Dynamics of semigroups of entire maps of $\mathbb{C}^k$

The goal of this paper is to study some basic properties of the Fatou and Julia sets for a family of holomorphic endomorphisms of $\mathbb{C}^k,\; k \ge 2$. We are particularly interested in studying these sets for semigroups generated by various classes of holomorphic endomorphisms of $\mathbb{C}^k,\; k \ge 2.$ We prove that if the Julia set of a semigroup $G$ which is generated by endomorphisms of maximal generic rank $k$ in $\mathbb{C}^k$ contains an isolated point, then $G$ must contain an element that is conjugate to an upper triangular automorphism of $\mathbb{C}^k.$ This generalizes a theorem of Fornaess-Sibony. Secondly, we define recurrent domains for semigroups and provide a description of such domains under some conditions.


Introduction
The purpose of this note is to study the Fatou-Julia dichotomy, not for the iterates of a single holomorphic endomorphism of C k , k ≥ 2, but for a family F of such maps. The Fatou set of F will be by definition the largest open set where the family is normal, i.e., given any sequence in F there exists a subsequence which is uniformly convergent or divergent on all compact subsets of the Fatou set, while the Julia set of F will be its complement.
We are particularly interested in studying the dynamics of families that are semigroups generated by various classes of holomorphic endomorphisms of C k , k ≥ 2. For a collection {ψ α } of such maps let G = ψ α denote the semigroup generated by them. The index set to which α belongs is allowed to be uncountably infinite in general. The Fatou set and Julia set of this semigroup G will be henceforth denoted by F (G) and J(G) respectively. The ψ α 's that will be considered in the sequel will belong to one of the following classes: • E k : The set of holomorphic endomorphisms of C k which have maximal generic rank k.
• I k : The set of injective holomorphic endomorphisms of C k .
• V k : The set of volume preserving biholomorphisms of C k .
• P k : The set of proper holomorphic endomorphisms of C k . The main motivation for studying the dynamics of semigroups in higher dimensions comes from the results of Hinkkanen-Martin [3] and Fornaess-Sibony [2]. While [3] considers the dynamics of semigroups generated by rational functions on the Riemann sphere, [2] puts forth several basic results about the dynamics of the iterates of a single holomorphic endomorphism of C k , k ≥ 2. Under such circumstances, it seemed natural to us to study the dynamics of semigroups in higher dimensions.
Section 2 deals with basic properties of F (G) and J(G) when G is generated by elements that belong to E k and P k . The main theorem in Section 3 states that if J(G) contains an isolated point, then G must contain an element that is conjugate to an upper triangular automorphism of C k . Finally we define recurrent domains for semigroups in Section 4 and provide a description of such domains under some conditions. All these results generalize the corresponding statements of Fornaess-Sibony [2] for the iterates of a single holomorphic endomorphism of C k , k ≥ 2.
Acknowledgement: We would like to thank Kaushal Verma for the valuable discussions and comments.
2. Properties of the Fatou set and Julia set for a semigroup G In this section we will prove some basic properties of the Fatou set and the Julia set for semigroups.
Proposition 2.1. Let G be a semigroup generated by elements of E k where k ≥ 2 and for any Proof. Note that φ ∈ G is an open map at any point z ∈ F (G) \ Σ φ . Since for any sequence ψ n ∈ G, the sequence ψ n • φ has a convergent subsequence around a neighbourhood of z (say V z ), ψ n also has a convergent subsequence on the open set φ(V z ) containing φ(z). Now if G is generated by elements of P k or I k then φ is an open map at every point in C k . Then the Fatou set is forward invariant and hence the Julia set is backward invariant in the range of φ.
A family of endomorphisms F in C k is said to be locally uniformly bounded on an open set Ω ⊂ C k if for every point there exists a small enough neighbourhood of the point (say V ⊂ Ω) such that F restricted to V is bounded i.e., for some M > 0 and for every f ∈ F.
. . , φ n , where each φ j ∈ E k and let Ω G be a Fatou component of G such that G is locally uniformly bounded on Ω G . Then for every φ ∈ G the image of Ω G under φ i.e., φ(Ω G ) is contained in Fatou set of G.
for every 1 ≤ i ≤ n and is contained inside a Fatou component say Ω i and G is locally uniformly bounded on each of Ω i for every 1 ≤ i ≤ n i.e., each Ω i is a Runge domain. Now, pick p ∈ Ω G ∩ Σ. Since Σ is a set with empty interior, there exists a sufficiently small disc centered at p say ∆ p such that ∆ p \{p} ⊂ Ω G \Σ. Then φ i (∆ p \{p}) ⊂ Ω i for every 1 ≤ i ≤ n and since, each Ω i is Runge φ i (p) ∈ Ω i i.e., φ i (Ω G ) is contained in the Fatou set for every 1 ≤ i ≤ n. Now for any φ ∈ G there exists a m > 0 such that where 1 ≤ n j ≤ n for every 1 ≤ j ≤ m. Thus applying the above argument repeatedly for each φ n j (Ω j ) where G is locally uniformly bounded onΩ j it follows that φ(Ω G ) is contained in the Fatou set of G.
Proof. Let φ ∈ G and let Σ φ denote the set of points in C k where the Jacobian of φ vanishes. Since Let {ψ n } be a sequence in G and without loss of generality it can be assumed that there exists a subsequence such that ψ n = f n • φ 1 . Now φ 1 (V p ) is a compact subset in Ω 1 and f n has a subsequence which either converges uniformly on φ 1 (V p ) or diverges to infinity. Thus V p is contained in the Fatou set of G which is a contradiction! The next observation is an extension of the fact that if φ ∈ P k , then F (φ) = F (φ n ) for every n > 0 for the case of semigroups.
The index of H in G is the smallest possible number m.
for every ψ ∈ G and for some 1 ≤ j ≤ m − 1. The index of H in G is the smallest possible number m.
Proposition 2.6. Let G be a semigroup generated by proper holomorphic endomorphisms of C k and H be a sub semigroup of G which has a finite (or co-finite) index in G. Then F (G) = F (H) and J(G) = J(H).
Proof. From the definition itself it follows that F (G) ⊂ F (H). To prove the other inclusion, pick So without loss of generality one can assume that there exists a subsequence say φ n k with the property Then H has a finite index in G and hence by Proposition 2.6 F (G) = F (H).
i.e., given two m−tuples p and q, F (G p ) = F (G q ).
Proof. Since G l has a finite index in G for every m−tuple l = (l 1 , l 2 , . . . , l m ), it follows that F (G l ) = F (G) and J(G l ) = J(G).
where a ∈ C such that |a| > 1. Then it is easy to check that Clearly this domain is forward invariant under both f and g. This shows that We claim that So inductively we get that Thus the Julia set of the semigroup G is not forward invariant and clearly from the above observations one can prove that 3. Isolated points in the Julia set of a semigroup G.
If the Julia set J(G) contains an isolated point (say a) then there exists a neighbourhood Ω a of a such that Ω a \ {a} ⊂ F (G) and ψ ∈ G which satisfy the following properties: Then A ⊂ F (G).
Claim: There exists a sequence φ n ∈ G such that φ n diverges to infinity on A.
Suppose not. Then for every sequence {φ n } ∈ G, there exists a subsequence {φ n k } which converges to a finite limit in A. By the maximum modulus principle φ n k B(0, ) < M.
By the Arzelá-Ascoli Theorem it follows that φ n k is equicontinuous on B(0, ), which contradicts that 0 ∈ J(G). By the same reasoning as above there exists a sequence {φ n } ∈ G such that it diverges uniformly to infinity on A but does not diverge uniformly to infinity on B(0, ), since it would again imply that B(0, ) is contained in the Fatou set of G. Thus there exists a sequence of points x n in B(0, ) such that φ n (x n ) is bounded i.e., Now φ n k (p) converges on A, then φ n k on A converges to a finite limit, and hence on A by the maximum modulus principle. This is a contradiction! Since φ n | ∂B(0, ) → ∞ for large n φ n ∂B(0, ) |q|. Thus for a sufficiently large R > 0 and n (B(0, )), then B(0, |q| + R) φ n (B(0, )) since B(0, ) ⊂ B(0, |q| + R) for large R > 0. Then there exists y n ∈ ∂B(0, ) such that |φ n (y n )| < |q| + R, which is not possible. Hence B(0, ) ⊂⊂ φ n (B(0, )) for sufficiently large n. Relabel this φ n as ψ and consider the neighbourhood Ω 0 as B(0, ). Since 0 ∈ B(0, ) ⊂ ψ(B(0, )), there exists α ∈ B(0, ) such that ψ(α) = 0. From Proposition 2.1 it follows that α = 0.
If the Julia set J(G) contains an isolated point, say a then there exists an element ψ ∈ G such that ψ is conjugate to an upper triangular automorphism.
Proof. Without loss of generality we can assume that a = 0. Now by Proposition 3.1 it follows that there exists a sufficiently small ball B(0, ) around 0 and an element ψ ∈ G such that B(0, ) ⊂⊂ ψ(B(0, )). Since ψ is injective map in C k , ψ(B(0, )) is biholomorphic to B(0, ) and hence we can consider the inverse i.e., Note that ψ(B(0, )) is bounded and B(0, ) is compactly contained in ψ(B(0, )). Therefore there exists an α > 1 such that the map defined by Proof. That the map φ has a fixed point p in U follows from Lemma 4.3 in [2]. Without loss of generality we can assume p = 0. Consider ψ(z) = φ(p + z) − p and Ω = {z − p : z ∈ U }. Then ψ is the required map with the properties Ω ⊂⊂ ψ(Ω) and 0 is a fixed point for ψ. Suppose ψ is not invertible at 0, i.e., A = Dψ(0) has a zero eigenvalue. Let λ i , 1 ≤ i ≤ k be the eigenvalues of A. Therefore there exist an α such that 0 < α < 1 and 1 < m ≤ k such that 0 = |λ i | < α for 1 ≤ i ≤ m and |λ i | > α for m < i ≤ k. Choose δ > 0 such that for z ∈ B(0, δ) and m < i ≤ k. Let Ψ be a Lipschitz map in C k such that and Ψ ≡ ψ on B(0, δ). Now W Ψ s := {z ∈ C k : |α n Ψ n (z)| is bounded } can be realized as a graph of a continuous function (See [6] s ∩ Ω is an infinite non-empty set containing 0. Also ψ n k |Ω → ψ 0 for some sequence n k and ψ 0 is holomorphic on the component (say F 0 ) of F (ψ) containing Ω. Let where ψ 0,i is the i−th coordinate function of ψ 0 . If W ψ 1 ∩ ∂Ω = ∅ then W ψ 1 ∩ Ω and hence W ψ s ∩ Ω will have to be finite which is not true. Thus there exists a positive integer n 0 such that ψ n 0 (∂Ω)∩Ω = ∅ but by assumption it follows that Ω ⊂⊂ ψ n (Ω) for all n ≥ 1, i.e., ψ n (∂Ω)∩Ω = ∅ for all n > 0. This proves that A has no zero eigenvalues.
Note that this observation also reveals that W ψ 1 ∩ Ω has to be a finite set, and since the backward orbit of 0 under ψ is finite. Now we can state and prove Theorem 3.2 for semigroups generated by the elements of E k .
If the Julia set J(G) contains an isolated point (say a) then there exists a ψ ∈ G such that ψ is conjugate to an upper triangular automorphism.
Proof. Assume a = 0. Then as before by Proposition 3.1 there exists a map ψ ∈ G and a domain Ω such that Ω ⊂⊂ ψ(Ω).
If 0 is in the Julia set of ψ then 0 is an isolated point in J(ψ) and by applying Theorem 4.2 in [2] ψ is conjugate to an upper triangular automorphism.
Applying Proposition 3.1, we have that ψ −1 (0) = 0 and there exists ψ ∈ G such that where Ω is a sufficiently small ball at 0 and R > 0 is a sufficiently large number. Now let ω is the component of ψ −1 (B(0, R)) in Ω containing the origin. Also from Proposition 3.4 it follows that 0 is a regular point of ψ, which implies that ψ is a biholomorphism on ω. Define Ψ β on ψ(ω) as Ψ β (z) = βψ −1 (z) and note that Ψ β is a self map of B(0, R) for some β > 1 with a fixed point at 0. Then the eigenvalues of D C Ψ β (0) are in the closed unit disc, i.e., ∩ ω. Now choose inductively a n ∈ ψ −1 (a n−1 ) \ {a n−1 } for n ≥ 2 and define S n = O − ψ (a n ) ∩ ω. Then S n ⊂ S n−1 and for every n ≥ 2. Note that a n / ∈ S n , otherwise there is a positive integer k n > 0 such that ψ kn (a n ) = a n i.e., a n is a periodic point of ψ, and ψ kn+m (a n ) = p for any m > n. Since O − ψ (p) ∩ ω is finite it follows that S n has to be empty for large n. This implies that there exists a n 0 ≥ 1 such that ψ −1 (a n 0 ) = a n 0 and a n 0 ∈ ω. But by Proposition 3.4 ψ is invertible at its fixed points which means that a n 0 is a regular value of ψ and #{ψ −1 (a n 0 )} = m ≥ 2 which is a contradiction! Hence the claim. Now by similar arguments as in the case of proper maps it follows that ψ is a biholomorphism from ω to B(0, R) and p is a repelling fixed point of ψ and hence lies in J(ψ) ⊂ J(G). Since ω ∩ J(G) = {0}, we have p = 0 which is an isolated point in the Julia set of ψ and hence ψ is conjugate to an upper triangular automorphism.

Recurrent and Wandering Fatou components of a semigroup G.
As discussed in Section 1 we will be studying the properties of recurrent and wandering Fatou components of semigroup generated by entire maps of maximal generic rank on C k . The wandering and the recurrent Fatou components for a semigroup G are defined as: where Σ φ is the set where the Jacobian of φ vanishes. A Fatou component is wandering if the set φ Ω : φ ∈ G contains infinitely many elements. Proof. Take any sequence {φ j } ⊂ G. Let there exist a subsequence {φ jm } and points {p jm } ⊂ K with K compact in Ω such that φ jm (p jm ) → p 0 ∈ Ω.
Without loss of generality we assume p jm → p 0 ∈ K. It is easy to show that φ jm (p 0 ) → p 0 ∈ Ω using the fact that any sequence of G is normal on the Fatou set of G.
If Ω is a recurrent Fatou component of G, then G is locally bounded on Ω. Moreover Ω is pseudoconvex and Runge.
Proof. Assume G is not locally bounded on Ω. Then there exists a compact set K ⊂ Ω and {g r } ⊆ G such that |g r (z r )| > r with z r ∈ K for every r ≥ 1. Clearly this can not be the case since Ω is a recurrent Fatou component, so we can always get a subsequence {g r k } from the sequence {g r } ∈ G such that it converges to a holomorphic function uniformly on compact set in Ω and in particular on K. From the proof of Proposition 2.2, it follows that local boundedness of G on Ω implies that Ω is polynomially convex. So Ω is pseudoconvex.
Assume that Ω is a recurrent Fatou component for G. Let φ ∈ G be such that φ(∂Ω) ⊂ ∂Ω. Then one of the following is true (i) There is an attracting fixed point p for φ.
(ii) There exists a closed connected complex submanifold M φ ⊂ Ω of dimension r with 1 ≤ r ≤ (k − 1) and an integer l ≥ 1 such that φ l is an automorphism of M φ . (iii) There exists a subsequence {φ n i } converging uniformly on compact sets of Ω to the identity.
Since G is locally bounded on Ω, without loss of generality we take ψ = lim φ l k i . Then we have φ r • ψ = Id on M . Pick any a ∈ M φ . Because of the fact that M φ is invariant under φ i , we have ψ(m) ∈ M φ . But φ r (M φ ) ∩ M φ = ∅ for 1 ≤ r ≤ (i − 1). This shows that r = 0.
Proposition 4.7. Let G = φ 1 , φ 2 , . . . , φ m where each φ i ∈ V k for every 1 ≤ i ≤ m and let Ω be an invariant Fatou component of G. Then either Ω is recurrent or there exists a sequence {φ n } ⊂ G converging to infinity. Proof.
If Ω is not recurrent, then there exists a sequence {φ n } ⊂ G such that {φ n } → ∂Ω ∪ {∞} uniformly on compact sets of Ω. Assume {φ n k } converges to a holomorphic function f on Ω .
This implies that f (Ω) ⊂ ∂Ω contradicting the assumption that each φ n k is volume preserving. Hence {φ n k } diverges to infinity uniformly on compact subsets of Ω. Proof.
Since Ω is wandering, one can choose a sequence {φ n } ⊂ G so that (4.1) φn Ω ∩ φm Ω = ∅ for n = m. If this sequence {φ n } does not diverge to infinity uniformly on compact subsets, some subsequence {φ n k } will converge to a holomorphic function h on Ω. By abuse of notation, we denote {φ n k } still by {φ n }. Fix z 0 ∈ Ω. Then for any given , there exists δ such that (4.2) |φ n 0 (z) − φ n (z)| < for all n ≥ n 0 and for all z ∈ B(z 0 , δ). From (4.2) it follows that vol(∪ n≥no φ n (B(z 0 , δ))) is finite. On the other hand, since each φ n is volume preserving and (4.1) holds, we get Vol n≥no φ n B(z 0 , δ) = +∞.
Hence we have proved the existence of a sequence in G converging to infinity.