Q and then pushing it forward as a current.) The exceptional set E in Theorem 1.1 should be taken to include the algebraic subvarieties E_{P^{k}} (which have codimension at least 2) so that the pull-back measures P^{k∗}ν are well-defined.

Theorem 1.1 gives an extension of the Lyubich-Freire-Lopes-Mañé Theorem to n variables with a pluripolar exceptional set E replacing the exceptional set of at most two points in the one-variable case. In general, one cannot expect E to be finite when n ≥ 2, since for the meromorphic case, E must contain all the algebraic subvarieties E_{P^{k}}, k ≥ 1. Simple examples of holomorphic maps give E as a finite union of hyperplanes and it is an open problem whether E can always be chosen to be algebraic if P is holomorphic. (See Example 1 and Question 1 in Section 2.)

By letting ν be the delta measure δ_{W} of a point W, we obtain the asymptotic equidistribution of the pre-images of points,

\frac{1}{λ^{k}} P^{k∗}δ_{W} → μ , W ∈ ℙ^{n} \ E ,  (1)

where P, μ, E are as in Theorem 1.1. If f is an L^{1} function on ℙ^{n}, we similarly obtain

\frac{1}{λ^{k}} P^{k∗}(fω^{n}) → c_{f}μ  (2)

where c_{f} = ∫ fω^{n}. In particular,

\frac{1}{λ^{k}} P^{k∗}ω^{n} → μ .  (3)

Write μ_{k} = λ^{−k}P^{k∗}ω^{n}; then P^{k∗}μ_{k} = λμ_{k+1}. If μ(E_{P}) = 0, we can then let k → ∞ to conclude that μ is invariant in the sense that P^{∗}μ = λμ. If μ(E) = 0, we can further conclude that μ is the unique invariant measure that does not charge E. It is an open problem whether the measure μ in Theorem 1.1 can charge the set E_{P}.

The Fatou set Ω_{P} of P can be defined to be the maximal open set in ℙ^{n} on which {P_{k}} is a normal family of holomorphic maps. One easily sees that a subsequence of {P^{k∗}ω^{n}} converges in Ω_{P} to a smooth 2n-form, and it then follows from (3) that μ vanishes on Ω_{P}. (For n ≥ 2, μ is usually supported on a proper subset of ℙ^{n} \ Ω_{P}; see Section 2.) Ueda [Ue] showed that Supp dd^{c}G = ℙ^{n} \ Ω_{P} for P holomorphic; however, equality does not always hold for meromorphic P (see Example 2 in Section 2).

Our proof of Theorem 1.1 is based on a study of equidistribution for sequences of rational mappings of projective spaces. This approach was introduced in 1992 by Sodin [So], who considered an arbitrary sequence {R_{k}} of rational functions (on ℙ^{1}) with rapidly increasing degrees and proved that the pre-images under {R_{k}} of points w outside an exceptional set E are equidistributed in a certain sense. Sodin’s result was generalized in [RS] to sequences of polynomial mappings ℂ^{n} → ℙ^{m}. The equidistribution phenomenon in [RS] can be described as follows:

Suppose that \{P_{k}\} is a sequence of polynomial mappings ℂ^{n} → ℙ^{m} such that ∑ \frac{1}{δ_{1}(P_{k})} < ∞. Then

(i) the pre-images of all but an exceptional pluripolar set of complex hyperplanes in ℙ^{m} are equidistributed with the pull-backs P_{k}^{∗}ω;

(ii) if all P_{k} are non-degenerate, then the pre-images of all but a pluripolar set of points in

ℙ^{m} are equidistributed with P_{k}^{∗}ω^{m} (m ≤ n). In this paper we also obtain more general equidistribution results for pre-images of linear

subspaces of any dimension. We let 𝔾(ℓ, m) denote the Grassmannian of projective-linear subspaces of codimension ℓ in ℙ^{m}. Note that 𝔾(m, m) = ℙ^{m}. We shall prove the following theorem on the equidistribution of pre-images under an arbitrary sequence of rational maps of projective spaces:

Theorem 1.2. Let {P_{k}} be a sequence of rational mappings from ℙ^{n} to ℙ^{m}. Let 1 ≤ ℓ ≤ min(n, m), and let {a_{k}} be a sequence of positive numbers such that

∑_{k=1}^{∞} \frac{δ_{ℓ−1}(P_{k})}{a_{k}} < +∞ .

Then there exists a pluripolar subset E of 𝔾(ℓ, m) such that

\frac{1}{a_{k}} (P_{k}^{∗}[W] − P_{k}^{∗}ω^{ℓ}) → 0

(weakly in D′^{ℓ,ℓ}(ℙ^{n})) as k → ∞, for all W ∈ 𝔾(ℓ, m) \ E.

The pull-back P_{k}^{∗}ω^{ℓ} is smooth off the indeterminacy locus of P_{k} and has locally integrable coefficients. For W ∈ 𝔾(ℓ, m), [W] denotes the current of integration over W, which is a closed positive (ℓ, ℓ)-current on ℙ^{m}. (If ℓ = m, [W] = δ_{W}.) For generic W, P_{k}^{∗}[W] is given by integration over P_{k}^{−1}(W). Precise definitions of the pull-back currents P_{k}^{∗}[W], P_{k}^{∗}ω^{ℓ} ∈ D′^{ℓ,ℓ}(ℙ^{n}) are given in Section 3. (The exceptional set E includes those W for which P_{k}^{∗}[W] is not defined.)

The proof of Theorem 1.2 uses the Levine current Λ_{W}^{ℓ}, which is a positive current that satisfies the equation dd^{c}Λ_{W}^{ℓ} = ω^{ℓ}−[W] ([Le, GK, Sh]). We consider the “proximity function”

m_{P}^{ℓ} (W) = (P^{∗}Λ_{W}^{ℓ} , ω^{n−ℓ+1})

and use the “First Main Theorem”

|(P^{∗}[W] − P^{∗}ω^{ℓ}, φ)| = |(P^{∗}Λ_{W}^{ℓ} , dd^{c}φ)| ≤ c_{φ}m_{P}^{ℓ} (W)

to define the exceptional set E in terms of the “proximity sequence” {m_{P_{k}}^{ℓ}} (see Section 6), as in [RS]. In order to prove that E is pluripolar, we apply a new technique based on an approach of Skoda [Sk1, Sk2] to obtain the estimate dd^{c}m_{P}^{ℓ} ≤ ℓδ_{ℓ−1}(P)ω on 𝔾(ℓ, m) (Lemma 5.1). This new explicit bound on the quasi-plurisubharmonicity of the function −m_{P}^{ℓ} is a crucial ingredient of our proof.

A consequence of the case ℓ = m of Theorem 1.2 is the following result on the equidistribution of the pull-backs of measures by sequences of mappings:

Corollary 1.3. Let {P_{k}} be a sequence of rational mappings from ℙ^{n} to ℙ^{m}, where n ≥ m. If

∑_{k=1}^{∞} \frac{δ_{m−1}(P_{k})}{δ_{m}(P_{k})} < +∞ ,

then there exists a pluripolar subset E of ℙ^{m} such that for all probability measures ν on ℙ^{m} with ν(E) = 0,

\frac{1}{δ_{m}(P_{k})} (P_{k}^{∗}ν − P_{k}^{∗}ω^{m}) → 0

as k → ∞,

The proof of Corollary 1.3 is given in Section 7. Remark: Convergence in Corollary 1.3 can be taken to be weak convergence in D′^{m,m}(ℙ^{n}), or since the positive currents δ_{m}(P_{k})^{−1}P_{k}^{∗}ν have mass 1 (see Section 6), we conclude more generally that

\frac{1}{δ_{m}(P_{k})} (P_{k}^{∗}ν − P_{k}^{∗}ω^{m}, φ) → 0

for all (n − m, n − m)-forms φ on ℙ^{n} with continuous coefficients. In the case m = n, the currents δ_{m}(P_{k})^{−1}P_{k}^{∗}ν are probability measures and we have weak convergence in the space of measures. However, in Theorem 1.2 (and in Corollary 1.4 below) we do not always have convergence in the space of measures; see the remark at the end of Section 7

We use Theorem 1.2 with ℓ = m in the proof of Theorem 1.1. For general ℓ, Theorem 1.2 yields the following result on the equidistribution of the iterated pre-images of linear subspaces:

Corollary 1.4. Let P : ℙ^{n}−→ℙ^{n} be a rational mapping, and let P^{k} denote the k-th iterate of P. Let 1 ≤ ℓ ≤ n. If a > λ_{ℓ−1}(P), then

\frac{1}{a^{k}} (P^{k∗}[W] − P^{k∗}ω^{ℓ}) → 0

(weakly in D′^{ℓ,ℓ}(ℙ^{n})) for all W ∈ 𝔾(ℓ, m) outside a pluripolar set.

This corollary follows immediately from Theorem 1.2 by noting that for ε > 0,

δ_{ℓ−1}(P^{k}) ≤ [λ_{ℓ−1}(P) + ε]^{k} for k ≫ 0 .  (4)

Acknowledgments: This work was begun when the first author was visiting Johns Hopkins University. Part of the work on this paper was done while both authors were at the Mathematical Sciences Research Institute in Berkeley. The authors would like to thank these organizations for their hospitality and support. The authors also wish to thank the referee for suggesting they consider the map given in Example 4. The second author would like to thank Steve Zelditch for helpful conversations.

2. Examples and open questions

We begin by describing several sequences of iterated mappings of the complex projective plane ℙ^{2} in order to illustrate our results. Besides equidistribution, we are also going to look at invariance properties of the limit currents and measures. The dynamics of a rational map in more than one variable is usually quite difficult to analyze, so we consider very simple examples of extensions to ℙ^{2} of proper polynomial mappings of ℂ^{2}. We also give a preliminary study of the properties of a particular map arising from Newton’s method (Example 4).

In this section, we identify (z, w) ∈ ℂ^{2} with (1, z, w) ∈ ℙ^{2} so that ℙ^{2} = ℂ^{2} ∪ H_{∞}, where H_{∞} is the hyperplane (or line) at infinity, {(0, z, w)}.

Example 1. (This example was discussed in [RS].) We first restrict our attention to ℂ^{2} and consider the iterates of the polynomial map P (z, w) = (z^{δ}, w^{δ}) : ℂ^{2} → ℂ^{2}, where δ is a

positive integer. It is easy to see that the uniform convergence \frac{1}{δ^{k}} log(1 + |P^{k}(z, w)|^{2})^{1/2} → G(z, w) takes place, where

G(z, w) = sup(log^{+} |z|, log^{+} |w|)

is the plurisubharmonic Green function of the unit bidisk with logarithmic growth at infinity. It follows that

2dd^{c}G = lim_{k→∞} \frac{1}{δ^{k}} P^{k∗}ω

(where d^{c} = (4π\sqrt{−1})^{−1}(∂ − \overline{∂}) ). The current T = 2dd^{c}G, which is positive and has unit mass, is concentrated on the boundary of the unit bidisk and on the set {|z| = |w| > 1}. This current is invariant under the mapping P in the sense that

P^{∗}T = δT .

(The pull-backs of closed positive (1, 1)-currents by holomorphic and meromorphic maps are defined in Section 3.) According to the results of [RS], this current is the limit of pre-images of all nonexceptional hyperplanes (lines).

Every hyperplane of the form z = c or w = c has pre-images of similar form. These preimages tend towards the cylinder |z| = 1 or |w| = 1, respectively. Thus all such hyperplanes are exceptional. A third family of exceptional hyperplanes consists of those passing through the origin. Pre-images of these hyperplanes tend towards the “cone” {|z| = |w|}. We let ℙ^{2∗} ≈ ℙ^{2} denote the parameter space of hyperplanes in ℙ^{2}; the point (ζ_{0}, ζ_{1}, ζ_{2}) ∈ ℙ^{2∗} represents the hyperplane {(z, w) ∈ ℂ^{2} : ζ_{0} + ζ_{1}z + ζ_{2}w = 0}. (The point (1, 0, 0) represents the hyperplane at infinity, H_{∞}.) Thus we see that the set of exceptional hyperplanes consists of the three pencils in ℙ^{2∗} : {ζ_{0} = 0}, {ζ_{1} = 0}, {ζ_{2} = 0}. We note that, besides T, there are at least five more (linearly independent) invariant closed (1, 1)-currents (on ℂ^{2}), T_{1} = dd^{c} log^{+} |z|^{2}, T_{2} = dd^{c} log^{+} |w|^{2}, T_{3} = dd^{c} log max(|z|^{2}, |w|^{2}), T_{4} = [H_{1}] = dd^{c} log |z|^{2}, T_{5} = [H_{2}] = dd^{c} log |w|^{2} (where H_{1} = {z = 0}, H_{2} = {w = 0}) with the same property

P^{∗}T_{i} = δT_{i}.

In fact, they are limits of pre-images of the corresponding exceptional hyperplanes. (These currents are invariant on ℙ^{2} as well as on ℂ^{2}, and we have the additional invariant current T_{6} = [H_{∞}] on ℙ^{2}.)

Now consider pre-images of points. The measure μ = T^{2} = (2dd^{c}G)^{2}, being the limit

of \frac{1}{δ^{2k}} P^{k∗}ω^{2}, is concentrated on the distinguished boundary of the unit bidisk; hence preimages of most points must tend to the torus by Theorem 2 of [RS]. However, all points of the form (0, c) or (c, 0) have pre-images of the form (0, c′) (respectively (c′, 0)) and are definitely exceptional (as well as the points of H_{∞}). So in this case the exceptional set is the same union of 3 hyperplanes in ℙ^{2∗}.

The measure μ possesses invariance properties, P^{∗}μ = δ^{2}μ. Note that we also have μ = T_{1} ∧ T_{2}.

The map P extends to a rational mapping Q(t, z, w) = (t^{δ}, z^{δ}, w^{δ}) : ℙ^{2} → ℙ^{2}, which we call the projectivization of P. For this example, Q is holomorphic. The situation then becomes symmetric with respect to all variables, and because of the absence of the indeterminacy set, no difficulty occurs. We consider the “projectivizations” of the currents defined on ℂ^{2}: instead of dd^{c} log^{+} |z|^{2} = dd^{c} log(|z|^{2} ∨ 1), one has dd^{c} log(|z|^{2} ∨ |t|^{2}), where ∨ stands for maximum, and so on. The projectivizations of the currents T, T_{1}, …, T_{5} (as well as [H_{∞}]), remain invariant on ℙ^{2}. There are 3 super-attracting fixed points (1, 0, 0), (0, 1, 0), (0, 0, 1) (and no other attracting periodic points), and their respective basins Ω_{1}, Ω_{2}, Ω_{3} comprise the entire Fatou set Ω_{Q}. Note that Supp μ = ⋂_{i} ∂Ω_{i} (the distinguished boundary of the bidisk), while Supp T = ⋃_{i} ∂Ω_{i} = ℙ^{2} \ Ω_{Q}.

Example 2. The situation changes if we consider the iterates of the mapping P (z, w) = (z^{d_{1}}, w^{d_{2}}) : ℂ^{2} → ℂ^{2} with d_{1} ≠ d_{2}. For simplicity, let P = (z^{2}, w^{3}). It is easy to see that in

this case \frac{1}{3^{k}} log(1 + |P^{k}(z, w)|^{2})^{1/2} → G(z, w), where now G(z, w) = log^{+} |w|. It follows as before that

2dd^{c}G = lim_{k→∞} \frac{1}{3^{k}} P^{k∗}ω ;

however, this time the current T = 2dd^{c}G is concentrated on the cylinder {|w| = 1}. It is invariant for the mapping P:

P^{∗}T = 3T .

Applying the results of [RS], we see that this current is the limit of the pre-images of all nonexceptional hyperplanes.

The set of exceptional hyperplanes in ℂ^{2} consists of the two pencils in ℙ^{2∗} : {ζ_{0} = 0}, {ζ_{2} = 0}. Besides T, there are at least three more invariant currents: T_{1} = dd^{c} log^{+} |z|^{2}, T_{2} = dd^{c} log |z|^{2}, T_{3} = dd^{c} log |w|^{2}. However this time

P^{∗}T_{1} = 2T_{1} .

(Note that for any smooth form S on ℂ^{2}, we have P_{∗}P^{∗}S = 6S; by smoothing and taking limits, one sees that this identity is also valid if S is a positive (1, 1)-current on ℂ^{2}. Hence P_{∗}T_{1} = 3T_{1}, P_{∗}T = 2T.) The currents T_{i} are the limits of pre-images of the corresponding families of exceptional hyperplanes. The coefficient 2 for T_{1} makes the situation somewhat different as we shall see.

Now consider pre-images of points. The results of [RS] and [FS4] do not provide useful information since now T^{2} = (2dd^{c}G)^{2} ≡ 0 on ℂ^{2}. According to Theorem 1.1, this is not a surprise, since δ_{2}(P) = 6, not 9, and we have a limit

μ = lim_{k→∞} \frac{1}{6^{k}} P^{k∗}ω^{2},

which is the same measure (concentrated on the distinguished boundary of the unit bidisk) as in Example 1. So pre-images of most points must concentrate there. Again, the exceptional set for points is the same union of 3 hyperplanes in ℙ^{2}. The measure μ possesses the invariance property P^{∗}μ = 6μ. Note that we also have μ = T ∧ T_{1}.

Consider the projectivization of this mapping: Q(t, z, w) = (t^{3}, tz^{2}, w^{3}) : ℙ^{2} → ℙ^{2} and the sequence Q^{k} of its iterations. Note that although P is holomorphic, Q is only meromorphic and has one indeterminacy point (0, 1, 0) which is the reason for all “anomalies.” Since H_{∞} is contracted by Q to the fixed point (0, 0, 1), the map Q is “generic” in the sense of Fornaess and Sibony [FS3, FS4]. Also, the graph of the mapping Q is singular, and thus it is necessary to resolve the singularities of the graph in order to define all our currents correctly (see Section 3). Note that we have δ_{1}(Q^{k}) = 3^{k}, δ_{2}(Q^{k}) = 6^{k} < δ_{1}^{2}(Q^{k}). In accordance with the results of [FS3, FS4], there is an invariant (1, 1)-current T = dd^{c} log(|w|^{2} ∨ |t|^{2}) which is the projectivization of the above mentioned current on ℂ^{2}. The projectivization of the other invariant current, T_{1}, is no longer invariant, since now Q^{∗}T_{1} = 2T_{1} + [H_{∞}].

There are only 2 super-attracting fixed points (1, 0, 0), (0, 0, 1). As noted in [FS4, Example 2.1], the Fatou set Ω_{Q} consists of the basins of attraction

Ω_{1} = {|z| < |t|, |w| < |t|} (the unit bidisk) , Ω_{2} = {|t| < |w|}

of these points together with a third component Ω′ = {|w| < |t| < |z|} attracted by the indeterminacy point (0, 1, 0), and thus

Supp T = ∂Ω_{2} ⊂_{≠} ∂Ω_{1} ∪ ∂Ω_{2} = ℙ^{2} \ Ω_{Q} .

The measure μ can be written homogeneously on ℙ^{2} as μ = [dd^{c} log(|z|^{2} ∨|w|^{2} ∨|t|^{2})]^{2}. We have μ = T ∧ T_{1} as before, and μ is invariant on ℙ^{2} (Q^{∗}μ = 6μ) although T_{1} is not invariant. We now compute the product T^{2} on ℙ^{2}. First we note that outside the indeterminacy point (0, 1, 0), T^{2} = lim 9^{−k}P^{k∗}ω^{2} = 0. The Green function (on ℂ^{3}) of Q is given by G = log(|w| ∨ |t|); hence, using the coordinates w′ = w/z, t′ = t/z in ℙ^{2} \ {z = 0} ≈ ℂ^{2}, we see that T^{2} = [2dd^{c} log(|w′| ∨ |t′|)]^{2} = [(0, 1, 0)]. (Since G is locally bounded off the line {(0, z, 0)} ⊂ ℂ^{3}, the product T^{2} is well-defined according to [De, Cor. 2.11] or [FS5, Cor. 3.6].) Remark: The current T = 2dd^{c}G in Example 2 has mass 1, i.e., ∥T ∥(ℙ^{2}) = (T, ω) = 1 (see Section 6). It follows that ∥T^{2}∥(ℙ^{2}) = 1, which gives an alternate proof that T^{2} = [(0, 1, 0)]. In fact, if T ∈ D′^{1,1}(ℙ^{n}), U ∈ D′^{q,q}(ℙ^{n}) (1 ≤ q < n) are closed positive currents of mass 1 such that the product T ∧ U is well-defined as in [De] or [FS5], then T ∧ U has mass 1 by [FS5, Th. 4.4]. A simple proof of this “general Bezout theorem” is as follows: Since (T, ω^{n−1}) = 1, we can write T = dd^{c}u + ω. The (quasi-plurisubharmonic) function u can be realized as the pointwise limit of smooth functions u_{k} ≥ u on ℙ^{n} such that dd^{c}u_{k} ≥ −ω. Let T_{k} = dd^{c}u_{k} + ω. Then on any affine open set, T_{k} = dd^{c}(u_{k} + h), where dd^{c}h = ω. By [De] or [FS5], T_{k} ∧ U → T ∧ U. Since (T_{k} ∧ U, ω^{n−q−1}) = (U, T_{k} ∧ ω^{n−q−1}) = (U, ω^{n−q}) = 1, we conclude that

∥T ∧ U∥(ℙ^{n}) = (T ∧ U, ω^{n−q−1}) = lim(T_{k} ∧ U, ω^{n−q−1}) = 1 .

Example 3. We now consider the sequence of iterations of the map P (z, w) = (w^{3}, z^{2}). This example is not generic in the sense of Fornaess and Sibony [FS3, FS4], since H_{∞} is contracted by (the projectivization of) P to the indeterminacy point (0, 1, 0), so we cannot conclude from [FS3, FS4] that the sequence of currents \frac{1}{δ_{1}(P^{k})}P^{k∗}ω converges. Instead we have two subsequences

{P^{2k} = (z^{6^{k}}, w^{6^{k}})} , {P^{2k+1} = (w^{3·6^{k}}, z^{2·6^{k}})} .

The first subsequence is the same as in Example 1 with δ = 6, and the corresponding current subsequence thus converges to

T_{1} = dd^{c} log^{+}(|z|^{2} ∨ |w|^{2}).

The second subsequence of currents converges to

T_{2} = dd^{c} log^{+}(|w|^{2} ∨ |z|^{4/3}) .

These currents are responsible for the distribution of pre-images of hyperplanes. As for the invariance properties, neither T_{1} nor T_{2} are invariant since P^{∗}T_{1} = 3T_{2}, P^{∗}T_{2} = 2T_{1}. However, the current S = dd^{c} log(|z|^{2} ∨ |w|^{\sqrt{6}}) is invariant on ℂ^{2} with P^{∗}S = \sqrt{6}S, but S does not extend to an invariant current on ℙ^{2}. (The current [H_{∞}] seems to be the only invariant closed positive (1, 1)-current on ℙ^{2}.) Note that δ_{1}(P) = 3 but λ_{1}(P) = \sqrt{6} < δ_{1}(P), which is another way to conclude that P is not Fornaess-Sibony generic (see Section 4).

For the codimension two case, we have δ_{2}(P^{k}) = δ_{2}^{k}(P) = 6^{k}. So there is a limit measure which is the same as in the two previous examples and is invariant.

Example 4. One of the early motivations for the study of iterations came from Newton’s method, which provides an interesting example of a rational mapping of ℙ^{2}, as follows. If one applies Newton’s method to find the zeros of the function F : ℂ^{2} → ℂ^{2} given by F (z, w) = (z − w^{2}, w − z^{2}), one then must iterate the map

(z, w) ↦ (\frac{w(w + 2z^{2})}{4zw − 1}, \frac{z(z + 2w^{2})}{4zw − 1}) ,

which extends to the rational mapping of ℙ^{2},

Q(t, z, w) = (t(4zw − t^{2}), w(tw + 2z^{2}), z(tz + 2w^{2})) .

The mapping Q has 4 super-attracting fixed points (which are the zeros of F) and 5 indeterminacy points: I_{Q} = I′ ∪ I_{∞}, where I′ = {(−2, α, α^{2}) : α^{3} = 1} and I_{∞} = {(0, 1, 0), (0, 0, 1)}. (The restriction of Q to H_{∞} \ I_{∞} is the identity map, and thus the points of H_{∞} \ I_{∞} are non-attracting fixed points of Q.) It is interesting to note that Q does not contract any curves in ℙ^{2}. Precisely,

\begin{aligned}Q^{−1}(1, a, b) & = & \{(1, a ± \sqrt{a^{2} − b}, b ± \sqrt{b^{2} − a})\} , \\ Q^{−1}(0, a, b) & = & \{(0, a, b), (2ab, b^{2}, a^{2}), (0, 1, 0), (0, 0, 1)\} . \end{aligned}  (5)

Since no curves are contracted, λ_{1}(Q) = δ_{1}(Q) = 3. We see from (5) that δ_{2}(P) = 4. (Note that δ_{1}(P)^{2} − δ_{2}(P) equals the number of indeterminacy points; see the comment following Lemma 4.4.)

Let Ω_{j}, j = 1, …, 4, denote the respective basins of attraction of the 4 super-attracting fixed points. By (5), each fixed point is completely invariant, so the Ω_{j} are connected. Since the Ω_{j} are components of the Fatou set Ω_{Q}, μ does not charge the Ω_{j}. (It is an open question whether Ω_{Q} contains any other components.) It then follows from Theorem 1.1 applied to the delta measures of (non-exceptional) points of Ω_{j} that

Supp μ ⊂ ⋂_{j=1}^{4} ∂Ω_{j} .

In particular the Ω_{j} have common boundary points.

Since E_{Q} = ∅, μ is invariant: Q^{∗}μ = 4μ. The 4 super-attracting fixed points, as well as the 2 infinite indeterminacy points, are exceptional. In fact all the points of H_{∞} are apparently exceptional. To see this, consider a point W = (0, a, b), ab ≠ 0. We iterate (5) to conclude that

\frac{1}{4^{k}} Q^{k∗}δ_{W} (I_{∞}) = \frac{2(4^{k} − 1)}{3 · 4^{k}} → \frac{2}{3} .

On the other hand, from computer experiments using Maple, it appears that μ does not charge I_{∞} (although it looks like I_{∞} ⊂ Supp μ), as we expect (see Question 2 below), and thus W is exceptional. We have not found any other exceptional points. In particular, if we assume that the finite pre-image (2ab, b^{2}, a^{2}) of W is nonexceptional, we can conclude from (5) that

\frac{1}{4^{k}} Q^{k∗}δ_{W} → \frac{1}{3}[I_{∞}] + \frac{1}{3}μ .

Let us assume that the points of I′ are nonexceptional, which appears to be the case from computer experiments. It then follows that the backwards orbit of each point of I′ is an infinite set with closure containing the support of μ. We can write I_{Q^{k}} = I_{∞} ∪ I′_{k} where I′_{k} = I′ ∪ Q^{−1}(I′) ∪ ⋯ ∪ Q^{−(k−1)}(I′), and it follows that the indeterminacy sets of the iterates of Q accumulate along Supp μ.

Let G_{k} = 3^{−k}dd^{c} log |Q^{k}|. According to [FS3, FS4], {G_{k}} converges to a Green function G, and {Q^{k∗}ω} converges to an invariant (1,1)-current T = 2dd^{c}G. One can show that

(2dd^{c}G_{k})^{2} = \frac{1}{9^{k}} (Q^{k∗}ω^{2} + [I′_{k}] + n_{k}[I_{∞}]) ,  (6)

where n_{k} = \frac{1}{2}(9^{k}+1−2·4^{k}). The discrete part of the measure in (6) is obtained by noting that the points of I′_{k} are simple indeterminacy points (see the comment following Lemma 4.4); the value of n_{k} is determined by the requirement that (2dd^{c}G_{k})^{2} have mass 1. (See the remark following Example 2.) Letting k → ∞ in (6), we conclude that (2dd^{c}G_{k})^{2} → \frac{1}{2}[I_{∞}].

We state here some open problems:

Question 1. In the Lyubich-Freire-Lopes-Mañé Theorem, the exceptional set contains at most two points; in Examples 1–3, the exceptional sets are unions of at most 3 hyperplanes in ℙ^{2}. Can we further describe the exceptional set in Theorem 1.1? For a holomorphic map, is the exceptional set algebraic?

Question 2. As mentioned in Section 1, if μ(E_{P}) = 0, then P^{∗}μ would be well defined and we would have P^{∗}μ = λμ. Does μ(E_{P}) = 0 for all rational maps satisfying the hypotheses of Theorem 1.1? Indeed, can μ charge an algebraic subvariety of ℙ^{n}?

Question 3. If P : ℙ^{n} → ℙ^{n} is holomorphic, then {δ_{ℓ}(P)^{−k}P^{k∗}ω^{ℓ}} converges to (2dd^{c}G)^{ℓ}, where G is the Green function. Fornaess and Sibony [FS4] showed that if P is a meromorphic map of ℙ^{2} with δ_{1}(P) = λ_{1}(P), then {δ_{1}(P)^{−k}P^{k∗}ω} converges to a positive (1, 1)-current of mass 1 (provided that G ≢ −∞). Can one interpolate between this result of Fornaess-Sibony [FS4] and Theorem 1.1, as in the holomorphic case? In particular, does {δ_{ℓ}(P)^{−k}P^{k∗}ω^{ℓ}} converge if δ_{ℓ}(P) = λ_{ℓ}(P) > λ_{ℓ−1}(P), for 1 < ℓ < n?

Question 4. Theorem 1.2 easily generalizes to the case where [W] is replaced with \frac{1}{d}[A], where A is a (nonexceptional) complete intersection of degree d; the conclusion also holds with W replaced with a product of closed positive smooth (1,1)-forms of mass 1. Does Theorem 1.2 (or Corollary 1.4) hold for [W] replaced with a “general” closed positive (ℓ, ℓ)-current of mass 1? (For the case of P : ℙ^{2} → ℙ^{2} holomorphic and ℓ = 1, Fornaess and Sibony [FS4, Th. 4.16] gave a sufficient condition on P for Corollary 1.4 to hold for all closed positive (1,1)-currents of mass 1.)

3. Notation and terminology

We let E^{p,q}(X), D^{p,q}(X), D′^{p,q}(X) denote the spaces of (complex-valued) C^{∞} forms, compactly supported C^{∞} forms, and currents, respectively, of bidegree (p, q) on a complex manifold X and we use the standard differentials d = ∂ + \overline{∂}, d^{c} = (4π\sqrt{−1})^{−1}(∂ − \overline{∂}). Points in complex projective n-space ℙ^{n} are identified with their representations z = (z_{0}, z_{1}, …, z_{n}) in homogeneous coordinates. We shall regard the Grassmannian 𝔾(ℓ, m) of projective linear subspaces of codimension ℓ in ℙ^{m} as a subvariety of ℙ(^{⋀m+1−ℓ}ℂ^{m+1}).

If P : ℙ^{n}−→ℙ^{m} is a non-constant meromorphic map, it is a well-known consequence of Chow’s theorem that P must be rational, i.e., P can be written in the form P = (P_{0}, …, P_{n}) where P_{j} ∈ ℂ[z_{0}, …, z_{n}] and deg P_{0} = ⋯ = deg P_{m}. We can assume that the P_{j} have no common factors; we then say that deg P = deg P_{0}. (This notion of degree should not be confused with the topological degree of an equidimensional rational map, which is the number of points in the pre-image of a generic point in the range.) We let I_{P} ⊂ ℙ^{n} denote the indeterminacy locus of P (the points where P is not holomorphic); I_{P} is an algebraic subvariety of codimension ≥ 2.

Suppose f : Y → X is a holomorphic mapping of complex manifolds. If α is a current on X, f^{∗}α is not always defined. However, we shall define f^{∗}α in two special cases: First, we suppose α = uγ where γ ∈ E^{p,q}(X) is a smooth form and u is the difference of plurisubharmonic functions. Assume further that f(Y) is not contained in the ±∞ locus of u. Then u ∘ f is the difference of plurisubharmonic functions on Y and hence is in L_{loc}^{1}(Y). We define f^{∗}α = (u ∘ f)f^{∗}γ, which is clearly independent of the representation α = uγ. The second case we consider is that of a current of the form [D] ∧ γ, where [D] is the current of integration over a divisor D on X and γ is a smooth form as before. We assume also that f(Y) ⊄ Supp D so that f^{∗}D is a divisor on Y. We then define f^{∗}([D] ∧ γ) = [f^{∗}D] ∧ f^{∗}γ. These two definitions are consistent in the following way. Suppose α = log |g|^{2} · γ where g is a meromorphic function on X such that neither the zeroes nor the poles of g contain f(Y) and γ is a closed (p, q)-form on X. Then dd^{c}α = [D] ∧ γ where D = Div(g). Hence

f^{∗}dd^{c}α = [f^{∗}D] ∧ f^{∗}γ = dd^{c}(log |g ∘ f|^{2} · f^{∗}γ) = dd^{c}f^{∗}α .  (7)

If ψ is a current on Y and f|_{Supp ψ} is proper, one defines the push-forward current f_{∗}ψ by (f_{∗}ψ, φ) = (ψ, f^{∗}φ); if ψ is of order 0, then so is f_{∗}ψ. Furthermore, if ψ has L_{loc}^{1} coefficients, then so does f_{∗}ψ, provided that f is surjective. (The latter statement follows from the fact that the coefficients of ψ are in L_{loc}^{1} if and only if ψ is of order 0 and the total variation measure ∥ψ∥ is absolutely continuous with respect to volume measure. Note that if ψ is a smooth form, then the coefficients of f_{∗}ψ are generally not smooth and may even be unbounded.)

Let P : ℙ^{n}−→ℙ^{m} be a rational map. For a smooth (p, q)-form η ∈ E^{p,q}(ℙ^{m}) we define the pull-back current P^{∗}η ∈ D′^{p,q}(ℙ^{m}) as follows: We let G_{P} ⊂ ℙ^{n} × ℙ^{m} denote the graph of P (which is an irreducible algebraic subvariety of ℙ^{n} × ℙ^{m}) and we consider a desingularization G̃ →^{ρ} G_{P}. We have the commutative diagram:

G̃ ; π_{1} ↙  ↘ π_{2} ; ℙ^{n}  −→^{P} ℙ^{m}  (8)

We then define

P^{∗}η = π_{1∗}π_{2}^{∗}η .

The pull-back P^{∗}η has coefficients in L_{loc}^{1}, since π_{2}^{∗}η is smooth. Furthermore P^{∗}η|_{ℙ^{n}\I_{P}} is the usual (smooth) pull-back (P |_{ℙ^{n}\I_{P}})^{∗}η, and thus P^{∗}η does not depend on the choice of desingularization of G_{P}.

We consider the current of integration [W] ∈ D′^{ℓ,ℓ}(ℙ^{m}) and define the pull-back π_{2}^{∗}[W] to be the current of integration over the algebraic (n − ℓ)-cycle π_{2}^{∗}W on G̃ (using the diagram (8)), whenever dim π_{2}^{−1}(W) = n − ℓ. If we represent W as the intersection of hyperplanes H_{1}, …, H_{ℓ} in ℙ^{m}, then π_{2}^{∗}W is the intersection of divisors π_{2}^{∗}H_{1} ∩⋯∩π_{2}^{∗}H_{ℓ}. For a definition of this intersection, which is a formal sum of the irreducible components of π_{2}^{−1}(W) with positive integer coefficients, see [Ha, Appendix A] or Definitions 2.3 and 2.4.2 (or Example 7.1.10) in [Fu]. This pull-back, or intersection, can also be defined analytically as follows. Let g_{j} be a local defining function for π_{2}^{∗}H_{j}, for 1 ≤ j ≤ ℓ, and write g = (g_{1}, …, g_{ℓ}). Then by Griffiths and King [GK, 1.10] (see also [Sh, I.12, Th. 3]), we have the local formula

π_{2}^{∗}[W] = dd^{c} (log |g|^{2}(dd^{c} log |g|^{2})^{ℓ−1}) .

(Alternately, π_{2}^{∗}[W] = dd^{c} log |g_{1}|^{2} ∧ … ∧ dd^{c} log |g_{ℓ}|^{2}, where the existence of this product of currents is guaranteed by Demailly [De].)

We now state the Poincaré-Lelong formula for linear subspaces of ℙ^{m} and describe its pull-backs by a rational map P : ℙ^{n}−→ℙ^{m}. Let W ∈ 𝔾(ℓ, m) be an (m − ℓ)-plane in ℙ^{m}. For each W ∈ 𝔾(ℓ, m), we consider the Levine current

Λ_{W}^{ℓ} = log \frac{|ζ|^{2}|W |^{2}}{|ζ ∧ W |^{2}} ∑_{j=0}^{ℓ−1}(d_{ζ}d_{ζ}^{c} log |ζ ∧ W |^{2})^{j} ∧ ω_{ζ}^{ℓ−1−j} ∈ D′^{ℓ−1,ℓ−1}(ℙ_{ζ}^{m}) ,

which (by definition) has locally integrable coefficients. We have the Poincaré-Lelong formula for W [Le] (see also [GK, 1.15] or [Sh, II.6, pp. 68-69]),

dd^{c}Λ_{W}^{ℓ} = ω^{ℓ} − [W] .  (9)

Now let W = H_{1} ∩ ⋯ ∩ H_{ℓ} such that dim π_{2}^{−1}(W) = n − ℓ, where we use the notation of (8). In order to “pull back” the identity (9) by the meromorphic map P, we first apply the generalized Poincaré-Lelong formula ([GK, 1.15] or [Sh, II.6, pp. 68-69]) to the divisors π_{2}^{∗}H_{1}, …, π_{2}^{∗}H_{ℓ} of the lifted hyperplane-section bundle π_{2}^{∗}O_{ℙ^{m}}(1) with Chern form π_{2}^{∗}ω to obtain

dd^{c}π_{2}^{∗}Λ_{W}^{ℓ} = π_{2}^{∗}ω^{ℓ} − π_{2}^{∗}[W] ,  (10)

where π_{2}^{∗}Λ_{W}^{ℓ} ∈ D′^{ℓ−1,ℓ−1}(G̃) is given by

(π_{2}^{∗}Λ_{W}^{ℓ})(z̃) =^{def} log \frac{|π_{2}(z̃)|^{2}|W |^{2}}{|π_{2}(z̃) ∧ W |^{2}} ∑_{j=0}^{ℓ−1}(dd^{c} log |π_{2}(z̃) ∧ W |^{2})^{j} ∧ π_{2}^{∗}ω^{ℓ−1−j} ∈ D′^{ℓ−1,ℓ−1}(G̃). (11)

In particular, π_{2}^{∗}Λ_{W}^{ℓ} has L_{loc}^{1} coefficients and is smooth on G̃\π_{2}^{−1}(W). We define the currents on ℙ^{n}

\begin{aligned}P^{∗}Λ_{W}^{ℓ} & = & π_{1∗}π_{2}^{∗}Λ_{W}^{ℓ} , \\ P^{∗}[W] & = & π_{1∗}π_{2}^{∗}[W] . \end{aligned}  (12)

The current P^{∗}Λ_{W}^{ℓ}, being the push-forward by π_{1} of a current with L_{loc}^{1} coefficients, also has coefficients in L_{loc}^{1}. By applying π_{1∗} to (10), we then obtain the Poincaré-Lelong formula for P^{∗}[W],

dd^{c}P^{∗}Λ_{W}^{ℓ} = P^{∗}ω^{ℓ} − P^{∗}[W].  (13)

Note that for generic W, π_{2}^{∗}[W] has multiplicity identically 1 and contains no components inside the exceptional locus of π_{1}, and thus P^{∗}[W] is the current of integration over the closure of (P |_{ℙ^{n}\I_{P}})^{−1}(W).

4. The intermediate degrees of a rational map

In this section, we give some properties of the intermediate degrees δ_{ℓ}(P) of a rational map P, which we also describe analytically and topologically. We use the following consequence of Bertini’s theorem:

Lemma 4.1. Let Y be a projective algebraic manifold and let f : Y → ℙ^{m} be a nonconstant holomorphic map. Then for a generic hyperplane H ⊂ ℙ^{m}, the divisor f^{∗}H is smooth and has multiplicity 1.

Proof: Apply Bertini’s theorem (see, for example, [GH, p. 137]) to the complete linear system of f^{∗}H.

We now denote the normalized Fubini-Study Kähler form on ℙ^{m} by ω_{m}. For a subvariety V ⊂ ℙ^{m}, we write P^{−1}(V) = π_{1}(π_{2}^{−1}(V)) (using the notation in (8)). We let #(S) denote the cardinality of a set S. We begin with a formula for the integral of certain singular forms on ℙ^{n}:

Lemma 4.2. Let P_{j} : ℙ^{n}−→ℙ^{m_{j}}, 1 ≤ j ≤ n, be rational maps, and let I = I_{P_{1}} ∪ ⋯ ∪ I_{P_{n}}. i) For generic hyperplanes H_{1} ⊂ ℙ^{m_{1}}, …, H_{n} ⊂ ℙ^{m_{n}},

∫_{ℙ^{n}\I} P_{1}^{∗}ω_{m_{1}} ∧ ⋯ ∧ P_{n}^{∗}ω_{m_{n}} = # ( ⋂_{j=1}^{n} P_{j}^{−1}(H_{j}) \ I) ,

ii)

∫_{ℙ^{n}\I} P_{1}^{∗}ω_{m_{1}} ∧ ⋯ ∧ P_{n}^{∗}ω_{m_{n}} ≤ ∏_{j=1}^{n} deg P_{j} .

To verify Lemma 4.2, we first give a topological description of the integral. Let X = ℙ^{m_{1}} × ⋯ × ℙ^{m_{n}} and write P = (P_{1}, …, P_{n}) : ℙ^{n}−→X; then I = I_{P}. Consider the commutative diagram

G̃
π_{1} ↙  ↘ π_{2}
ℙ^{n}  −→^{P}  X

where G̃ is a desingularization of the graph of P and π_{1}, π_{2} are the projections. Let p_{j} : X → ℙ^{m_{j}} denote the projection to the j-th factor and let P̃_{j} = p_{j} ∘ π_{2} : G̃ → ℙ^{m_{j}}, for 1 ≤ j ≤ n. Let t_{m} denote the positive generator of H^{2}(ℙ^{m}, ℤ), and write

t_{m_{1}} × ⋯ × t_{m_{n}} = p_{1}^{∗}t_{m_{1}} • ⋯ • p_{n}^{∗}t_{m_{n}} ∈ H^{2n}(X, ℤ) ,

where • denotes the cup product in the cohomology ring.

Lemma 4.3. Using the notation of Lemma 4.2,

∫_{ℙ^{n}\I} P_{1}^{∗}ω_{m_{1}} ∧ ⋯ ∧ P_{n}^{∗}ω_{m_{n}} = (π_{2}^{∗}(t_{m_{1}} × ⋯ × t_{m_{n}}), G̃)
= (P̃_{1}^{∗}t_{m_{1}} • ⋯ • P̃_{n}^{∗}t_{m_{n}}, G̃) ∈ ℤ .

Proof: Since the de Rham class of the Kähler form ω_{m} on ℙ^{m} equals t_{m}, we have by Section 3,

∫_{ℙ^{n}\I} P_{1}^{∗}ω_{m_{1}} ∧ ⋯ ∧ P_{n}^{∗}ω_{m_{n}} = ∫_{ℙ^{n}\I_{P}} P^{∗}(ω_{m_{1}} × ⋯ × ω_{m_{n}})
= (π_{1∗}π_{2}^{∗}(ω_{m_{1}} × ⋯ × ω_{m_{n}}), 1)
= (π_{2}^{∗}(ω_{m_{1}} × ⋯ × ω_{m_{n}}), 1)
= (π_{2}^{∗}(t_{m_{1}} × ⋯ × t_{m_{n}}), G̃) .

The second equality follows from our definitions. Proof of Lemma 4.2: Let E ⊂ G̃ be the exceptional locus of π_{1}. By Lemma 4.1 applied to P̃_{1} : G̃ → ℙ^{m_{1}}, there is a hyperplane H_{1} ⊂ ℙ^{m_{1}} such that the divisor π_{2}^{∗}(H_{1} ×ℙ^{m_{2}} ×⋯×ℙ^{m_{n}}) is a smooth hypersurface Y_{1} ⊂ G̃ of multiplicity 1 with dim Y_{1} ∩ E < n − 1. By Lemma 4.1, we can inductively find hyperplanes H_{2} ⊂ ℙ^{m_{2}}, …, H_{n} ⊂ ℙ^{m_{n}} such that, writing

Y_{j} = π_{2}^{−1}(H_{1} × ⋯ × H_{j} × ℙ^{m_{j+1}} × ⋯ × ℙ^{m_{n}}) ,

Y_{j} is a smooth submanifold of G̃ of dimension n − j, dim Y_{j} ∩ E < n − j, and the divisor (P̃_{j}|_{Y_{j−1}})^{∗}H_{j} (on Y_{j−1}) has multiplicity 1, or equivalently, P̃_{j}^{−1}(H_{j}) intersects Y_{j−1} transversely. In particular, dim Y_{n} = 0 and Y_{n} ∩ E = ∅. Write H′_{j} = p_{j}^{−1}(H_{j}), for 1 ≤ j ≤ n, so that

Y_{j} = π_{2}^{−1}(H′_{1} ∩ ⋯ ∩ H′_{j}) = π_{2}^{−1}(H′_{j}) ∩ Y_{j−1} .

A codimension j submanifold S of a complex manifold Y determines the current of integration [S] ∈ D′^{j,j}(Y) given by ([S], φ) = ∫_{S} φ for a test form φ. We also let [S] denote the de Rham class in H^{2j}(Y, ℝ) containing the current [S]. (If Y is compact of dimension n, then the cohomology class [S] is the Poincaré dual of the (2n − 2j)-cycle S.) If two submanifolds S_{1}, S_{2} intersect transversely, then [S_{1} ∩ S_{2}] = [S_{1}] • [S_{2}] in the cohomology ring of Y. In particular, [H_{j}] = t_{j} ∈ H^{2}(ℙ^{m_{j}}, ℤ) ⊂ H^{2}(ℙ^{m_{j}}, ℝ) and

[H_{1} × ⋯ × H_{n}] = [H′_{1} ∩ ⋯ ∩ H′_{n}] = t_{m_{1}} × ⋯ × t_{m_{n}} .

Furthermore, our construction of the H_{j}, Y_{j} implies that

[Y_{j}] = [Y_{j−1}] • π_{2}^{∗}[H′_{j}] = π_{2}^{∗}[H′_{1}] • ⋯ • π_{2}^{∗}[H′_{j}] = π_{2}^{∗}[H′_{1} ∩ ⋯ ∩ H′_{j}] .

Therefore,

π_{2}^{∗}(t_{m_{1}} × ⋯ × t_{m_{n}}) = π_{2}^{∗}[H_{1} × ⋯ × H_{n}] = [Y_{n}]  (14)

(where the points of Y_{n} have multiplicity 1).

Let A_{j} : ℂ^{m_{j}+1} → ℂ be a linear map defining the hyperplane H_{j}, and consider the polynomial

Q_{j} = A_{j}(P_{j0}, …, P_{jm_{j}}) ∈ ℂ[z_{0}, …, z_{n}] ,

where P_{j} = (P_{j0}, …, P_{jm_{j}}), for 1 ≤ j ≤ n. We then have

{z ∈ ℙ^{n} : Q_{j}(z) = 0} = π_{1} (π_{2}^{−1}(H′_{j})) ⊃ I_{P_{j}} .  (15)

Since Y_{j} ⊂ G̃ \ E and π_{1} maps G̃ \ E bijectively to ℙ^{n} \ I_{P}, we have

π_{1}(Y_{n}) = π_{1} (π_{2}^{−1}(H′_{1} ∩ ⋯ ∩ H′_{n})) \ I_{P}
= ⋂_{j=1}^{n} π_{1} (π_{2}^{−1}(H′_{j})) \ I_{P} = ⋂_{j=1}^{n} P_{j}^{−1}(H_{j}) \ I_{P}
= {z ∈ ℙ^{n} \ I_{P} : Q_{1}(z) = ⋯ = Q_{n}(z) = 0} .

Thus by (14),

(π_{2}^{∗}(t_{m_{1}} × ⋯ × t_{m_{n}}), G̃) = ([Y_{n}], G̃) = #(Y_{n}) .

By Bézout’s Theorem,

#(Y_{n}) = #(π_{1}(Y_{n})) = # ( ⋂_{j=1}^{n} P_{j}^{−1}(H_{j}) \ I_{P} ) ≤ ∏_{j=1}^{n} deg Q_{j} = ∏_{j=1}^{n} deg P_{j} .

The conclusion follows from Lemma 4.3. Definition: Let P : ℙ^{n}−→ℙ^{m} be a rational map. We define the intermediate degrees δ_{ℓ}(P) of P by the formula

δ_{ℓ}(P) = ∫_{ℙ^{n}\I_{P}} P^{∗}ω_{m}^{ℓ} ∧ ω_{n}^{n−ℓ}

for 0 ≤ ℓ ≤ min(n, m).

We shall show in Lemma 4.4 below that the intermediate degrees are also given by the geometric definition in the introduction; in particular, if m = n, then δ_{n}(P) is the topological degree of P, which is defined as the cardinality of P^{−1}(x), for a generic point x ∈ ℙ^{n}. Note that δ_{0}(P) = 1. Clearly, δ_{ℓ}(P) > 0 if and only if rank P ≥ ℓ. It is easy to verify that δ_{1}(P) = deg P, and if P is holomorphic (this can happen only if m ≥ rank P = n), then δ_{ℓ}(P) = (deg P)^{ℓ}, for ℓ ≤ n.

It follows from Lemma 4.2 applied to the maps P_{1} = ⋯ = P_{ℓ} = P , P_{ℓ+1} = ⋯ = P_{n} = Id_{ℙ^{n}} that in general,

δ_{ℓ}(P) ≤ (deg P)^{ℓ} , for 1 ≤ ℓ ≤ min(n, m) .  (16)

We shall give a more general inequality in Lemma 4.7 below.

Lemma 4.4. Let P : ℙ^{n}−→ℙ^{m} be a rational map, and let 1 ≤ ℓ ≤ min(n, m). Then

δ_{ℓ}(P) = deg P^{−1}(W) ≤ (deg P)^{ℓ}

for generic W ∈ 𝔾(ℓ, m), with equality if and only if codim I_{P} > ℓ. In particular, δ_{n}(P) ≤ (deg P)^{n}, with equality if and only if P is holomorphic.

Proof: We shall apply Lemma 4.2 with P_{1} = ⋯ = P_{ℓ} = P , P_{ℓ+1} = ⋯ = P_{n} = Id_{ℙ^{n}}. By part (ii) of the lemma, δ_{ℓ}(P) ≤ (deg P)^{ℓ}. For generic W = H_{1} ∩ ⋯ ∩ H_{ℓ}, π_{2}^{−1}(W) is of codimension ℓ and has no components contained in the exceptional locus of π_{1}, and thus P^{−1}(W) has pure dimension n − ℓ and dim P^{−1}(W) ∩ I_{P} < n − ℓ. Hence for generic hyperplanes H_{1}, …, H_{n}, we have

⋂_{j=1}^{n} P_{j}^{−1}(H_{j}) \ I_{P} = P^{−1}(W) ∩ H_{ℓ+1} ∩ ⋯ ∩ H_{n} ,

where W = H_{1} ∩ ⋯ ∩ H_{ℓ}, and thus by part (i),

δ_{ℓ}(P) = # (P^{−1}(W) ∩ H_{ℓ+1} ∩ ⋯ ∩ H_{n}) = deg P^{−1}(W) .

Furthermore, using the notation in the proof of Lemma 4.2,

{z ∈ ℙ^{n} : Q_{1}(z) = ⋯ = Q_{ℓ}(z) = 0} = P^{−1}(W) ∪ I_{P} .

Since deg Q_{1} = ⋯ = deg Q_{ℓ} = deg P, it follows from Bézout’s theorem (see for example [Fu, Example 8.4.6]) that δ_{ℓ}(P) < (deg P)^{ℓ} if dim I_{P} ≥ n − ℓ. If dim I_{P} < n − ℓ, then P^{−1}(W) ⊃ I_{P} (by dimension considerations) and Bézout’s theorem gives equality.

In fact, if n = 2 in Lemma 4.4, then δ_{2}(P) = (deg P)^{2} − q, where q is the number of points of I_{P} counting multiplicity. This is illustrated by the following example.

Example: Let P : ℙ^{2}−→ℙ^{2} be the “quadratic transform” given by

P (z_{0}, z_{1}, z_{2}) = (z_{1}z_{2}, z_{0}z_{2}, z_{0}z_{1}) = (\frac{1}{z_{0}} , \frac{1}{z_{1}} , \frac{1}{z_{2}} ) .

Then δ_{1}(P) = 2, δ_{2}(P) = 1. Note that in this example I_{P} consists of the three points (1, 0, 0), (0, 1, 0), (0, 0, 1).

Lemma 4.5. Suppose that P : ℙ^{n}−→ℙ^{m} is a rational map and L : ℂ^{m+1} → ℂ^{M+1} is a linear map such that Image P ⊄ ℙ(L^{−1}(0)). Let P_{L} = L̂ ∘ P : ℙ^{n}−→ℙ^{M}, where L̂ : ℙ^{m}−→ℙ^{M} is the map induced from L. Then

∫_{ℙ^{n}\I}(P_{L}^{∗}ω)^{k} ∧ (P^{∗}ω)^{ℓ−k} ∧ ω^{n−ℓ} ≤ δ_{ℓ}(P)

for 1 ≤ k ≤ ℓ ≤ n, where I = I_{P} ∪ I_{P_{L}}.

Proof: We can assume without loss of generality that M = m. We first consider the case where L is nonsingular and thus L̂ is biholomorphic. Therefore

∫_{ℙ^{n}\I}(P_{L}^{∗}ω)^{k} ∧ (P^{∗}ω)^{ℓ−k} ∧ ω^{n−ℓ} = ∫_{ℙ^{n}\I}(P^{∗}ω′)^{k} ∧ (P^{∗}ω)^{ℓ−k} ∧ ω^{n−ℓ}

where ω′ = L̂^{∗}ω. Since ω′ and ω are in the same de Rham class, it follows from Section 3 (or by the proof of Lemma 4.3) that

∫_{ℙ^{n}\I}(P^{∗}ω′)^{k} ∧ (P^{∗}ω)^{ℓ−k} ∧ ω^{n−ℓ} = ∫_{ℙ^{n}\I}(P^{∗}ω)^{ℓ} ∧ ω^{n−ℓ} = δ_{ℓ}(P) .

We now suppose that L is singular. Choose a sequence {L_{ν}} of nonsingular linear operators on ℂ^{m+1} such that L_{ν} → L. We can write

(P_{L_{ν}}^{∗}ω)^{k} ∧ (P^{∗}ω)^{ℓ−k} ∧ ω^{n−ℓ} = f_{ν}ω^{n} , (P_{L}^{∗}ω)^{k} ∧ (P^{∗}ω)^{ℓ−k} ∧ ω^{n−ℓ} = fω^{n}

where f_{ν}, f are non-negative C^{∞} functions on ℙ^{n} \ I. Then f_{ν} → f pointwise on ℙ^{n} \ I, and hence by Fatou’s Lemma,

∫_{ℙ^{n}\I}(P_{L}^{∗}ω)^{k} ∧ (P^{∗}ω)^{ℓ−k} ∧ ω^{n−ℓ} = ∫_{ℙ^{n}\I} fω^{n} ≤ lim inf_{ν→∞} ∫_{ℙ^{n}\I} f_{ν}ω^{n} = δ_{ℓ}(P) .

Lemma 4.6. Let P : ℙ^{n}−→ℙ^{m} , Q : ℙ^{m}−→ℙ^{r} be rational maps. Then

δ_{ℓ}(Q ∘ P) ≤ δ_{ℓ}(P)δ_{ℓ}(Q) .

Proof: Let η = Q^{∗}ω^{ℓ} ∈ D′^{ℓ,ℓ}(ℙ^{n}). We smooth η by an approximate identity {ψ_{ε}}_{ε>0} with respect to a Haar measure h on GL(n + 1, ℂ) to obtain

η_{ε} =^{def} ∫_{GL(n+1,ℂ)}(g^{∗}η)ψ_{ε}(g)dh(g) ∈ E^{ℓ,ℓ}(ℙ^{n}) .

Then η_{ε} → η pointwise as ε → 0, η_{ε} ≥ 0, and we have the identity in de Rham cohomology,

[η_{ε}] = [η] = δ_{ℓ}(Q)[ω^{ℓ}] ∈ H^{2ℓ}(ℙ^{n}, ℤ) .

Using the commutative diagram (8), we then have

∫_{ℙ^{n}} P^{∗}η_{ε} ∧ ω^{n−ℓ} = ∫_{G} π_{2}^{∗}η_{ε} ∧ π_{1}^{∗}ω^{n−ℓ} = ∫_{G} δ_{ℓ}(Q)π_{2}^{∗}ω^{ℓ} ∧ π_{1}^{∗}ω^{n−ℓ} = δ_{ℓ}(Q)δ_{ℓ}(P) .

Therefore, by Fatou’s lemma,

δ_{ℓ}(Q ∘ P) = ∫_{ℙ^{n}\I_{Q∘P}} P^{∗}η ∧ ω^{n−ℓ} ≤ lim inf_{ε→0} ∫_{ℙ^{n}} P^{∗}η_{ε} ∧ ω^{n−ℓ} = δ_{ℓ}(Q)δ_{ℓ}(P) .

In particular, if P : ℙ^{n}−→ℙ^{n}, we have δ_{ℓ}(P^{j+k}) ≤ δ_{ℓ}(P^{j})δ_{ℓ}(P^{k}). Hence the sequence c_{k} = log δ_{ℓ}(P^{k}), k = 1, 2, …, is subadditive and thus c_{k}/k → inf_{k} c_{k}/k (see, for example, [PS, I.98]). This enables us to introduce the following definition. Definition: Let P : ℙ^{n}−→ℙ^{n} be a meromorphic map, and let P^{k} denote the k-th iterate of P. We define the intermediate dynamic degrees λ_{ℓ}(P) of P by

λ_{ℓ}(P) = lim_{k→∞} δ_{ℓ}(P^{k})^{1/k} = inf_{k≥1} δ_{ℓ}(P^{k})^{1/k} ,

for 1 ≤ ℓ ≤ n.

The intermediate dynamic degrees were considered by Friedland [Fr]; λ_{1}(P) is called the “dynamic degree” in [Di]. We note that λ_{ℓ}(P) ≤ δ_{ℓ}(P), and λ_{n}(P) ≤ δ_{n}(P). If P is holomorphic, then λ_{ℓ}(P) = δ_{ℓ}(P). The map P : ℙ^{2}−→ℙ^{2} is “generic” in the sense of Fornaess and Sibony [FS3, FS4] if and only if λ_{1}(P) = δ_{1}(P) [FS4, Di].

Lemma 4.7. Let P : ℙ^{n}−→ℙ^{m} be a rational map. Then

δ_{k+ℓ}(P) ≤ δ_{k}(P)δ_{ℓ}(P) .

Proof: Let η = P^{∗}ω^{ℓ} ∈ D′^{ℓ,ℓ}(ℙ^{m}) and consider the smooth forms η_{ε} as in the above proof. As before,

[η_{ε}] = δ_{ℓ}(P)[ω^{ℓ}] ∈ H^{2ℓ}(ℙ^{n}, ℤ) ,

and

∫_{ℙ^{n}} P^{∗}ω^{k} ∧ η_{ε} ∧ ω^{n−k−ℓ} = ∫_{G} π_{2}^{∗}ω^{k} ∧ π_{1}^{∗}(η_{ε} ∧ ω^{n−k−ℓ})
= δ_{ℓ}(P) ∫_{G} π_{2}^{∗}ω^{k} ∧ π_{1}^{∗}ω^{n−k} = δ_{ℓ}(P)δ_{k}(P) .

The conclusion follows as above by letting ε → 0 and applying Fatou’s lemma. Note that it follows immediately from Lemma 4.7 that

λ_{k+ℓ}(P) ≤ λ_{k}(P)λ_{ℓ}(P) .

5. The proximity function

Let P : ℙ^{n}−→ℙ^{m} be a rational map, and let 1 ≤ ℓ ≤ min(n, m). Recall that if codim π_{2}^{−1}(W) = ℓ, where π_{2} : G̃ → ℙ^{m} is given by the commutative diagram (8), then equations (11), (12) define a current P^{∗}Λ_{W}^{ℓ} ∈ D′^{ℓ−1,ℓ−1}(ℙ^{n}) with locally integrable coefficients. Hence we can define the proximity function m_{P}^{ℓ} : 𝔾(ℓ, m) → [0, +∞] by

m_{P}^{ℓ} (W) = \{ \begin{matrix}(P^{∗}Λ_{W}^{ℓ} , ω^{n−ℓ+1}) = ∫_{ℙ^{n}\[I_{P} ∪P^{−1}(W)]} P^{∗}Λ_{W}^{ℓ} ∧ ω^{n−ℓ+1} & if codim π_{2}^{−1}(W) = ℓ \\ +∞ & if codim π_{2}^{−1}(W) < ℓ \end{matrix}

for W ∈ 𝔾(ℓ, m). We give 𝔾(ℓ, m) the Kähler metric ω induced from the natural embedding 𝔾(ℓ, m) ⊂ ℙ(⋀^{m+1−ℓ} ℂ^{m+1}).

The following key estimate is used in our proof of Theorem 1.2.

Lemma 5.1. If P : ℙ^{n}−→ℙ^{m} is a rational map, then m_{P}^{ℓ} ∈ L^{1}(𝔾(ℓ, m)) and

dd^{c}m_{P}^{ℓ} ≤ ℓδ_{ℓ−1}(P)ω ,

for 1 ≤ ℓ ≤ min(n, m).

Before proving Lemma 5.1, we give a brief outline of our approach, which was inspired by Skoda [Sk1, Sk2]. We consider the (n, n)-current on ℙ_{z}^{n} × 𝔾(ℓ, m)_{W},

Ω = log \frac{|P (z)|^{2}|W |^{2}}{|P (z) ∧ W |^{2}} ω_{z}^{n−ℓ+1} ∧ ∑_{j=0}^{ℓ−1}(dd^{c} log |P (z) ∧ W |^{2})^{j} ∧ (dd^{c} log |P (z)|^{2})^{ℓ−1−j} . (17)

We shall show that Ω has locally integrable coefficients and that

m_{P}^{ℓ} = π_{∗}Ω ,  (18)

where π : ℙ^{n} × 𝔾(ℓ, m) → 𝔾(ℓ, m) is the projection. We then differentiate (17) and obtain the inequality

dd^{c}m_{P}^{ℓ} = π_{∗}dd^{c}Ω
≤ π_{∗} [ω_{W} ∧ ω_{z}^{n−ℓ+1} ∧ ∑_{0}^{ℓ−1}(dd^{c} log |P (z) ∧ W |^{2})^{j} ∧ (dd^{c} log |P (z)|^{2})^{ℓ−1−j}] .

Finally, we apply Lemma 4.5 to obtain the desired bound on dd^{c}m_{P}^{ℓ}.

In order to avoid various difficulties with multiplying and pulling back currents, we shall define Ω as the pull-back (by a rational map) of a current Ω̂ on a larger product space X. The details are as follows. Proof of Lemma 5.1: We assume first that ℓ ≥ 2. (The estimate of Lemma 5.1 is straightforward for the hyperplane case ℓ = 1; we give the argument for this case at the end of this proof. See also [RS] for a complete treatment of pre-images of hyperplanes.) Write E = ⋀^{m+1−ℓ} ℂ^{m+1}, Ẽ = ⋀^{m+2−ℓ} ℂ^{m+1} and let λ_{0} : E → ℂ, λ_{1} : Ẽ → ℂ be linear functions of unit norm. For ζ ∈ ℂ^{m+1}, W ∈ E, we define the augmented exterior product

ζ ˜∧W = (ζ_{0}W_{0}, ζ ∧ W) ∈ ℂ ⊕ Ẽ

where W_{0} = λ_{0}(W). By making a linear change of coordinates in ℙ^{m} we can assume without loss of generality that Image P ⊄ {ζ ∈ ℙ^{m} : ζ_{0} = 0}; we also assume that λ_{1} is chosen so that λ_{1}(P (z) ∧ W) ≢ 0.

We write

X = ℙ_{z}^{n} × 𝔾(ℓ, m)_{W} × ℙ_{ζ}^{m} × ℙ(Ẽ)_{θ} × ℙ(ℂ ⊕ Ẽ)_{η} .

(The subscripts z, W, ζ, θ, η serve to identify the variables used in this discussion.) Let Q : ℙ^{n} × 𝔾(ℓ, m)−→X be the meromorphic (rational) map given by

Q(z, W) = (z, W, P (z), P (z) ∧ W, P (z)˜∧W) .  (19)

Let I_{Q} ⊂ ℙ^{n} × 𝔾(ℓ, m) denote the indeterminacy locus of Q; I_{Q} is an algebraic subvariety of codimension ≥ 2. Write U = ℙ^{n} × 𝔾(ℓ, m) \ I_{Q}, and let Q_{0} = Q|_{U} : U → X.

We further write θ_{1} = λ_{1}(θ) for θ ∈ Ẽ. If η = (c, θ) ∈ ℂ ⊕ Ẽ, we write η_{0} = c, η_{1} = θ_{1} = λ_{1}(θ). We consider the current

Ω̂ = log \frac{|ζ|^{2}|W |^{2}|θ_{1}|^{2}|η_{0}|^{2}}{|ζ_{0}|^{2}|W_{0}|^{2}|θ|^{2}|η_{1}|^{2}} ω_{z}^{n−ℓ+1} ∧ ∑_{j=0}^{ℓ−1} ω_{θ}^{j} ∧ ω_{ζ}^{ℓ−1−j} ∈ D′^{n,n}(X) .

Since

\frac{|ζ||W ||θ_{1}||η_{0}|}{|ζ_{0}||W_{0}||θ||η_{1}|} ∘ Q = \frac{|P (z)||W |}{|P (z) ∧ W |} ,

it follows that Q_{0}^{∗}Ω̂ is the smooth form on U given by the right side of equation (17).

We now show that Q_{0}^{∗}Ω̂ extends to a current Ω with locally integrable coefficients on all of ℙ^{n} × 𝔾(ℓ, m). Let

Y →^{ρ} Image Q ⊂ X

be a desingularization of the image of Q. (Note that Image Q is an algebraic subvariety of X and can be identified with the graph of Q.) Consider the projection π′ : X → ℙ^{n} × 𝔾(ℓ, m) and write ρ_{1} = π′ ∘ ρ, ρ_{2} = π ∘ ρ_{1}, so that we have the commutative diagram:

ℙ^{n}×𝔾(ℓ, m) ⟵^{ρ_{1}} Y

Q ↓^{|} ρ ↙ ↓ ρ_{1} ↘ ρ_{2}

X ⟶^{π′} ℙ^{n}×𝔾(ℓ, m) →^{π} 𝔾(ℓ, m)

We then define Ω = ρ_{1∗}ρ^{∗}Ω̂, where ρ^{∗}Ω̂ is given as in Section 3. Since ρ_{1} maps ρ_{1}^{−1}(U) biholomorphically onto U, we have Q_{0} = ρ ∘ ρ_{1}^{−1}|_{U}, and therefore

Ω|_{U} = (ρ_{1}^{−1}|_{U})^{∗}ρ^{∗}Ω̂ = Q_{0}^{∗}Ω̂ .

Since Ω has coefficients in L_{loc}^{1}, it is the extension to ℙ^{n} × 𝔾(ℓ, m) of Q_{0}^{∗}Ω̂ with zero mass on I_{Q}.

We now verify equation (18):

Lemma 5.2. m_{P}^{ℓ} = ρ_{2∗}ρ^{∗}Ω̂ = π_{∗}Ω ∈ L^{1}(𝔾(ℓ, m)).

Proof: By the definition of Ω we have

ρ_{2∗}ρ^{∗}Ω̂ = π_{∗}ρ_{1∗}ρ^{∗}Ω̂ = π_{∗}Ω .  (20)

Since Ω has coefficients in L_{loc}^{1}, π_{∗}Ω ∈ L^{1}(𝔾(ℓ, m)) and

(π_{∗}Ω)(W) = ∫_{ℙ_{z}^{n}×{W }\I_{Q}} Ω < +∞ , for a.a. W ∈ 𝔾(ℓ, m) .  (21)

Therefore

m_{P}^{ℓ} (W) = ∫_{ℙ^{n}\[I_{P} ∪P^{−1}(W)]} P^{∗}Λ_{W}^{ℓ} ∧ ω^{n−ℓ+1} = ∫_{ℙ_{z}^{n}×{W }\I_{Q}} Ω  (22)

whenever codim π_{2}^{−1}(W) = ℓ and thus for a.a. W ∈ 𝔾(ℓ, m). The desired identity follows from (20), (21), and (22).

We are now ready to compute dd^{c}m_{P}^{ℓ}. We let H_{ζ}^{0}, H_{W}^{0} , H_{θ}^{1}, H_{η}^{0}, H_{η}^{1} denote the hyperplanes in X given by the divisors of ζ_{0}, W_{0}, θ_{1}, η_{0}, η_{1} respectively, and we let

D = H_{ζ}^{0} + H_{W}^{0} − H_{θ}^{1} + H_{η}^{1} − H_{η}^{0} = Div (\frac{ζ_{0}W_{0}η_{1}}{θ_{1}η_{0}} ) .

We have

\begin{aligned}dd^{c}Ω̂ & = (ω_{ζ} + ω_{W} − ω_{θ} − D) ∧ ω_{z}^{n−ℓ+1} ∧ ∑_{j=0}^{ℓ−1}ω_{θ}^{j} ∧ ω_{ζ}^{ℓ−1−j} \\ & = [ω_{ζ}^{ℓ} − ω_{θ}^{ℓ} + (ω_{W} − D) ∧ ∑_{j=0}^{ℓ−1}ω_{θ}^{j} ∧ ω_{ζ}^{ℓ−1−j}] ∧ ω_{z}^{n−ℓ+1} . \end{aligned}  (23)

Lemma 5.3. The divisor ρ^{∗}D is effective (i.e., is locally the divisor of a holomorphic function).

Proof: (Our choices of ζ_{0} and θ_{1} guarantee that ρ(Y) ⊄ Supp D so that ρ^{∗}D is defined.) Let y_{0} ∈ Y be arbitrary, and let

X′ = {(z, W, ζ, θ, η) ∈ X : z_{i} ≠ 0, λ(W) ≠ 0, ζ_{j} ≠ 0, λ′(θ) ≠ 0}

where 0 ≤ i ≤ n, 0 ≤ j ≤ m and λ : E → ℂ, λ′ : Ẽ → ℂ are linear functions chosen so that ρ(y_{0}) ∈ X′. Let

g = \frac{ζ̃_{0}W̃_{0}}{θ̃_{1}} \frac{η_{1}}{η_{0}} ∈ Mer(X′)

where ζ̃_{0} = ζ_{0}/ζ_{j}, W̃_{0} = W_{0}/λ(W) = (λ_{0}/λ)(W), θ̃_{1} = θ_{1}/λ′(θ) = (λ_{1}/λ′)(θ). Then Div(g) = D|_{X′}. We must show that g ∘ ρ is holomorphic at y_{0}. Now

\begin{matrix}g ∘ Q = \frac{P_{0}(z)}{P_{j}(z)} \frac{W_{0}}{λ(W)} \frac{λ′(P (z) ∧ W)}{λ_{1}(P (z) ∧ W)} \frac{λ_{1}(P (z) ∧ W)}{P_{0}(z)W_{0}} \\ = \frac{λ′(P (z) ∧ W)}{P_{j}(z)λ(W)} = \frac{λ′(P (z̃) ∧ W̃)}{P_{j}(z̃)} \end{matrix}

where z̃ = z_{i}^{−1}z ∈ ℂ^{n+1}, W̃ = λ(W)^{−1}W ∈ E. Write, for y ∈ Y,

ρ_{1}(y) = (z(y), W (y)) ,

z̃(y) = z_{i}^{−1}(y)z(y), W̃(y) = λ(W (y))^{−1}W (y). Since ρ is holomorphic and ρ(y_{0}) ∉ Div(ζ_{j}), there is a neighborhood Y_{0} of y_{0} so that P_{μ}(z̃(y)) = φ(y)f_{μ}(y) for y ∈ Y_{0} and 0 ≤ μ ≤ m, where φ, f_{0}, …, f_{m} ∈ O(Y_{0}), φ ≢ 0, and f_{j}(y_{0}) ≠ 0. Thus

g ∘ ρ = g ∘ Q ∘ ρ_{1} = \frac{λ′(φF ∧ W̃)}{φf_{j}} = \frac{λ′(F ∧ W̃)}{f_{j}}

on Y_{0}, where F = (f_{0}, … f_{m}). Since f_{j}(y_{0}) ≠ 0, g ∘ ρ is holomorphic at y_{0}.

We now complete the proof of Lemma 5.1: By Lemma 5.3 and (23), we have

ρ^{∗}dd^{c}Ω̂ ≤ ρ^{∗} [(ω_{ζ}^{ℓ} + ω_{W} ∧ ∑_{j=0}^{ℓ−1} ω_{θ}^{j} ∧ ω_{ζ}^{ℓ−1−j}) ∧ ω_{z}^{n−ℓ+1}] .

By Section 3, ρ^{∗}dd^{c}Ω̂ = dd^{c}ρ^{∗}Ω̂. Hence by Lemma 5.2,

\begin{aligned}dd^{c}m_{P}^{ℓ} & = dd^{c}(ρ_{2∗}ρ^{∗}Ω̂) = ρ_{2∗}ρ^{∗}dd^{c}Ω̂ \\ & ≤ ρ_{2∗}ρ^{∗} [(ω_{ζ}^{ℓ} + ω_{W} ∧ ∑_{0}^{ℓ−1} ω_{θ}^{j} ∧ ω_{ζ}^{ℓ−1−j}) ∧ ω_{z}^{n−ℓ+1}] . \end{aligned}

For any smooth form η on X, we define as before Q^{∗}η = ρ_{1∗}ρ^{∗}η, which is the unique extension with L_{loc}^{1} coefficients of the form Q_{0}^{∗}η, and we have ρ_{2∗}ρ^{∗}η = π_{∗}Q^{∗}η. Since

Q_{0}^{∗}(ω_{ζ}^{ℓ} ∧ ω_{z}^{n−ℓ+1}) = (dd^{c} log |P (z)|^{2})^{ℓ} ∧ ω_{z}^{n−ℓ+1} = 0 ,

we obtain

\begin{aligned}dd^{c}m_{P}^{ℓ} & ≤ ∑_{j=0}^{ℓ−1} π_{∗}Q^{∗} (ω_{W} ∧ ω_{θ}^{j} ∧ ω_{ζ}^{ℓ−1−j} ∧ ω_{z}^{n−ℓ+1}) \\ & = ∑_{j=0}^{ℓ−1} (∫_{ℙ_{z}^{n}×\{W \}\I_{Q}}(dd^{c} log |P (z) ∧ W |^{2})^{j} ∧ (P^{∗}ω_{ζ})^{ℓ−1−j} ∧ ω_{z}^{n−ℓ+1}) ω_{W} . \end{aligned}

By Lemma 4.5 with L : ℂ^{m+1} → Ẽ given by L(ζ) = ζ ∧ W,

∫_{ℙ_{z}^{n}×{W }\I_{Q}}(dd^{c} log |P (z) ∧ W |^{2})^{j} ∧ (P^{∗}ω_{ζ})^{ℓ−1−j} ∧ ω_{z}^{n−ℓ+1} ≤ δ_{ℓ−1}(P) ,

and the desired bound on dd^{c}m_{P}^{ℓ} follows.

We now modify (and simplify) the above argument for the case ℓ = 1: Identify

G(1, m) = ℙ(^{⋀m}ℂ^{m+1}) = ℙ(ℂ^{m+1∗}) = ℙ^{m∗} .

Let X = ℙ_{z}^{n} × ℙ_{W}^{m∗} × ℙ_{ζ}^{m} and consider the current

Ω̂ = log \frac{|ζ|^{2}|W |^{2}}{|(W, ζ)|^{2}} ω_{z}^{n} ∈ D′^{n,n}(X) .

Then

dd^{c}Ω̂ = (ω_{ζ} + ω_{W} − D) ∧ ω_{z}^{n} ,

where D = Div(W, ζ). Then m_{P}^{1} = π_{∗}Q^{∗}Ω̂, where

Q = (z, W, P (z)) : ℙ_{z}^{n} × ℙ_{W}^{m∗} → X .

We conclude as before that dd^{c}m_{P}^{1} = π_{∗}Q^{∗}dd^{c}Ω̂ ≤ ω_{W}. Remark: We could use (11), (12) to define m_{P}^{ℓ} (W) for all W ∈ ℙ(⋀^{m+1−ℓ} ℂ^{m+1}); then Lemma 5.1 remains valid on ℙ(⋀^{m+1−ℓ} ℂ^{m+1}).

6. Description of the exceptional set

In this section we prove Theorem 1.2, giving a description of the exceptional set E in terms of the proximity function as follows. For a rational map P : ℙ^{n} → ℙ^{m}, we write

B_{P}^{ℓ} = {W ∈ 𝔾(ℓ, m) : codim π_{2}^{−1}(W) < ℓ} ,  (24)

where π_{2} is given by (8). The set B_{P}^{ℓ} is a proper algebraic subvariety (possibly empty) of 𝔾(ℓ, m). Recall that P^{∗}Λ_{ℓ}^{W} and P^{∗}[W] are defined by (12) and the Poincaré-Lelong formula (13) holds whenever W ∈ 𝔾(ℓ, m) \ B_{P}^{ℓ}. By the definition of the proximity function at the beginning of Section 5, m_{P}^{ℓ} (W) = +∞ if and only if W ∈ B_{P}^{ℓ}. Now let P = {P_{k}} be a sequence of rational mappings from ℙ^{n} to ℙ^{m} as in Theorem 1.2 and fix a sequence A = {a_{k}} of positive numbers. We define the exceptional set

E_{ℓ}(P, A) = \{W ∈ 𝔾(ℓ, m) : lim sup_{k→∞} \frac{m_{P_{k}}^{ℓ}(W)}{a_{k}} > 0\} .  (25)

(In particular, E_{ℓ}(P, A) contains the set ⋂_{k} ⋃_{j=k}^{∞} B_{P_{j}}^{ℓ}.)

The following two propositions yield Theorem 1.2 with E = E_{ℓ}(P, A).

Proposition 6.1. Let P = {P_{k}} be a sequence of rational mappings from ℙ^{n} to ℙ^{m} and let A = {a_{k}} be a sequence of positive numbers. Let 1 ≤ ℓ ≤ min(n, m). Then for all W ∈ 𝔾(ℓ, m) \ E_{ℓ}(P, A),

\frac{1}{a_{k}} (P_{k}^{∗}[W] − P_{k}^{∗}ω^{ℓ}) → 0

as k → ∞.

Proposition 6.2. Let P = {P_{k}}, A = {a_{k}} be as in Proposition 6.1. If

∑_{k=1}^{∞} \frac{δ_{ℓ−1}(P_{k})}{a_{k}} < +∞ ,

then E_{ℓ}(P, A) is pluripolar in 𝔾(ℓ, m).

Before proving Propositions 6.1 and 6.2, we note a corollary to Theorem 1.2 on the equidistribution of pre-images for subsequences of a given sequence of rational mappings. This corollary uses the following terminology: For a current T ∈ D′^{p,p}(ℙ^{n}) of order 0, we let ∥T ∥ denote the total variation measure of T, which is the regular measure on ℙ^{n} given by

∥T ∥(U) = sup {|(T, φ)| : φ ∈ D^{n−p,n−p}(U), ∥φ∥ ≤ 1}

for U open in ℙ^{n}. Here, ∥φ∥ denotes the comass norm of a compactly supported form φ (see Federer [Fe, 1.8.1, 4.1.7]). The quantity ∥T ∥(ℙ^{n}) is called the mass of T. If T is positive, then it follows from Wirtinger’s inequality (see for example, [Fe, 1.8.2]) that ∥T ∥(ℙ^{n}) = (T, ω^{n−p}). In particular, ∥ω^{p}∥(ℙ^{n}) = 1.

Corollary 6.3. Let P = {P_{k}} be as in Theorem 1.2, let 1 ≤ ℓ ≤ min(n, m), and suppose that

\frac{δ_{ℓ−1}(P_{k})}{δ_{ℓ}(P_{k})} → 0 .

Let M ⊂ D′^{ℓ,ℓ}(ℙ^{n}) be the set of weak cluster points of \{\frac{1}{δ_{ℓ}(P_{k})}P^{k∗}ω^{ℓ}\}. Then M ≠ ∅, and for every current η ∈ M,

i) η is a positive current ii) η has mass 1 iii) there is a subsequence {P′_{k}} of {P_{k}} and a pluripolar set E such that

\frac{1}{δ_{ℓ}(P′_{k})}P′_{k}^{∗}[W] → η

for all W ∈ 𝔾(ℓ, m) \ E.

Proof (assuming Theorem 1.2): Since

∥\frac{1}{δ_{ℓ}(P_{k})}P_{k}^{∗}ω^{ℓ}∥(ℙ^{n}) = \frac{1}{δ_{ℓ}(P_{k})} ∫ P_{k}^{∗}ω^{ℓ} ∧ ω^{n−ℓ} = 1,

it follows that M ≠ ∅. Let

η = lim_{k→∞} \frac{1}{δ_{ℓ}(P′_{k})}P_{k}^{′∗}ω^{ℓ} ∈ M

for some subsequence {P′_{k}} of P. Then (i) is obvious, and (ii) follows from

∥η∥(ℙ^{n}) = (η, ω^{ℓ−1}) = lim_{k→∞} \frac{1}{δ_{ℓ}(P′_{k})}(P_{k}^{′∗}ω^{ℓ}, ω^{n−ℓ}) = 1 .

Finally, choose a subsequence \{P′′_{k}\} of \{P′_{k}\} such that ∑ \frac{δ_{ℓ−1}(P′′_{k})}{δ_{ℓ}(P′′_{k})} < +∞ , and apply

Theorem 1.2 with a_{k} = δ_{ℓ}(P′′_{k}) to obtain (iii). Proof of Proposition 6.1: Let φ ∈ D^{n−ℓ,n−ℓ}(ℙ^{n}) be an arbitrary real form and choose a constant c_{φ} such that −c_{φ}ω^{n−ℓ+1} ≤ dd^{c}φ ≤ c_{φ}ω^{n−ℓ+1}. Then for all W ∈ 𝔾(ℓ, m) \ B_{P_{k}}^{ℓ}, we have by the Poincaré-Lelong formula (13),

|(P_{k}^{∗}[W] − P_{k}^{∗}ω^{ℓ}, φ)| = |(P_{k}^{∗}Λ_{W}^{ℓ} , dd^{c}φ)| ≤ c_{φ}m_{P_{k}}^{ℓ}(W) .  (26)

The conclusion follows from the definition of E_{ℓ}(P, A). Proof of Proposition 6.2: Let P : ℙ^{n}−→ℙ^{m} be a rational map. By (22) we can write

P^{∗}Λ_{W}^{ℓ} ∧ ω_{z}^{n−ℓ+1} = f(z, W)ω_{z}^{n} ,

where i) 0 ≤ f(z, W) < +∞, ii) f ∈ L^{1}(ℙ^{n} × 𝔾(ℓ, m)), iii) f is continuous on ℙ^{n} × 𝔾(ℓ, m) \ I_{Q} . Here Q is the map given by equation (19) in Section 5. We set f(z, W) = 0 for (z, W) ∈ I_{Q} so that f is lower semi-continuous on ℙ^{n} × 𝔾(ℓ, m), and we write

m̂_{P}^{ℓ} (W) = ∫_{ℙ_{z}^{n}} f(z, W)ω_{z}^{n} ,

for W ∈ 𝔾(ℓ, m). Since f is lower semi-continuous, it follows immediately from Fatou’s Lemma that

m̂_{P}^{ℓ} (W_{0}) ≤ lim inf_{W →W_{0}} m̂_{P}^{ℓ} (W)

for W_{0} ∈ 𝔾(ℓ, m), and thus m̂_{P}^{ℓ} is lower semi-continuous on 𝔾(ℓ, m). By the definition of the proximity function at the beginning of Section 5,

m_{P}^{ℓ} (W) = m̂_{P}^{ℓ} (W) , for W ∈ 𝔾(ℓ, m) \ B_{P}^{ℓ} .

A function u on a complex manifold Y (with values in ℝ ∪ −{∞}) is said to be quasiplurisubharmonic if u is locally equal to the sum of a C^{∞} function and a plurisubharmonic function; an element ũ ∈ L_{loc}^{1} (has a representative that) is quasi-plurisubharmonic if and only if dd^{c}ũ is bounded below by a continuous real (1, 1)-form. By Lemma 5.1, −m_{P}^{ℓ} is equal almost everywhere to a (pointwise-defined) quasi-plurisubharmonic function −m̃_{P}^{ℓ} on 𝔾(ℓ, m). Since −m̂_{P}^{ℓ} is upper semi-continuous and equal almost everywhere to −m̃_{P}^{ℓ}, it follows that −m̂_{P}^{ℓ} ≥ −m̃_{P}^{ℓ}. (If an upper semi-continuous function g is equal almost everywhere to a quasi-plurisubharmonic function g̃, then it follows immediately by reducing to the plurisubharmonic case that g ≥ g̃.)

We now let P = \{P_{k}\}, A = \{a_{k}\} be as in Proposition 6.2 and suppose that ∑ \frac{δ_{ℓ−1}(P_{k})}{a_{k}} < +∞. Let

E′_{ℓ}(P, A) = \{W ∈ 𝔾(ℓ, m) : ∑_{k→∞} \frac{m_{P_{k}}^{ℓ}(W)}{a_{k}} = +∞\} ⊃ E_{ℓ}(P, A) .

We shall show that E′_{ℓ}(P, A) is pluripolar; it suffices to show that it is pluripolar in an arbitrary affine open set

G′ =^{def} {W ∈ 𝔾(ℓ, m) : λ(W) ≠ 0}

given by a linear function λ : ⋀^{m+1−ℓ} ℂ^{m+1} → ℂ.

We write

m_{k} = m̃_{P_{k}}^{ℓ} ≥ m̂_{P_{k}}^{ℓ}

and we consider the functions u_{k} : G′ → ℝ ∪ {−∞} given by

u_{k} = v − \frac{m_{k}}{ℓδ_{ℓ−1}(P_{k})} ,  (27)

where

v(W) = log \frac{|W |^{2}}{|λ(W)|^{2}} .  (28)

We assume that ∥λ∥ = 1, so v(W) ≥ 0. By Lemma 5.1, u_{k} is plurisubharmonic on G′. Next, construct the series

u = ∑_{k=1}^{∞} \frac{δ_{ℓ−1}(P_{k})}{a_{k}}u_{k} = Sv − \frac{1}{ℓ} ∑_{k=1}^{∞} \frac{m_{k}}{a_{k}} ,  (29)

where

S = ∑_{k=1}^{∞} \frac{δ_{ℓ−1}(P_{k})}{a_{k}} .

Since

m_{k}(W) ≥ m_{P_{k}}^{ℓ}(W) for W ∈ 𝔾(ℓ, m) \ B_{P_{k}}^{ℓ} ,  (30)

it follows from (29) that

E′_{ℓ}(P, A) ∩ G′ ⊂ {W ∈ G′ : u(W) = −∞} ∪ (⋃_{k} B_{P_{k}}^{ℓ}) .

Since the sets B_{P_{k}}^{ℓ} are pluripolar it suffices to show that u is plurisubharmonic. We can represent u as a limit of the sequence

τ_{k} = Sv − \frac{1}{ℓ} ∑_{j=1}^{k} \frac{m_{j}}{a_{j}} .

Since m_{k} ≥ 0, {τ_{k}} is a decreasing sequence of plurisubharmonic functions on G′, so the limit u is either plurisubharmonic or identically −∞. To see that the latter case is impossible, we shall average u(W) over all W with respect to Haar probability measure σ on 𝔾(ℓ, m). It is well known (e.g., see [Sh, Ch. 2, § 4, Th. 7]) that

∫_{𝔾(ℓ,m)} Λ_{W}^{ℓ} (z)dσ(W) = c_{ℓ,m}ω^{ℓ−1}(z)  (31)

for some constant c_{ℓ,m}. Therefore, by Fubini’s theorem and the pull-back of (31) by P_{k},

∫_{𝔾(ℓ,m)} m_{P_{k}}^{ℓ}(W)dσ(W) = ∫_{𝔾(ℓ,m)} ( ∫_{ℙ^{n}} P_{k}^{∗}Λ_{W}^{ℓ} ∧ ω^{n−ℓ+1}) dσ(W)

= c_{ℓ,m} ∫_{ℙ^{n}} P_{k}^{∗}ω^{ℓ−1} ∧ ω^{n−ℓ+1} = c_{ℓ,m}δ_{ℓ−1}(P_{k}) .

Hence

∫ u(W)dσ(W) ≥ −\frac{1}{ℓ} ∑_{k=1}^{∞} \frac{1}{a_{k}} ∫ m_{k}(W)dσ(W)

= −\frac{1}{ℓ} ∑_{k=1}^{∞} \frac{1}{a_{k}} ∫ m_{P_{k}}^{ℓ}(W)dσ(W)

= −\frac{1}{ℓ} ∑_{k=1}^{∞} \frac{c_{ℓ,m}δ_{ℓ−1}(P_{k})}{a_{k}} = −\frac{c_{ℓ,m}}{ℓ} S > −∞ .

Thus u is plurisubharmonic.

7. Limit measures for iterates of rational maps

In this section we prove Theorem 1.1 and Corollary 1.3. We also show that the iterated pull-backs of certain measures converge in the distribution sense at an exponential rate (Proposition 7.1). We continue to use the notation from Section 6. Proof of Corollary 1.3: Let {P_{k}} be given as in the corollary, and let E be the exceptional pluripolar set of Theorem 1.2 with ℓ = m and a_{k} = δ_{m}(P_{k}). We can assume that E ⊃ ⋃_{k} B_{P_{k}}^{m}, where B_{P_{k}}^{m} is given by (24). Let ν be a probability measure with ν(E) = 0. We pull back the tautological identity

ν = ∫_{ℙ^{n}}[W]dν(W)

to obtain

P_{k}^{∗}ν = ∫_{ℙ^{n}\B_{P_{k}}^{m}} P_{k}^{∗}[W]dν(W) .

Let φ ∈ D^{n−m,n−m}(ℙ^{n}) be arbitrary. We then have

\frac{1}{δ_{m}(P_{k})} (P_{k}^{∗}ν − P_{k}^{∗}ω^{m}, φ) = ∫_{ℙ^{n}\E} f_{k}dν ,  (32)

where

f_{k}(W) = \frac{1}{δ_{m}(P_{k})} (P_{k}^{∗}[W] − P_{k}^{∗}ω^{m}, φ)

for W ∈ ℙ^{m} \ E. By (13), for W ∈ ℙ^{m} \ B_{P_{k}}^{m} we have

(P_{k}^{∗}[W], ω^{n−m}) = (P_{k}^{∗}ω^{m}, ω^{n−m}) = δ_{m}(P_{k}) ,

and thus

|f_{k}(W)| ≤ \frac{1}{δ_{m}(P_{k})} (|(P_{k}^{∗}[W], φ)| + |(P_{k}^{∗}ω^{m}, φ)|) ≤ 2 sup ∥φ∥ .  (33)

By Theorem 1.2, f_{k} → 0 pointwise, and thus by (33) we can let k → ∞ in (32) to conclude that

\frac{1}{δ_{m}(P_{k})} (P_{k}^{∗}ν − P_{k}^{∗}ω^{m}, φ) → 0 .

Proof of Theorem 1.1: Assume that P : ℙ^{n}−→ℙ^{n}, λ = δ_{n}(P) are as in Theorem 1.1. Since δ_{n−1}(P^{k}) ≤ [λ_{n−1}(P) + o(1)]^{k} and δ_{n}(P^{k}) = λ^{k}, by Corollary 1.3 (and the remark following the corollary) it suffices to show that the sequence of measures {λ^{−k}P^{k∗}ω^{n}} converges weakly to a measure μ. Since the measures λ^{−k}P^{k∗}ω^{n} are probability measures, it suffices to show that the sequence converges weakly in D′^{n,n}(ℙ^{n}).

Let

h = \frac{1}{λ}P^{∗}ω^{n} ∈ D′^{n,n}(ℙ^{n}).

By the definition of the topological degree δ_{n}(P), ∫_{ℙ^{n}} h = 1. We claim that if f is a quasiplurisubharmonic function on ℙ^{n}, then

∫_{ℙ^{n}} fh > −∞ .  (34)

(Note that h ≥ 0; since f is bounded above, the claim is equivalent to saying that fh is L^{1}.) To verify the claim, we again consider the commutative diagram (8). We then have

∫ fh = \frac{1}{λ} ∫_{G}(f ∘ π_{1})π_{2}^{∗}ω^{n} > −∞

since f ∘ π_{1} is quasi-plurisubharmonic on G̃ and hence is in L^{1}, verifying (34). Choose a > 0 such that λ_{n−1}(P) < a < λ. By (4), the sum

S =^{def} ∑_{k=1}^{∞} \frac{δ_{n−1}(P^{k})}{a^{k}} < +∞.

Applying Proposition 6.2 with ℓ = n, P = {P^{k}}, and A = {a^{k}}, we conclude that

\frac{m_{P^{k}}^{n}(W)}{a^{k}} → 0  (35)

for points W of ℙ^{n} outside a pluripolar set. Since pluripolar sets have Lebesgue measure zero, (35) is valid for a.a. W ∈ ℙ^{n}. Write

f = ∑_{k=1}^{∞} \frac{m_{P^{k}}^{n}}{a^{k}} .

By the proof of Proposition 6.2, f is equal almost everywhere to the function

f̃ =^{def} nSv − nu ,

where v, u are given by (28),(29); the function u is plurisubharmonic and v is C^{∞} on an (arbitrarily chosen) affine open set in 𝔾(ℓ, m). Hence −f̃ is quasi-plurisubharmonic on 𝔾(ℓ, m). Therefore, by (34),

∫_{ℙ^{n}} fh = ∫_{ℙ^{n}} f̃h < +∞  (36)

Let φ ∈ D^{0,0}(ℙ^{n}) be arbitrary. Since ∫_{ℙ^{n}} h = 1,

\frac{1}{a^{k}} (P^{k∗}ω^{n} − P^{k∗}h, φ) = \frac{1}{a^{k}} ∫ (P^{k∗}ω^{n} − P^{k∗}[W], φ)h(W)
= ∫ \frac{1}{a^{k}} (P^{k∗}Λ_{W}^{n} , dd^{c}φ)h(W) ,  (37)

where the second equality follows from the Poincaré-Lelong formula (13). By (26),

|\frac{1}{a^{k}} (P^{k∗}Λ_{W}^{n} , dd^{c}φ)| ≤ \frac{c_{φ}m_{P^{k}}^{n}(W)}{a^{k}} ≤ c_{φ}f(W) ,  (38)

and hence by (35),

\frac{1}{a^{k}} (P^{k∗}Λ_{W}^{n} , dd^{c}φ) → 0 for h-a.a. W .  (39)

By (36), (38), and (39), we can let k → +∞ in (37) and apply Lebesgue’s dominated convergence theorem to conclude that

\frac{1}{a^{k}} (P^{k∗}ω^{n} − P^{k∗}h, φ) → 0  (40)

as k → +∞.

We note that

\frac{1}{λ^{k+1}} (P^{k+1})^{∗}ω^{n} = \frac{1}{λ^{k}} P^{k∗}h,

as an identity of currents with L_{loc}^{1} coefficients. Hence by (40),

|(\frac{1}{λ^{k+1}} (P^{k+1})^{∗}ω^{n} − \frac{1}{λ^{k}} P^{k∗}ω^{n}, φ)| = \frac{1}{λ^{k}} |(P^{k∗}h − P^{k∗}ω^{n}, φ)| ≤ o ((a/λ)^{k}) . (41)

Therefore the sequence \{(\frac{1}{λ^{k}} P^{k∗}ω^{n}, φ)\} is Cauchy. Since φ is arbitrary, it follows that the sequence \{\frac{1}{λ^{k}} P^{k∗}ω^{n}\} converges weakly in D′^{n,n}(ℙ^{n}).

It follows from the above proof that for each smooth test function φ on ℙ^{n}, the sequence {(λ^{−k}P^{k∗}δ_{W} − μ, φ)} converges to 0 at an exponential rate. Precisely, we have the following result:

Proposition 7.1. Let P, λ > λ_{n−1}(P), μ be as in Theorem 1.1. Then there is a pluripolar set E ⊂ ℙ^{n} such that for all C^{2} functions φ on ℙ^{n}:

i) if W ∈ ℙ^{n} \ E, then

lim sup_{k→∞} |(λ^{−k}P^{k∗}δ_{W} − μ, φ)|^{1/k} ≤ \frac{λ_{n−1}(P)}{λ} ;

ii) if g ∈ L^{∞}(ℙ^{n}), then

lim sup_{k→∞} |(λ^{−k}P^{k∗}(gω^{n}) − c_{g}μ, φ)|^{1/k} ≤ \frac{λ_{n−1}(P)}{λ}

where c_{g} = ∫ gω^{n}.

Proof: By Theorem 1.2, we can find a pluripolar set E such that (35) holds for all a > λ_{n−1}(P) and for all W ∈ ℙ^{n} \ E. Let a > λ_{n−1}(P) be arbitrary. We conclude from (41) that

|(\frac{1}{λ^{k}} P^{k∗}ω^{n} − μ, φ)| ≤ o ((a/λ)^{k}) .  (42)

Part (i) of the proposition follows immediately from Corollary 1.4 and (42). To prove part (ii), we can assume without loss of generality that g ≥ 0 and c_{g} = 1. By repeating the part of the proof of Theorem 1.1 from equation (36) to equation (40) with h replaced by gω^{n}, we conclude that

\frac{1}{a^{k}} (P^{k∗}ω^{n} − P^{k∗}(gω^{n}), φ) → 0 ,  (43)

and (ii) then follows from (42) and (43). Remark: Proposition 7.1 does not hold for continuous test functions φ. For example, let P : ℂ → ℂ be the map P (z) = z^{λ}, λ ≥ 2. Let φ be the continuous function on ℂ ∪ {∞} given by φ(z) = ψ(|z|) where

ψ(r) = (| log | log r|| + 1)^{−1} .

Then μ is Haar measure on the circle and (μ, φ) = 0. Let w ∈ ℂ \ {0} with |w| ≠ 1, and write μ_{w}^{k} = λ^{−k}P^{k∗}δ_{w}. We have

(μ_{w}^{k} − μ, φ) = (μ_{w}^{k}, φ) = ψ (|w|^{1/λ^{k}}) = \frac{1}{k log λ + c}

for k ≫ 0, and thus (i) does not hold for this C^{0} function φ. This example also shows that convergence in the space of measures does not always hold in Corollary 1.4 if a < λ_{ℓ}(P) or in Theorem 1.2.

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[Sk2] H. Skoda. Nouvelle méthode pour l’étude des potentiels associés aux ensembles analytiques. Séminaire P. Lelong (Analyse) 1972 - 1973. Lecture Notes in Math., 410, Springer-Verlag, New York, 1974, 117–141.

[So] M. L. Sodin. Value distribution of sequences of rational mappings. Advances in Soviet Math., 11, Entire and Subharmonic Functions, B. Ya. Levin, ed., 1992.

[Ue] T. Ueda. Fatou sets in complex dynamics on projective spaces. J. Math. Soc. Japan, 46, 1994, 545–554.

Alexander Russakovskii, Theory of Functions Department, Institute for Low Temperature Physics, 310164 Kharkov, Ukraine

Current address: Department of Mathematics, Stanford University, Stanford, CA 94305, USA

  E-mail address: russakov@msri.org

Bernard Shiffman, Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA

  E-mail address: shiffman@math.jhu.edu