Algebraic degrees for iterates of meromorphic self-maps of $\P^k$

We first introduce the class of quasi-algebraically stable meromorphic maps of $\P^k.$ This class is strictly larger than that of algebraically stable meromorphic self-maps of $\P^k.$ Then we prove that all maps in the new class enjoy a recurrent property. In particular, the algebraic degrees for iterates of these maps can be computed and their first dynamical degrees are always algebraic integers.


Introduction
Let f : P k −→ P k be a meromorphic self-map. It can be written f := [F ] := [F 0 : . . . : F k ] in homogeneous coordinates where the F j 's are homogeneous polynomials in the k + 1 variables z 0 , . . . , z k of the same degree d with no nontrivial common factor. The polynomial F will be called a lifting of f in C k+1 . The number d(f ) := d will be called the algebraic degree of f. Moreover f is said to be dominating if it is generically of maximal rank k, in other words, its jacobian determinant does not vanish identically (in any local chart). The indeterminacy locus I(f ) of f is the set of all points of P k where f is not holomorphic, or equivalently the common zero set of k + 1 component polynomials F 0 , . . . , F k . Observe that I(f ) is a subvariety of codimension at least 2. From now on, we always consider dominating meromorphic self-maps f of P k with k ≥ 2. For such a map f, its first dynamical degree λ 1 (f ) is given by (1.1) λ 1 (f ) := lim n→∞ d(f n ) 1 n .
Computing λ 1 (f ) and other dynamical degrees associated to f (see, for example, [5] for a definition of the latter degrees) is a fundamental problem in Complex Dynamics. Indeed, this is a necessary step to determine the topological entropy of f (see [5], [10]). Recall the following definition (see [8,9,11]) Definition 1. A meromorphic self-map f : P k −→ P k is said to be algebraically stable (or AS for short) if there is no hypersurface of P k which would be sent, by some iterate f N , to I(f ).
In other words, f is AS if and only if d(f n ) = d(f ) n , n ∈ N.
For all AS maps f with d(f ) > 1, Sibony proves in [11] that the following limit in the sense of current lim n→∞ (f n ) * ω d(f n ) exists, where ω denotes the Fubiny-Study Kähler form on P k so normalized that P k ω k = 1 (see Section 2 below). This limit is called the Green current associated to f. The Green current contains many important dynamical informations of the corresponding map. We address the reader to the recent works of Sibony [11], Dinh-Sibony [3,4], Fornaess-Sibony [8,9], and Favre [6] for further explanations.
However, the situation becomes much harder in the case of non AS maps. In general, for a non AS map f, (f m ) * • (f n ) * = (f m+n ) * when these operators act on currents. One of the first works in this direction is the article of Bonifant-Fornaess [2] where some special non AS maps are thoroughly studied. In her thesis [1] Bonifant constructs an appropriate Green current for these maps and then writes down the functional equation. Favre and Jonsson [7] have studied the case of polynomial maps in C 2 . These works show that there is a deep connection between the construction of a good invariant current for a non AS meromorphic self-map f and the sequence (d(f n )) ∞ n=1 of algebraic degrees for iterates of f. Moreover, this sequence is, in general, very complicated.
The purpose of this paper is to study the sequence (d(f n )) ∞ n=1 for all f in a new class of meromorphic self-maps of P k : the class of quasi-algebraically stable selfmaps (or QAS for short). This class contains strictly that of all AS self-maps. The QAS self-maps share a recurrent property with the AS ones. Let us explain this more explicitly. For an AS self-map f we may define a sequence of polynomial maps (F n ) ∞ n=0 : C k+1 −→ C k+1 such that F n is a lifting of f n , n ≥ 0, in the following recurrent way: where F 1 , F 0 are arbitrarily fixed liftings of f, f 0 := Id respectively. Following the same pattern, the recurrent law for a QAS self-map f which is not AS may be stated as follows: for all n > n 0 . Here n 0 ≥ 1 is an integer and H 0 is a homogeneous polynomial. The recurrent phenomenon happens when the orbits of the hypersurfaces which are sent to I(f ) by some iterate f N (see Definition 1 above) are, in some sense, not so complicated. That is the main point of our observation.
This paper is organized as follows.
We begin Section 2 by collecting some background and introducing some notation. This preparatory is necessary for us to state the results afterwards.
Section 3 starts with the definition of quasi-algebraically stable meromorphic selfmaps. Then we provide some examples illustrating this definition.
The last section is devoted to the formulation and the proof of the main theorem. Some examples are also analyzed in the light of this theorem. Finally, we conclude the paper with some remarks.
Acknowledgment. The author is supported by The Alexander von Humboldt Foundation and The Max-Planck Institut für Mathematik in Bonn (Germany) during the preparation of this paper. He wishes to express his gratitude to these organisations. He would like to thank N. Sibony and T.-C. Dinh for introducing him to Complex Dynamics and for many help. He is also grateful to C. Favre whose suggestions helped to clarify the exposition. The first version of this article circulated in a form of a preprint in 2002.

Background and notation
Let f be a dominating meromorphic self-map of P k , Γ(f ) the graph of f in P k ×P k and π 1 , π 2 the natural projections of Γ(f ) ⊂ P k × P k onto its factors. Then I(f ) is exactly the set of points z ∈ P k where π 1 does not admit a local inverse. Let A be a (not necessarily irreducible) analytic subset of P k . We define the following analytic sets ). If g is another dominating meromorphic self-map of P k , the graph Γ(g • f ) of the composite map g • f is the closure of the set (z, g(f (z)) ∈ P k × P k : z ∈ I(f ) and f (z) ∈ I(g) .
Consequently, we can define the sequence (f n ) ∞ n=1 of n-iterates of f by the induction formula Γ(f n ) := Γ(f n−1 • f ), n ≥ 2.
In the sequel, codim(A) denotes the codimension of A. Moreover, we recall that a hypersurface is an analytic set of pure codimension 1 in P k . Let Crit(f ) denote the critical set of f (i.e. the hypersurface defined outside I(f ) by the zero set of the jacobian of f in any local coordinates). The following result is very useful.
1. Let f be as above. Then, for every irreducible analytic set A ⊂ P k such that A ⊂ I(f ), f (A) is also an irreducible analytic set.
Proof. Suppose in order to get a contradiction that f (A) = B 1 ∪B 2 , where B 1 , B 2 are analytic sets in P k , distinct from f (A). It follows that (A∩f −1 (B 1 ))∪(A∩f −1 (B 2 )) ⊂ A and the two sets A ∩ f −1 (B 1 ), A ∩ f −1 (B 2 ) are distinct analytic components of A. We therefore get the desired contradiction. This finishes the proof.
We denote by C + 1 (P k ) the set of positive closed currents of bidegree (1, 1) on P k . A current T ∈ C + 1 (P k ) can be written locally as T = dd c u for some plurisubharmonic function u (which is called a local potential of T ). The mass of T is defined by T := P k T ∧ ω k−1 . Fix a point a ∈ P k and local coordinates sending a to the origin in C k . Choose a local plurisubharmonic potential u for T defined around 0 in these coordinates. We can define the Lelong number of u at 0 as follows which is a finite nonnegative real number. We then set ν(T, a) := ν(u, 0), which does not depend on any choice we made.
For a current T ∈ C + 1 (P k ), we use local potentials to define the induced pull-back f * T ∈ C + 1 (P k ). More precisely, for any z ∈ P k \ I(f ), T has a local potential u in a neighborhood of f (z), and we define f * T := dd c (u • f ) in a neighborhood of z. This yields a well-defined, positive closed (1, 1)-current on the set P k \ I(f ). Since codim(I(f )) > 1, we can extend f * T to a current f * T ∈ C + 1 (P k ) by assigning zero mass on the set I(f ) to the coefficients measures of f * T .
Any hypersurface H of P k defines a current of integration [H] ∈ C + 1 (P k ), and For further information on this matter, the reader is invited to consult the works [8] and [11].

Quasi-algebraically stable meromorphic self-maps
In [9] Fornaess and Sibony establish the following definition. The following proposition gives us the structure of a non AS self-map. Proof. First, we give the construction of M and H j , n j , j = 1 . . . , M. To this end observe that every hypersurface H satisfying codim (f (H)) > 1 should be contained in Crit(f ). Therefore, one takes the family F of all degree lowering irreducible components H of Crit(f ) such that codim (f m (H)) > 1 for 1 ≤ m ≤ n, where n is the height of H. Let 1 ≤ n 1 < · · · < n M be all the heights of elements in F (it is easy to see that M is finite). Let H j be the (finite) union of all elements in F with the same height n j . Then properties (i) and (ii) are satisfied.
To prove (iii) let H be a degree lowering irreducible hypersurface with height h. In virtue of Lemma 2.1, let n be the greatest integer such that 0 ≤ n < h and f n (H) is a hypersurface. The choice of n implies that codim (f m (H)) > 1, m = n + 1, . . . , h. Consequently, in virtue of the above construction, we deduce that f n (H) ⊂ H j for some 1 ≤ j ≤ M. This proves (iii).
Since the uniqueness of M and H j , n j , j = 1, . . . , M, is almost obvious, it is therefore left to the reader. This completes the proof. We are now able to define the class of quasi-algebraically stable self-maps.
Definition 3.4. A meromorphic self-map f of P k is said to be quasi-algebraically stable (or QAS for short) if either it is AS or it satisfies the following properties: (i) there is only one primitive degree lowering hypersurface (let H 0 be this hypersurface and let n 0 be its height); It is worthy to remark that Proposition 3.2 allows us to check if a map is QAS. We conclude this section by studying some examples.
Example 3.5. Consider the following meromorphic self-map of P 2 : It can be checked that Therefore, {t = 0} is the unique primitive degree lowering hypersurface and its height is 1. Since [1 : 1 : 1] ∈ {t = 0} and f 2 ({t = 0}) is a hypersurface, f is QAS.
Example 3.6. For all integers d ≥ 2 and m ≥ 1, the following map is given by Bonifant-Fornaess in [2] f ([z : It can be checked that I Example 3.7. Consider the following meromorphic self-map of P 2 : It can be checked that Now it is not difficult to see that there is no hypersurface G which is not contained in H 0 := {(z + w + t) 2 (z 3 + w 3 + t 3 ) = 0} and which satisfies codim(f (G)) > 1. In other words, H 0 is the unique primitive degree lowering hypersurface and its height is 1. Since [1 : 1 : 1] ∈ H 0 , and f 2 ({z + w + t = 0}), f 2 ({z 3 + w 3 + t 3 = 0}) are hypersurfaces, it follows that f is QAS.

The main result
Now we are ready to formulate the main result of this article.
The Main Theorem. Let f be a QAS meromorphic self-map of P k which is not AS. Let H 0 be its unique primitive degree lowering hypersurface and let n 0 be its height. We define a sequence {F n : n ≥ 1} of maps C k+1 −→ C k+1 as follows : . . , f n 0 , f n 0 +1 respectively. Let H 0 be the unique homogeneous polynomial which verifies the equality Next we define F n for all n > n 0 + 1 as follows : Then H 0 = {H 0 (z) = 0}, and for any n ≥ 0, F n is a lifting of f n . Moreover, for any current T ∈ C + 1 (P k ), Prior to the proof of the theorem we need a preparatory result.
In summary, we have shown that for some q < r < p < s, • all the analytic sets f t (H) (r + 1 ≤ t ≤ s − 1) are of codimension strictly greater than 1, and none of them is contained in I(f ); Case 2c: f r (H) ⊂ I(f ), r = q + 1, . . . , p.
First we claim that there is a smallest integer s such that s > q and f s (H) ⊂ I(f ). Otherwise, using the fact that f q (H) is a hypersurface (by the choice of q), one would deduce that f n+t (H) = f n (f t (H)) for all t ≥ q and n ≥ 0. In particular, using (4.4) and the identity G = f p (H) we would have which is a contradiction. Hence, the above claim has been proved.
Using this claim, the assumption of Case 2c and the choice of q, we see that p < s, and all the analytic sets f t (H) (q + 1 ≤ t ≤ s − 1) are of codimension strictly greater than 1, and none of them is contained in I(f ), but f s (H) ⊂ I(f ). This implies that f q (H) is an irreducible component of H 0 , and s = q + n 0 . Since f p−q (f q (H)) = f p (H) ⊂ H 0 (by (4.4)) and 0 < p − q < s − q = n 0 , we obtain a contradiction with Definition 3.4 (ii). Hence, Case 2c cannot happen.
Hence Let q be the greatest integer such that 0 ≤ q ≤ m and f q (H) is a hypersurface. Clearly, q < m because of (4. We will prove (4.3) and the fact that F n is a lifting of f n by induction on n ≥ n 0 +1. For n = n 0 + 1, these assertions are immediate consequences of (4.1)-(4.2). Suppose them true for n − 1, we like to show them for n.
To this end let G be the homogeneous polynomial given by and let G be the hypersurface {G(z) = 0} . We may rewrite (4.6) as for any current T ∈ C + 1 (P k ) of mass 1. In virtue of (4.2) and (4.7), we only need to show that One breaks the proof of this identity into two steps.
Step I: Proof of the inclusion G ⊂ (f n−n 0 −1 ) −1 (H 0 ). Consider an arbitrary irreducible component F of G. Then we deduce from (4.7) that ν (f n−1 ) * (f * T ), z > 0 for any current T := [l(z) = 0], where {l(z) = 0} is a generic complex hyperplane in P k , and for a generic point z ∈ F . Since for any point z outside I(f ), we can choose T so that f * T vanishes in a neighborhood of z, it follows from the latter inequality that (f n−1 )(F ) ⊂ I(f ). Now let m be the greatest integer such that 0 ≤ m < n − 1 and f m (F ) is a hypersurface. Put H := f m (F ). Therefore, f m+1 (F ), . . . , f n−1 (F ) are analytic sets of codimension strictly greater than 1. Since we have shown that f n−1 (F ) ⊂ I(f ), there is a smallest integer p such that m + 1 ≤ p ≤ n − 1 and f p (F ) ⊂ I(f ). Using the hypothesis that f is QAS, one concludes that H := f m (F ) is an irreducible component of H 0 . Moreover, one has p = n 0 + m.
Recall from the previous paragraph that all the analytic sets f t (H) (1 ≤ t ≤ n − 1 − m) are of codimension strictly greater than 1. In addition, we have that Invoking Definition 3.4 (iii) and the choice of p, it follows that p − m = n−1 −m. This, combined with the equality p = n 0 + m, implies that m = n−n 0 −1. In summary, we have shown that f n−n 0 −1 (F ) = H ⊂ H 0 . Since F is an arbitrary component of G, we deduce that G ⊂ (f n−n 0 −1 ) −1 (H 0 ). This completes Step I.
Step II: Proof of identity (4.8). 1 In what follows T is a current in C + 1 (P k ) of mass 1, and d := d(f ). Moreover, we make the following convention (f m ) * [H 0 ] := 0 for all m < 0. Next, we apply the hypothesis of induction (i.e. identity (4.3)) for n−1, . . . , n−n 0 repeatedly by taking into account the identity (f m ) * T = d m , m = 1, . . . , n 0 , (see (2.1)). Consequently, one gets (4.9) On the one hand, applying Lemma 4.1 to the right-hand side of (4.9), we deduce that We combine the latter two equalities with (4.7) and taking into account the result of Step I. Consequently, (4.8) follows. This completes Step II. The proof of the theorem is thereby finished.
Theorem 4.2. Let f be a QAS meromorphic self-map of P k . Then its first dynamical degree λ 1 (f ) is an algebraic integer. Moreover, Proof. If f is AS, then the theorem is trivial since d(f n ) = d(f ) n and λ 1 (f ) = d(f ).
Let σ 1 , . . . , σ s be all the roots of P with multiplicities m 1 , . . . , m s respectively ( s j=1 m j = n 0 + 1). Since P ∈ R[σ], we remark that for every p : 1 ≤ p ≤ s, there is a unique q such that σ p = σ q and m p = m q . Then the Newton formula yields where P j is a complex polynomial of one variable and deg(P j ) ≤ m j −1, j = 1, . . . , s. There are two cases to consider.
Case 1: all roots of P are distinct.
We like to prove the following Claim 1. If σ q = σ p for some 1 ≤ p < q ≤ n 0 + 1, then c q = c p .
Hence, c 1 = 0. Next, we show that σ 1 is a real number. Suppose in order to get a contradiction that this is not the case. Without loss of generality we assume that σ 2 = σ 1 . Then, by Claim 1, c 2 = c 1 . Write σ 1 := |σ 1 |e iφ and c 1 = |c 1 |e iψ for 0 ≤ φ, ψ < 2π, φ > 0 and φ = π. Then formula (4.13) yields that Recall from Claim 2 and (4.15) that |σ 1 | > |σ j | for all j > 2 with c j = 0. Letting n to infinity in the above formula, we see that any accumulation point of the left side is a positive number, on the other hand the sum in the right side tends to zero. Thus any accumulation point of the set {cos (nφ + ψ) : n ∈ N} is also a positive number. We consider two subcases • (a) φ 2π is rational. Since 0 < φ < 2π and φ = π, there is a subsequence (n k ) ∞ k=1 such that cos (n k φ + ψ) = c < 0 for all k. This contradicts our observation above.
2π is irrational. It is clear that the set {cos (nφ + ψ) : n ∈ N} is dense on the interval [−1, 1]. Therefore we obtain a contradiction.
In summary, we have shown that the algebraic integer σ 1 is a real number. Applying Claim 2 we see that |σ 1 | > |σ j | for any j ≥ 2 with c j = 0. Then it follows from (1.1) and (4.13) that σ 1 > 0, λ 1 (f ) = σ 1 and that (4.10) holds. This completes the proof of Case 1. Case 2: the polynomial P has a multiple root.
Since h ≥ 1, it can be checked that P (σ) = P ′ (σ) = 0 if and only if σ = σ n 0 := dn 0 n 0 +1 and h = h 0 := d n 0 +1 n 0 +1 n n 0 0 . Moreover, P ′′ (σ n 0 ) = 0 for h = h 0 . Consequently, we conclude that h = h 0 and the only multiple root of P is σ n 0 , and it is a double root. In the remaining part of the proof we may assume without loss of generality that σ n 0 +1 = σ n 0 .
Since the sum in the right side tends to zero as n → ∞, it follows that λ 1 (f ) = σ n 0 and that (4.10) holds. Hence, the proof of Case 2 follows.
This finishes the proof.
Applications. We apply the Main Theorem to the examples given in Section 3. First consider Example 3.5. Put H(z, w, t) := t and let F : C 3 −→ C 3 be the lifting of f given by the right-hand side of (3.1). In the light of the Main Theorem, a sequence of liftings F n of f n may be defined as follows F 0 := Id, F 1 := F, F n := F • F n−1 H(F n−2 ) , n ≥ 2.
As a consequence, one obtains the equation d(f n ) − 2 d(f n−1 ) + d(f n−2 ) = 0. Therefore, a straightforward computation shows that d(f n ) = n + 1 and λ 1 (f ) = 1. Next consider Example 3.7. Put H(z, w, t) := (z + w + t) 2 (z 3 + w 3 + t 3 ) and let F : C 3 −→ C 3 be the lifting of f given by the right-hand side of (3.2). Using the Main Theorem, a sequence of liftings F n of f n may be defined as follows Concluding remarks. One may widen the class of QAS self-maps by weakening considerably the conditions in Definition 3.4. Of course the recurrent law would be then more complicating. One might also seek to • for any given k, d ≥ 2, find many families of QAS (but non AS) self-maps of P k with the algebraic degree d; • generalize the Main Theorem to meromorphic self-maps in compact Kähler manifolds; • construct an appropriate Green current for every QAS self-map f with λ 1 (f ) > 1. We hope to come back these issues in a future work.