Open Access

Dynamics of semigroups of entire maps of \(\mathbb C^k\)

Complex Analysis and its Synergies20162:2

https://doi.org/10.1186/s40627-016-0006-x

Received: 26 February 2016

Accepted: 11 August 2016

Published: 5 September 2016

Abstract

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. Second, we define recurrent domains for semigroups and provide a description of such domains under some conditions.

Keywords

Semigroups Entire maps in \(\mathbb C^k\) Fatou–Julia dichotomy

Mathematics Subject Classification

Primary 32H02 Secondary 32H50

1 Background

The purpose of this note is to study the Fatou–Julia dichotomy, not for the iterates of a single holomorphic endomorphism of \(\mathbb C^k, \; k \ge 2\), but for a family \(\mathcal {F}\) of such maps. The Fatou set of \(\mathcal {F}\) will be by definition the largest open set where the family is normal, i.e., given any sequence in \(\mathcal {F}\) there exists a subsequence which is uniformly convergent or divergent on all compact subsets of the Fatou set, while the Julia set of \(\mathcal {F}\) 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 \(\mathbb C^k,\; k \ge 2.\) For a collection \(\{\psi _{\alpha }\}\) of such maps let
$$\begin{aligned} G=\langle \psi _{\alpha }\rangle \end{aligned}$$
denote the semigroup generated by them. The index set to which \(\alpha\) 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. Also for a holomorphic endomorphism \(\phi\) of \(\mathbb C^k ,\) \(F(\phi )\) and \(J(\phi )\), will denote the Fatou set and Julia set for the family of iterations of \(\phi .\) The \(\psi _{\alpha }\) that will be considered in the sequel will belong to one of the following classes:
  • \(\mathcal {E}_k{:}\;\) The set of holomorphic endomorphisms of \(\mathbb C^k\) which have maximal generic rank k.

  • \(\mathcal {I}_k{:}\;\) The set of injective holomorphic endomorphisms of \(\mathbb C^k.\)

  • \(\mathcal {V}_k{:}\;\) The set of volume preserving biholomorphisms of \(\mathbb C^k.\)

  • \(\mathcal {P}_k{:}\;\) The set of proper holomorphic endomorphisms of \(\mathbb C^k.\)

The main motivation for studying the dynamics of semigroups in higher dimensions comes from the results of Hinkkanen–Martin [7] and Fornaess–Sibony [5]. While [7] considers the dynamics of semigroups generated by rational functions on the Riemann sphere, [5] puts forth several basic results about the dynamics of the iterates of a single holomorphic endomorphism of \(\mathbb C^k,\; k \ge 2.\) 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 \(\mathcal {E}_k\) and \(\mathcal {P}_k.\) The main theorem in Sect. 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 \(\mathbb C^k\) and in Sect. 4, we discuss a few interesting examples of Julia set of a semigroup. Finally, we define recurrent domains for semigroups in Sect. 5 and provide a classification of such domains under some conditions which are generalizations of the corresponding statements of Fornaess–Sibony [5] for the iterates of a single holomorphic endomorphism of \(\mathbb C^k,\; k\ge 2.\) The classification for recurrent Fatou components for the iterates of holomorphic endomorphisms of \(\mathbb P^2\) and \(\mathbb P^k\) is studied in [4] and [3], respectively. In [4], Fornaess–Sibony also gave a classification of recurrent Fatou components for iterations of Hénon maps inside \(K^+\), which was initially considered by Bedford–Smillie in [1]. A classification for non-recurrent, non-wandering Fatou components of \(\mathbb P^2\) is given in [11], whereas a classification of invariant Fatou components for nearly dissipative Hénon maps is studied in [9].

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 \(\mathcal {E}_k\) where \(k \ge 2\) and for any \(\phi \in G\) define
$$\begin{aligned} \Sigma _{\phi }=\{z \in \mathbb C^k{:}\; \det {\phi (z)}=0\}. \end{aligned}$$
Then, for every \(\phi \in G\)
  1. (i)

    \(\phi (F(G)\ \Sigma _{\phi })\subset F(G).\)

     
  2. (ii)

    \(J(G)\cap \phi (\mathbb C^k)\subset \phi (J(G)),\) if G is generated by elements of \(\mathcal {P}_k\) or \(\mathcal {I}_k.\)

     

Proof

Note that \(\phi \in G\) is an open map at any point \(z \in F(G)\ \Sigma _{\phi }.\) Since for any sequence \(\psi _n \in G\), the sequence \(\psi _n \circ \phi\) has a convergent subsequence around a neighbourhood of z (say \(V_z\)), \(\psi _n\) also has a convergent subsequence on the open set \(\phi (V_z)\) containing \(\phi (z).\)

Now if G is generated by elements of \(\mathcal {P}_k\) or \(\mathcal {I}_k,\) then \(\phi\) is an open map at every point in \(\mathbb C^k.\) Then, the Fatou set is forward invariant and hence the Julia set is backward invariant in the range of \(\phi .\) \(\square\)

A family of endomorphisms \(\mathcal {F}\) in \(\mathbb C^k\) is said to be locally uniformly bounded on an open set \(\Omega \subset \mathbb C^k\) if for every point there exists a small enough neighbourhood of the point (say \(V\subset \Omega\)) such that \(\mathcal {F}\) restricted to V is bounded, i.e.,
$$\begin{aligned} \Vert f\Vert _{V}=\sup _{V}|f(z)|<M \end{aligned}$$
for some \(M>0\) and for every \(f \in \mathcal {F}.\)

Proposition 2.2

Let \(G=\langle \phi _1,\phi _2,\ldots ,\phi _n\rangle ,\) where each \(\phi _j \in \mathcal {E}_k\) and let \(\Omega _G\) be a Fatou component of G such that G is locally uniformly bounded on \(\Omega _G.\) Then for every \(\phi \in G\) the image of \(\Omega _G\) under \(\phi ,\) i.e., \(\phi (\Omega _G)\) is contained in Fatou set of G.

Proof

Let \(K \subset \subset \Omega _G\), i.e., K is a relatively compact subset of \(\Omega _G,\) then

Claim \(\Omega _G\) is a Runge domain, i.e., \(\hat{K} \subset \Omega _G\) where
$$\begin{aligned} \hat{K}{:}\; =\{z \in \mathbb C^k{:}\; |P(z)| \le \sup _{K}|P| ~\text{ for } \text{ every } \text{ polynomial }~ P\}. \end{aligned}$$
Let \(K_{\delta }=\{z \in \mathbb C^k{:}\; ~\text{ dist }(z,K)\le \delta \}\). Choose \(\delta >0\) such that \(K_{\delta } \subset \subset \Omega _G.\) Now note that \(\hat{K_\delta }\subset \subset \mathbb C^k\) , \(\hat{K_{\delta }}\supset \hat{K}\) and G is uniformly bounded on \(K_{\delta }.\) Pick \(\phi \in G.\) Then, there exists a polynomial endomorphism \(P_{\phi }\) of \(\mathbb C^k\) such that
$$\begin{aligned} |\phi (z)-P_{\phi }(z)| \le \epsilon \quad ~\text{ for } \text{ every }~z \in \hat{K_{\delta }},\\ \text{ i.e., } |P_{\phi }(z)|-\epsilon \le |\phi (z)| \le |P_{\phi }(z)|+\epsilon . \end{aligned}$$
Hence
$$\begin{aligned} |\phi (z)|&\le |P_{\phi }(z)|+\epsilon \le \sup _{K_\delta }|P_{\phi }(z)|+\epsilon \\&\le \sup _{K_\delta }|{\phi }(z)|+2 \epsilon \le M+2\epsilon \end{aligned}$$
for every \(z \in \hat{K_{\delta }}\) and some constant \(M > 0.\) So G is uniformly bounded on \(\hat{K_{\delta }}\) and \(\hat{K} \subset \Omega _G.\)
Let
$$\begin{aligned} \Sigma _i=\{z \in \mathbb C^k{:}\; \det {\phi _i(z)}=0\} \end{aligned}$$
for every \(1 \le i \le n\) and
$$\begin{aligned} \Sigma =\bigcup _{i=1}^n \Sigma _i. \end{aligned}$$
Thus \(\phi _i\) for every i, where \(1 \le i \le n\) is an open map in \(\Omega _G\ \Sigma\). Hence \(\phi _i(\Omega _G\ \Sigma )\) is contained inside a Fatou component say \(\Omega _i\) and G is locally uniformly bounded on each of \(\Omega _i\) for every \(1 \le i \le n,\) i.e., each \(\Omega _i\) is a Runge domain.
Now pick \(p \in \Omega _G \cap \Sigma .\) Since \(\Sigma\) is a set with empty interior, there exists a sufficiently small disc centred at p say \(\Delta _p\) such that \(\overline{\Delta }_p\ \{p\} \subset \Omega _G \ \Sigma .\) Then, \(\phi _i(\overline{\Delta }_p\ \{p\}) \subset \Omega _i\) for every \(1\le i \le n\) and since each \(\Omega _i\) is Runge \(\phi _i(p) \in \Omega _i,\) i.e., \(\phi _i(\Omega _G)\) is contained in the Fatou set for every \(1 \le i \le n.\) Now for any \(\phi \in G\) there exists a \(m > 0\) such that
$$\begin{aligned} \phi =\phi _{n_1}\circ \phi _{n_2}\circ \cdots \circ \phi _{n_m} \end{aligned}$$
where \(1 \le n_j \le n\) for every \(1 \le j \le m.\) Thus, applying the above argument repeatedly for each \(\phi _{n_j}(\tilde{\Omega }_j)\) where G is locally uniformly bounded on \(\tilde{\Omega }_j\) it follows that \(\phi (\Omega _G)\) is contained in the Fatou set of G. \(\square\)

Proposition 2.3

If \(G=\langle \phi _1,\phi _2,\ldots ,\phi _n\rangle\) where each \(\phi _i \in \mathcal {E}_k\) for every \(1 \le i \le n\) and let \(\Omega _G\) be a Fatou component of G. Then for any \(\phi \in G\) there exists a Fatou component of G, say \(\Omega _{\phi }\) such that \(\phi (\Omega _G) \subset \bar{\Omega }_{\phi }\) and
$$\begin{aligned} \partial \Omega _G\subset \bigcup _{i=1}^n \phi _i^{-1}(\partial \Omega _{\phi _i}). \end{aligned}$$

Proof

Let \(\phi \in G\) and let \(\Sigma _{\phi }\) denote the set of points in \(\mathbb C^k\) where the Jacobian of \(\phi\) vanishes. Since \(\Omega _G \ \Sigma _{\phi }\) is connected it follows that \(\phi (\Omega _G \ \Sigma _{\phi }) \subset \Omega _{\phi }\) where \(\Omega _{\phi }\) is a Fatou component of G and by continuity \(\phi (\Omega _G) \subset \bar{\Omega }_{\phi }.\)

Pick \(p \in \partial \Omega _G\) such that \(p \notin \partial \Omega _{\phi _i}\) for every \(1 \le i \le n.\) Since \(\phi _i(\Omega _G) \subset \bar{\Omega }_{\phi _i}\), \(\phi _i(p) \in \Omega _{\phi _i}\) for every \(1 \le i \le n.\) So there exists \(V_{\phi _i}\) an open neighbourhood of \(\phi _i(p)\) in \(\Omega _{\phi _i}\) for every i. Let \(V_p\) be a neighbourhood of p such that
$$\begin{aligned} \bar{V}_p\subset \bigcap _{i=1}^n \phi _i^{-1}(V_{\phi _i}). \end{aligned}$$
Let \(\{\psi _n\}\) be a sequence in G and without loss of generality it can be assumed that there exists a subsequence such that \(\psi _n=f_n \circ \phi _1.\) Now \(\phi _1(\bar{V}_p)\) is a compact subset in \(\Omega _1\) and \(f_n\) has a subsequence which either converges uniformly on \(\phi _1(\bar{V}_p)\) or diverges to infinity. Thus, \(V_p\) is contained in the Fatou set of G which is a contradiction! \(\square\)

The next observation is an extension of the fact that if \(\phi \in \mathcal {P}_k\), then \(F(\phi )=F(\phi ^n)\) for every \(n > 0\) for the case of semigroups.

Definition 2.4

Let G be a semigroup generated by endomorphisms of \(\mathbb C^k.\) A sub-semigroup H of G is said to have finite index if there is a finite collection of elements say \(\psi _1,\psi _2,\ldots ,\psi _ {m-1} \in G\) such that
$$\begin{aligned} G =\Big (\bigcup _{i=1}^{m-1} \psi _i \circ H\Big ) \cup H. \end{aligned}$$
The index of H in G is the smallest possible number m.

Definition 2.5

A sub-semigroup H of a semigroup G of endomorphisms of \(\mathbb C^k\) is of co-finite index if there is a finite collection of elements say \(\psi _1,\psi _2,\ldots ,\psi _{m-1} \in G\) such that either
$$\begin{aligned} \psi \circ \psi _{j} \in H \; \text{ or }\; \psi \in H \end{aligned}$$
for every \(\psi \in G\) and for some \(1 \le j \le m-1.\) 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 \(\mathbb C^k\) 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 \(F(G) \subset F(H).\) To prove the other inclusion, pick any sequence \(\{\phi _n \}\in G\). Since H has a finite index in G, there exists \(\psi _i\), \(1 \le i \le m-1\) such that
$$\begin{aligned} G =\left (\bigcup _{i=1}^{m-1} \psi _i \circ H\right) \cup H. \end{aligned}$$
So without loss of generality one can assume that there exists a subsequence say \(\phi _{n_k}\) with the property
$$\begin{aligned} \phi _{n_k}=\psi _1 \circ h_{n_k} \end{aligned}$$
where \(\{h_{n_k}\}\) is a sequence in H. Now on F(H), the sequence \(\{h_{n_k}\}\) has a convergent subsequence. Hence, so do \(\{\phi _{n_k}\}\) and \(\{\phi _n\}\) as \(\psi _1\) is a proper map in \(\mathbb C^k.\) \(\square\)
Let G be a semigroup
$$\begin{aligned} G=\langle \phi _1,\phi _2,\ldots ,\phi _m\rangle \end{aligned}$$
where \(\phi _i \in \mathcal {P}_k\), for every \(1 \le i \le m\) and each of these \(\phi _i\) commute with each other, i.e., \(\phi _i \circ \phi _j=\phi _j \circ \phi _i\) for \(i \ne j.\) Let H be a sub-semigroup of G defined as
$$\begin{aligned} H=\langle \phi _1^{l_1},\phi _2^{l_2},\ldots ,\phi _m^{l_m}\rangle \end{aligned}$$
where \(l_i>0\) for every \(1 \le i \le m.\) Then, H has a finite index in G and hence by Proposition 2.6 \(F(G)=F(H).\)

Corollary 2.7

Let \(\phi _i\) be elements in \(\mathcal {P}_k\) for \(1 \le i \le m\), \(l=(l_1,l_2,\ldots ,l_m)\) an \(m\) -tuple of positive integers and \(G_l=\langle \phi _1^{l_1},\phi _2^{l_2},\ldots , \phi _m^{l_m}\rangle .\) Then, \(F(G_l)\) and \(J(G_l)\) are independent of the \(m\) -tuple l, if \(\phi _i \circ \phi _j=\phi _j\circ \phi _i\) for every \(1 \le i,j\le m,\) 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 \(l=(l_1,l_2,\ldots ,l_m)\), it follows that \(F(G_l)=F(G)\) and \(J(G_l)=J(G).\) \(\square\)

Example 2.8

Let \(G=\langle f,g \rangle\) where \(f(z_1,z_2)=(z_1^2,z_2^2)\) and \(g(z_1,z_2)=(z_1^2/a,z_2^2)\) for \(a \in \mathbb C\) such that \(|a|>1.\) Then, it is easy to check that
$$\begin{aligned} J(f)=\big \{|z_1|=1\big \} \times \big \{|z_2| \le 1 \big \} \cup \big \{|z_1|\le 1\big \}\times \big \{|z_2| =1\big \} \end{aligned}$$
and
$$\begin{aligned} J(g)=\big \{|z_1|=|a|\big \} \times \big \{|z_2| \le 1\big \} \cup \big \{|z_1|\le |a|\big \}\times \big \{|z_2| =1\big \}. \end{aligned}$$
Now consider the bidisc \(\{|z_1 |< 1, |z_2 | < 1\}.\) Clearly, this domain is forward invariant under both f and g. This shows that \(\{|z_1 |< 1, |z_2 | < 1\} \subset F(G)\). Similarly observe that
$$\begin{aligned} \{|z_2 |> 1\} \cup \{|z_1 | > |a|\} \subset F(G). \end{aligned}$$
We claim that
$$\begin{aligned} \big \{ 1 \le |z_1| \le |a|\big \} \times \big \{ |z_2| \le 1 \big \} \subset J(G). \end{aligned}$$
Note that \(\{|z_1|=|a|,|z_2| \le 1\}\) is contained inside J(G) and since J(G) is backward invariant it follows that
$$\begin{aligned} \{|z_1|=|a|^{1/2},|z_2| \le 1\} \subset f^{-1}(\{|z_1|=|a|,|z_2| \le 1\}) \subset J(G). \end{aligned}$$
So inductively we get that
$$\begin{aligned} \{|z_1|=|a|^t,|z_2| \le 1\} \subset J(G) \end{aligned}$$
for any \(t=k2^{-n}\) where \(1 \le k \le 2^n\) and \(n \ge 1.\) As \(\{k2^{-n}{:}\; 1 \le k \le 2^n,\; n \ge 1\}\) is dense in [0, 1], it follows that \(\{ 1 \le |z_1| \le |a|\} \times \{ |z_2| \le 1 \} \subset J(G).\) Thus, the Julia set of the semigroup G is not forward invariant and clearly from the above observations one can prove that
$$\begin{aligned} J(G)=\big \{ |z_1| \le 1 \big \} \times \big \{ |z_2| = 1 \big \}\cup \big \{ 1 \le |z_1| \le |a|\big \} \times \big \{ |z_2| \le 1 \big \}. \end{aligned}$$

Example 2.9

Let \(T_0 (z) = 1,\; T_1 (z) = z\) and \(T_{n+1}(z) = 2z T_n (z)-T_{n-1}(z)\) for \(n \ge 1\) and \(G =\langle f_0 , f_1 , f_2 , \ldots \rangle\), with \(f_i (z_1 , z_2 ) = (T_i (z_1 ), z_2^2 )\) for \(i \ge 0.\) Consider
$$\begin{aligned} G_1 =\langle T_0 (z_1 ), T_1 (z_1 ), T_2 (z_1 ), ... \rangle ,\; G_2 =\left\langle z_2^2 \right\rangle . \end{aligned}$$
Since any sequence in \(G_1\) is uniformly unbounded on the complement of \([-1,1],\) it follows that
$$\begin{aligned} J(G) = [-1, 1] \times \{|z_2 | \le 1\}. \end{aligned}$$
Also, as \(J(G_1)\subset \mathbb C\) is completely invariant so is J(G).

3 Isolated points in the Julia set of a semigroup G

Proposition 3.1

Let \(G=\langle \phi _1, \phi _2, \ldots \rangle\) where each \(\phi _i \in \mathcal E_k\) . If the Julia set J (G) contains an isolated point (say a), then there exists a neighbourhood \(\Omega _a\) of a such that \(\Omega _a\ \{a\} \subset F(G)\) and \(\psi \in G\) which satisfies \(\Omega _a \subset \subset \psi (\Omega _a).\) In particular, if G is a semigroup generated by proper maps, then \(\psi ^{-1}(a)=a\).

Proof

Assume \(a=0\) is an isolated point in the Julia set J(G). Then there exists a sufficiently small ball \(B(0,\epsilon )\) around 0 such that \(B(0,\epsilon )\ \{0\}\) is contained F(G). Let
$$\begin{aligned} A{:}\; =\{z{:}\; \epsilon /2\le |z|\le \epsilon \}. \end{aligned}$$
Then \(A \subset F(G).\)

Claim There exists a sequence \(\phi _n \in G\) such that \(\phi _n\) diverges to infinity on A.

Suppose not. Then for every sequence \(\{\phi _n\} \in G\), there exists a subsequence \(\{\phi _{n_k}\}\) which converges to a finite limit in A. By the maximum modulus principle
$$\begin{aligned} \Vert {\phi _{n_k}}\Vert _{B(0,\epsilon )}<M. \end{aligned}$$
By the Arzelá–Ascoli theorem, it follows that \(\phi _{n_k}\) is equicontinuous on \(B(0,\epsilon )\), which contradicts that \(0 \in J(G).\)
By the same reasoning as above there exists a sequence \(\{\phi _n\} \in G\) such that it diverges uniformly to infinity on A but does not diverge uniformly to infinity on \(B(0,\epsilon )\), since it would again imply that \(B(0,\epsilon )\) is contained in the Fatou set of G. Thus, there exists a sequence of points \(x_n\) in \(B(0,\epsilon )\) such that \(\phi _n(x_n)\) is bounded, i.e.,
$$\begin{aligned} |\phi _n(x_n)|< M \end{aligned}$$
for some large \(M>0.\) So we can choose a subsequence of this \(\{\phi _n\}\) and relabel it as \(\{\phi _n\}\) again such that it satisfies the following condition:
$$\begin{aligned} \phi _n(x_n) \rightarrow q \quad ~\text{ and } \quad~ x_n \rightarrow p \end{aligned}$$
where \(p \in \overline{B(0,\epsilon )}.\)

Claim \(p=0\).

Suppose not. Then \(\phi _n(p)\) is bounded. Let \(\widetilde{A}=\{z{:}\; \min (|p|, \epsilon /2)\le |z| \le \epsilon \}.\) Then \(\widetilde{A} \supseteq A.\) Now \(\phi _{n_k}(p)\) converges on \(\widetilde{A}\), then \(\phi _{n_k}\) on \(\widetilde{A}\) converges to a finite limit, and hence on A by the maximum modulus principle. This is a contradiction!

Since \({\phi _{n}|}_{\partial B(0,\epsilon )} \rightarrow \infty\) for large n
$$\begin{aligned} \Vert {\phi _n} \Vert _{\partial B(0,\epsilon )} \gg |q|. \end{aligned}$$
Thus for a sufficiently large \(R>0\) and n
$$\begin{aligned} B(0,|q|+R)\cap \phi _n(B(0,\epsilon ))\ne \emptyset . \end{aligned}$$
Now, if \(B(0,\epsilon ) \nsubseteq \phi _n(B(0,\epsilon ))\), then \(B(0,|q|+R)\nsubseteq \phi _n(B(0,\epsilon ))\) since \(B(0,\epsilon ) \subset B(0,|q|+R)\) for large \(R>0.\) Then there exists \(y_n \in \partial B(0,\epsilon )\) such that \(|\phi _n(y_n)| <|q|+R\), which is not possible. Hence \(B(0,\epsilon ) \subset \subset \phi _n(B(0,\epsilon ))\) for sufficiently large n. Relabel this \(\phi _n\) as \(\psi\) and consider the neighbourhood \(\Omega _0\) as \(B(0,\epsilon ).\)

Since \(0 \in B(0,\epsilon )\subset \psi (B(0,\epsilon ))\), there exists \(\alpha \in B(0,\epsilon )\) such that \(\psi (\alpha )=0.\) From Proposition 2.1 it follows that \(\alpha =0.\) \(\square\)

Theorem 3.2

Let \(G=\langle \phi _1, \phi _2, \ldots \rangle\) where each \(\phi _i \in \mathcal I_k.\) If the Julia set J(G) contains an isolated point, say a then there exists an element \(\psi \in G\) such that \(\psi\) 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 \(B(0,\epsilon )\) around 0 and an element \(\psi \in G\) such that \(B(0,\epsilon ) \subset \subset \psi (B(0,\epsilon )).\) Since \(\psi\) is injective map in \(\mathbb C^k\), \(\psi (B(0,\epsilon ))\) is biholomorphic to \(B(0,\epsilon )\) and hence we can consider the inverse, i.e.,
$$\begin{aligned} \psi ^{-1}{:}\; \psi (B(0,\epsilon )) \rightarrow B(0,\epsilon ). \end{aligned}$$
Note that \(\psi (B(0,\epsilon ))\) is bounded and \(B(0,\epsilon )\) is compactly contained in \(\psi (B(0,\epsilon )).\) Therefore, there exists an \(\alpha >1\) such that the map defined by
$$\begin{aligned} \psi _{\alpha }=\alpha \psi ^{-1}(z) \end{aligned}$$
is a self-map of the bounded domain \(\psi (B(0,\epsilon ))\) with a fixed point at 0. Then by the Carathéodory–Cartan–Kaup–Wu Theorem (see Theorem 11.3.1 in [8]), it follows that all the eigenvalues of \(\psi _\alpha\) are contained in the unit disc. Hence 0 is a repelling fixed point for \(\psi\) and also is an isolated point in the Julia set of \(\psi .\)
Since \(B(0,\epsilon )\ \{0\} \in J(G)\), \(B(0,\epsilon )\ \{0\}\) is also contained in the Fatou set of \(\psi\) and using the same argument as in the Proposition 3.1, there exists a subsequence (say \(n_k\)) such that
$$\begin{aligned} \Vert \psi ^{n_k}\Vert _{\partial B(0,\epsilon )} \rightarrow \infty \end{aligned}$$
uniformly. Thus for any given \(R>0,\) there exists \(k_0\) large enough such that \(B(0,R) \subset \psi ^{n_{k_0}}(B(0,\epsilon )).\) Hence \(\psi\) is an automorphism of \(\mathbb C^k\) and the basin of attraction of \(\psi ^{-1}\) at 0 is all of \(\mathbb C^k.\) Now by the result of Rosay–Rudin ([10]) \(\psi\) is conjugate to an upper triangular map. \(\square\)

Remark 3.3

The proof here shows that there exists a sequence \(\phi _n \in G\) such that each \(\phi _n\) is conjugate to an upper triangular map.

Recall that a domain \(\omega\) is holomorphically homotopic to a point in a domain \(\Omega\) if there exists a continuous map \(h{:}\; [0,1]\times \bar{\omega }\rightarrow \Omega\) with \(h(1,z)=z\) and \(h(0,z)=p\) where \(p \in \omega\) and \(h(t,\cdot)\) is holomorphic in \(\omega\) for every \(t \in [0,1].\)

Proposition 3.4

Let \(\phi\) be a non-constant endomorphism of \(\mathbb C^k\) such that on a bounded domain \(U \subset F(\phi )\) , the map \(\phi\) is proper onto its image, \(U \subset \subset \phi (U)\) and U is holomorphically homotopic to a point in \(\phi (U)\) then
  1. (i)

    \(\phi\) has a fixed point, say p in U.

     
  2. (ii)

    \(\phi\) is invertible at its fixed points.

     
  3. (iii)
    The backward orbit of \(\phi\) at the fixed point in U is finite, i.e., \(O^- (p) \cap U\) is finite where
    $$\begin{aligned} O^-_{\phi } (p)=\{ z \in \mathbb C^k{:}\; \phi ^n(z)=p, n \ge 1\}. \end{aligned}$$
     

Proof

That the map \(\phi\) has a fixed point p in U follows from Lemma 4.3 in [5].

Without loss of generality we can assume \(p=0\). Consider \(\psi (z)=\phi (p+z)-p\) and \(\Omega =\{z-p{:}\; z \in U \}.\) Then, \(\psi\) is the required map with the properties \(\Omega \subset \subset \psi (\Omega )\) and 0 is a fixed point for \(\psi .\)

Suppose \(\psi\) is not invertible at 0,  i.e., \(A=D\psi (0)\) has a zero eigenvalue. Let \(\lambda _i\), \(1 \le i \le k\) be the eigenvalues of A. Therefore, there exist an \(\alpha\) such that \(0< \alpha < 1\) and \(1 <m \le k\) such that \(0=|\lambda _i|< \alpha\) for \(1 \le i \le m\) and \(|\lambda _i|> \alpha\) for \(m< i \le k.\) Choose \(\delta >0\) such that
$$\begin{aligned} 0< \Vert D_{\mathbb C}\psi (z)-A\Vert < \epsilon _0=\min \Big \{\alpha ,\big ||\lambda _i|-\alpha \big |\Big \} \end{aligned}$$
for \(z \in B(0, \delta )\) and \(m< i \le k.\) Let \(\Psi\) be a Lipschitz map in \(\mathbb C^k\) such that
$$\begin{aligned} Lip(\Psi )=\Vert A\Vert +\epsilon _0 \end{aligned}$$
and
$$\begin{aligned} \Psi \equiv \psi \;\; \text{ on } \; \; B(0,\delta ). \end{aligned}$$
Now
$$\begin{aligned} W_s^{\Psi }{:}\; =\{z \in \mathbb C^k{:}\;|\alpha ^n \Psi ^n(z)|\;\text{ is } \text{ bounded }\} \end{aligned}$$
can be realized as a graph of a continuous function (see [12]) \(G_{\Psi }{:}\; \mathbb C^m \rightarrow \mathbb C^{k-m}\) such that \(G_{\Psi }(0)=0.\) Since
$$\begin{aligned} {W_s^{\Psi }}={W_s^{\psi }} \; \quad \text {on} \; B(0,\delta /2) \end{aligned}$$
\(W_s^{\psi } \cap \Omega\) is an infinite non-empty set containing 0. Also \({\psi ^{n_k}}_{|\bar{\Omega }} \rightarrow \psi _0\) for some sequence \(n_k\) and \(\psi _0\) is holomorphic on the component (say \(F_0\)) of \(F(\psi )\) containing \(\Omega\). Let
$$\begin{aligned} W_1^{\psi }=\{z \in F_0{:}\;\psi ^{n_k}(z)\rightarrow 0 \;\text{ as }\; k \rightarrow \infty \}. \end{aligned}$$
Then \(W_s^{\psi } \cap F_0 \subset W_1^{\psi }\) and
$$\begin{aligned} W_1^{\psi }=\bigcap _{i=1}^k {\psi _{0,i}}^{-1}(0) \end{aligned}$$
where \({\psi _{0,i}}\) is the ith coordinate function of \(\psi _0.\) If \(W_1^{\psi } \cap \partial \Omega =\emptyset\) then \(W_1^{\psi } \cap \Omega\) and hence \(W_s^{\psi } \cap \Omega\) will have to be finite which is not true. Thus, there exists a positive integer \(n_0\) such that \(\psi ^{n_0}(\partial \Omega ) \cap \Omega \ne \emptyset\) but by assumption it follows that \(\Omega \subset \subset \psi ^n(\Omega )\) for all \(n \ge 1,\) i.e., \(\psi ^{n}(\partial \Omega ) \cap \Omega =\emptyset\) for all \(n > 0.\) This proves that A has no zero eigenvalues.
Note that this observation also reveals that \(W_1^{\psi } \cap \Omega\) has to be a finite set, and since
$$\begin{aligned} O^-_{\psi }(0) \subset W_1^{\psi } \end{aligned}$$
the backward orbit of 0 under \(\psi\) is finite. \(\square\)

Now we can state and prove Theorem 3.2 for semigroups generated by the elements of \(\mathcal {E}_k.\)

Theorem 3.5

Let \(G=\langle \phi _1, \phi _2, \ldots \rangle\) where each \(\phi _i \in \mathcal {E}_k.\) If the Julia set J(G) contains an isolated point (say a) then there exists a \(\psi \in G\) such that \(\psi\) is conjugate to an upper triangular automorphism.

Proof

Assume \(a=0.\) Then, as before by Proposition 3.1 there exists a map \(\psi \in G\) and a domain \(\Omega\) such that \(\Omega \subset \subset \psi (\Omega ).\)

If 0 is in the Julia set of \(\psi ,\) then 0 is an isolated point in \(J(\psi )\) and by applying Theorem 4.2 in [5], it follows that \(\psi\) is conjugate to an upper triangular automorphism.

Suppose \(\Omega \subset F(\psi ).\) By Proposition 3.4, \(\psi\) has a fixed point in \(\Omega ,\) i.e., \(\{\psi ^n\}\) has a convergent subsequence in \(\bar{\Omega }\).

Case 1 Suppose that \(G=\langle \phi _1,\phi _2, \ldots \rangle\) where each \(\phi _i \in \mathcal P_k.\)

Applying Proposition 3.1, we have that \(\psi ^{-1}(0)=0\) and there exists \(\psi \in G\) such that
$$\begin{aligned} \Omega \subset \subset B(0,R) \subset \subset \psi (\Omega ) \end{aligned}$$
(3.1)
where \(\Omega\) is a sufficiently small ball at 0 and \(R>0\) is a sufficiently large number. Now, let \(\omega\) is the component of \(\psi ^{-1}(B(0,R))\) in \(\Omega\) containing the origin. Also from Proposition 3.4 it follows that 0 is a regular point of \(\psi\), which implies that \(\psi\) is a biholomorphism on \(\omega .\) Define \(\Psi _\beta\) on \(\psi (\omega )\) as
$$\begin{aligned} \Psi _\beta (z)=\beta \psi ^{-1}(z) \end{aligned}$$
and note that \(\Psi _\beta\) is a self-map of B(0, R) for some \(\beta >1\) with a fixed point at 0. Then, the eigenvalues of \(D_{\mathbb C}{\Psi _{\beta }}(0)\) are in the closed unit disc, i.e.,
$$\begin{aligned} \beta |\lambda _i^{-1}| \le 1 \end{aligned}$$
where \(\lambda _i\) are eigenvalues of A. Hence 0 is a repelling fixed point for the map \(\psi\) and \(0 \notin F(\psi ).\) Since 0 is an isolated point in the Julia set of \(\psi\), by Theorem 4.2 in [5] \(\psi\) is conjugate to an upper triangular automorphism of \(\mathbb C^k.\)

Case 2 Suppose that \(G=\langle \phi _1,\phi _2, \ldots \rangle\) where each \(\phi _i \in \mathcal E_k.\)

As before by Proposition 3.1 there exists \(\psi \in G\) such that
$$\begin{aligned}\Omega \subset B(0,R) \subset \psi (\Omega )\end{aligned}$$
and let \(\omega\) be a component of \(\psi ^{-1} (B(0,R)) \subset \Omega .\) Then, \(\omega\) satisfies all the condition of Proposition 3.4 and hence there exists a fixed point p of \(\psi\) in \(\omega\) and \(O^-_{\psi }(p)\cap \omega\) is finite.

Claim \(\psi ^{-1}(p) \cap \omega =p\)

Suppose not, i.e.,
$$\begin{aligned} \#\{\psi ^{-1}(p)\}=\text {the cardinality of}\; \psi ^{-1}(p) =m \end{aligned}$$
and \(m \ge 2.\) Let \(a_1 \in \psi ^{-1}(p)\ \{p\}\) in \(\omega\) and define
$$\begin{aligned} S_1=O^-_{\psi }(a_1) \cap \omega . \end{aligned}$$
Then \(S_1 \subset O^-_{\psi }(p)\cap \omega .\) Now choose inductively \(a_n \in \psi ^{-1}(a_{n-1}) \ \{a_{n-1}\}\) for \(n \ge 2\) and define
$$\begin{aligned} S_n=O^-_{\psi }(a_n) \cap \omega . \end{aligned}$$
Then
$$\begin{aligned} S_n \subset S_{n-1} \; \; \text{ and } \; \; \bigcup _{i=1}^n S_i\subset O^-_{\psi }(p)\cap \omega \end{aligned}$$
for every \(n \ge 2.\) Note that \(a_n \notin S_n,\) otherwise there is a positive integer \(k_n >0\) such that \(\psi ^{k_n}(a_n)=a_n,\) i.e., \(a_n\) is a periodic point of \(\psi\), and
$$\begin{aligned} \psi ^{k_n+m}(a_n)=p \end{aligned}$$
for any \(m>n\). Since \(O^-_{\psi }(p)\cap \omega\) is finite it follows that \(S_n\) has to be empty for large n. This implies that there exists a \(n_0 \ge 1\) such that \(\psi ^{-1}(a_{n_0})=a_{n_0}\) and \(a_{n_0} \in \omega .\) But by Proposition 3.4 \(\psi\) is invertible at its fixed points which means that \(a_{n_0}\) is a regular value of \(\psi\) and
$$\begin{aligned} \#\{\psi ^{-1}(a_{n_0})\}=m \ge 2 \end{aligned}$$
which is a contradiction! Hence the claim.

Now by similar arguments as in the case of proper maps it follows that \(\psi\) is a biholomorphism from \(\omega\) to B(0, R) and p is a repelling fixed point of \(\psi\) and hence lies in \(J(\psi ) \subset J(G).\) Since \(\omega \cap J(G)=\{0\}\), we have \(p=0\) which is an isolated point in the Julia set of \(\psi\) and hence \(\psi\) is conjugate to an upper triangular automorphism. \(\square\)

4 Examples of semigroups and their Julia sets

Example 4.1

Consider the following lower triangular maps in \(\mathbb {C}^2\):
$$\begin{aligned} F_1(z,w)=(\lambda z, tw+p(z)), \ F_2(z,w)=(\mu z, sw+q(z)) \end{aligned}$$
where p and q are polynomials of degree d fixing the origin and \(|\lambda |, |\mu |,|s|, |t |>\theta >1\). Let \(G=\langle F_1,F_2\rangle\).
Note that for any sequence \(\{f_n\}\subseteq G\) and \((z,w)\ne 0\), \(|f_n(z,w)|\rightarrow \infty\) as \(n\rightarrow \infty\). It also can be checked that
$$\begin{aligned} \{(z,w)\in \mathbb {C}^2{:}\; z\ne 0\}\subseteq F(G) \quad \text { and } \quad 0\in J(G). \end{aligned}$$
Claim \(J(G)=\{0\}\).
If not, then there exists a point in J(G) apart from the origin and it must be of the form \((0,w_0)\) with \(w_0\ne 0\). Therefore, there exists a sequence \(\{(z_n,w_n)\}\) converging to \((0,w_0)\), \(\{f_n\}\subseteq G\) and \(M\ge 1\) such that
$$\begin{aligned} |f_n(z_n,w_n)|\le M, \text { i.e., } (z_n,w_n)\in f_n^{-1}(B(0;M)) \end{aligned}$$
(4.1)
for all \(n\ge 1\).
Let \(\tilde{G}=\langle F_1^{-1}, F_2^{-1} \rangle\) be the semigroup generated by \(F_1^{-1}\) and \(F_2^{-1}.\) Then \(\tilde{G}\) can be realized as:
$$\begin{aligned} \tilde{G}= \bigcup _{k=1}^{\infty } G_k \end{aligned}$$
where \(G_k \subseteq \tilde{G}\) is of the following form:
$$\begin{aligned} G_k=\{h_k \circ h_{k-1} \circ \cdots \circ h_1 {:}\; h_i=F_1^{-1} \text { or }F_2^{-1} \text { for every } 1 \le i \le k \}. \end{aligned}$$
Without loss generality we assume that \(f_n^{-1}\in G_n\) for all \(n\ge 1\).
Now note that \(F_1^{-1}\) and \(F_2^{-1}\) are lower triangular polynomial maps of the form
$$\begin{aligned} F_1^{-1}(z,w)=(\lambda ^{-1} z, t^{-1} w+\tilde{p}(z)), \;\; F_2^{-1}(z,w)=(\mu ^{-1} z, s^{-1} w+ \tilde{q}(z)) \end{aligned}$$
where \(\tilde{p}\) and \(\tilde{q}\) are polynomials of degree d preserving the origin. Let
$$\begin{aligned} \tilde{p}(z)=\sum _{i=1}^d C_iz^i \quad \text { and } \quad \tilde{q}(z)=\sum _{i=1}^d D_i z^i. \end{aligned}$$
Then choose C such that
$$\begin{aligned} C> \max _{1 \le i \le d}\{|C_i|,|D_i|\}. \end{aligned}$$
Induction statement: For every \((z,w) \in B((0,0);M)\) and for each \(h \in G_k\), \(k \ge 1\)
$$\begin{aligned} |\pi _1 \circ h(z,w)| \le \theta ^{-k}M \quad \text { and } \quad |\pi _2 \circ h(z,w)| \le \theta ^{-k}M + kC \theta ^{-(k-1)} M^d d . \end{aligned}$$
(4.2)
Clearly when \(k=1\) and \((z,w) \in B((0,0);M)\),
$$\begin{aligned} |\tilde{p}(z)|\le \sum _{i=1}^d|C_i| M^i< C M^d d. \end{aligned}$$
Similarly \(|\tilde{q}(z)|<C M^d d.\) Thus for \(h \in G_1\) as \(|\lambda |^{-1},|\mu |^{-1},|s|^{-1},|t|^{-1}< \theta ^{-1}<1\)
$$\begin{aligned} |\pi _1 \circ h(z,w)| \le \theta ^{-1}M \quad \text { and } \quad |\pi _2 \circ h(z,w)| \le \theta ^{-1}M + C M^d d. \end{aligned}$$
Hence the induction statement is true for \(k=1.\) Now assuming it to be true for some k we will show that it is true for \(k+1.\)
Let \(h \in G_{k+1}\) then \(h=F_1^{-1} \circ \tilde{h}\) or \(h = F_2^{-1}\circ \tilde{h}\) where \(\tilde{h} \in G_k.\) So we have
$$\begin{aligned} |\pi _1 \circ \tilde{h}(z,w)| \le \theta ^{-k}M \quad \text { and } \quad |\pi _2 \circ \tilde{h}(z,w)| \le \theta ^{-k}M + kC \theta ^{-(k-1)} M^d d. \end{aligned}$$
Assume that \(h=F_1^{-1}\circ \tilde{h}\) then
$$\begin{aligned} \pi _1 \circ h(z,w)&= \lambda ^{-1}\big (\pi _1 \circ \tilde{h}(z,w) \big )\nonumber \\ \pi _2 \circ h(z,w)&= t^{-1}\big (\pi _2 \circ \tilde{h}(z,w)\big )+ \tilde{p}\circ \pi _1 \circ \tilde{h}(z,w). \end{aligned}$$
(4.3)
Then clearly from the above observation if \((z,w) \in B((0,0);M)\) then
$$\begin{aligned} |\pi _1 \circ h(z,w)| \le \theta ^{-k-1}M. \end{aligned}$$
Since \(\theta ^{-1} <1\) and \(M>1\)
$$\begin{aligned} |\tilde{p}\circ \pi _1 \circ \tilde{h}(z,w)| \le \sum _{i=1}^d |C_i| (\theta ^{-k}M)^i \le C \theta ^{-k} M^d d. \end{aligned}$$
Now substituting this estimate on equation (4.3) we have
$$\begin{aligned} |\pi _2 \circ h(z,w)|&\le |\theta ^{-1}\big (\pi _2 \circ \tilde{h}(z,w)\big )|+ |\tilde{p}\circ \pi _1 \circ \tilde{h}(z,w)|\\&\le \theta ^{-k-1}M+kC\theta ^{-k} M^d d+C \theta ^{-k} M^d d \\&\le \theta ^{-k-1}M+(k+1)C\theta ^{-k} M^d d . \end{aligned}$$
Similarly if \(h=F_2^{-1} \circ \tilde{h}.\) Hence, the induction statement is true.
Now since \(f_k^{-1} \in G_k\), it follows from the induction statement (4.2) that for every \((z,w) \in B(0;M)\)
$$\begin{aligned} |\pi _1 \circ f_k^{-1}(z,w)| \le \theta ^{-k}M \text \quad { and } \quad |\pi _2 \circ f_k^{-1}(z,w)| \le \theta ^{-k}M + kC \theta ^{-(k-1)} M^d d . \end{aligned}$$
This implies that \((z_k,w_k)\rightarrow 0\) as \(k\rightarrow \infty\). Contradiction! Hence the claim follows.

Remark 4.2

Let \(G=\langle F_1, F_2,\cdots F_n\rangle\) for some \(n\ge 1\) where each \(F_i\) is a lower triangular polynomial map in \(\mathbb C^k\), \(k \ge 2\) having a repelling fixed point at the origin. Then using a similar set of arguments as above, it can be proved that \(J(G)=\{0\}\).

Remark 4.3

A large class of elementary polynomial automorphisms in the Friedland–Milnor classification ([6]) comprises of lower triangular polynomial automorphisms fixing the origin. Thus for a semigroup G which is finitely generated by such elementary maps, we get \(J(G)=\{0\}\).

Example 4.4

Let \(f_c\) denote the automorphism of \(\mathbb C^2\) of the form
$$\begin{aligned} f_c(z,w)=\big (z e^{ch(zw)}, we^{-ch(zw)}\big ) \end{aligned}$$
where \(c \in \mathbb C\) and h be a non-constant entire function on \(\mathbb {C}\). The Jacobian of \(f_c\) for every \(c\in \mathbb C\) is constant, i.e., \(Jf_c \equiv 1\) on \(\mathbb C^2.\) Consider the semigroup G:
$$\begin{aligned} G=\langle f_c{:}\; 1<c<\infty \rangle . \end{aligned}$$
Observe that
$$\begin{aligned} f_{c_2} \circ f_{c_1}(z,w)=\big (z e^{(c_1+c_2) h(zw)}, w e^{-(c_1+c_2) h(zw)}\big )=f_{c_1+c_2}(z,w). \end{aligned}$$
Hence, corresponding to any element \(f \in G\), there exists \(c_f>1\) such that
$$\begin{aligned} f(z,w)=\big (z e^{c_f h(zw)},we^{-c_f h(zw)}\big ). \end{aligned}$$
Since \(Jf \equiv 1\) for every \(f \in G\), no point is a repelling fixed point for any element of G. The proof of Theorem 3.2 shows that if the Julia set J(G) has an isolated point it should be a repelling fixed point for some element of G which is clearly not the case here. Thus, the Julia set J(G) should be perfect.
Claim If \(\text {Re }h(0)<0\), then the Julia set J(G) is exactly the following perfect set:
$$\begin{aligned} \{(z,w) \in \mathbb C^2{:}\; \text {Re }h(zw)=0 \} \cup \{(z,w) \in \mathbb C^2{:}\; w=0\}. \end{aligned}$$
Consider \(\{f_n\} \subset G\). Then each \(f_n\) can be thought of as
$$\begin{aligned} f_n(z,w)=\big (z e^{c_n h(zw)},we^{-c_n h(zw)}\big ). \end{aligned}$$
If there exists a subsequence \(c_{n_k}\rightarrow c \in \mathbb R^+\) then \(f_{n_k} \rightarrow f_c\) on compact subsets, otherwise \(c_n \rightarrow \infty\) as \(n\rightarrow \infty\).
Case 1 If \(\text {Re }h(zw)=0\) then \(\{f_{n}\}\) does not diverge to infinity as,
$$\begin{aligned} \Vert f_{n}(z,w) \Vert =\Vert (z,w)\Vert . \end{aligned}$$
But \(\pi _1(f_{n}(z,w))\) or \(\pi _2(f_{n}(z,w))\) diverges to infinity uniformly on a small enough neighbourhood of such a point, depending on whether \(\text {Re }h(zw)>0\) or \(\text {Re }h(zw)<0,\) respectively.
Case 2 If \(w=0\), then
$$\begin{aligned} \Vert f_{n}(z,0)\Vert =|z|e^{c_{n}\alpha }, \end{aligned}$$
i.e., \(\Vert f_{n}(z,0)\Vert \rightarrow 0\) as \(n \rightarrow \infty\) since \(\alpha =\text {Re }h(0)<0\). Now for every sufficiently small neighbourhood \(B_z\) around any (z, 0) there exists \((z,w')\in B_z\) such that \(w'\ne 0\) and \(\text {Re }h(zw')<0\). Therefore, \(\pi _2(f_{n}(z,w'))\) diverges to infinity as \(n\rightarrow \infty\).

Hence the claim follows.

Recall Examples 2.8 and 2.9. In each case G is a semigroup generated by maps of maximal generic rank in \(\mathbb C^2.\) So by Theorem 3.5 they should be perfect since none of the elements in the semigroup is conjugated to an upper triangular automorphism of \(\mathbb C^2\), which is exactly the case.

Example 4.5

Let \(f_1\) and \(f_2\) be the following maps in \(\mathbb C^2\) of maximal generic rank:
$$\begin{aligned} F_1(z,w)=(z,w^2), \ F_2(z,w)=(z,wz). \end{aligned}$$
Let G be the semigroup generated by them, i.e.,
$$\begin{aligned} G=\langle F_1,F_2\rangle . \end{aligned}$$
Then by Theorem 3.5, the Julia set for the semigroup G, i.e., J(G) should be perfect.
Claim The Julia set J(G) for the semigroup G is: (this is illustrated in Fig. 1)
$$\begin{aligned} \tilde{J}=\{(z,w) \in \mathbb C^2{:}\; 0 \le |z|\le 1, |w|\ge 1\} \cup \{(z,w) \in \mathbb C^2{:}\; 0 \le |w|\le 1, |z| \ge 1\}. \end{aligned}$$
Fig. 1

The Julia set J(G)

Suppose \(\{f_n\}\) is a sequence from G. Then, there exist sequences of positive integers \(\{a_n\}\) and \(\{b_n\}\) such that
$$\begin{aligned} f_n(z,w)=(z,z^{a_n}w^{b_n}) \end{aligned}$$
and at least one of them is unbounded. Let \(\Delta ^2(0;1)\) denote the unit polydisc. Then we prove that the Fatou set
$$\begin{aligned} F(G) \supseteq \Delta ^2(0;1) \cup \{ (z,w) \in \mathbb C^2{:}\; |z|>1 \text{ and } |w|>1 \}. \end{aligned}$$
Case 1 In \(\Delta ^2(0;1)\)
$$\begin{aligned} \Vert {f_n}_{|\Delta ^2(0;1)}\Vert _{\infty }< 1. \end{aligned}$$
Thus G is locally uniformly bounded on \(\Delta ^2(0;1)\), and hence there exists a subsequence which converges uniformly on its compact subsets. So \(\Delta ^2(0;1) \subset F(G).\)

Case 2 Suppose \(|z|>1\) and \(|w|>1.\)

Then without loss of generality one can assume that there exists a subsequence \(\{a_{n_k}\}\) of \(\{a_n\}\) which diverges to \(\infty\) as \(k \rightarrow \infty .\) Thus
$$\begin{aligned} \Vert f_{n_k}(z,w)\Vert _{\infty } > |z|^{a_{n_k}} \rightarrow \infty , \end{aligned}$$
i.e., \(\{f_{n_k}\}\) diverges to \(\infty\) uniformly in a small enough neighbourhood of such a (zw). hence \((z,w) \in F(G).\)
Consider the set A defined as:
$$\begin{aligned} A=\big \{(z,w) \in \mathbb C^2{:}\; |z^{\frac{p}{2^q}} w|=1 \text { for some integers } p\ge 1 \text { and } q \ge 0\big \}. \end{aligned}$$
Since the set
$$\begin{aligned} \Big \{ \frac{p}{2^q}{:}\; p, q \text { integers with } p \ge 1 \text { and } q \ge 0 \Big \} \end{aligned}$$
is dense in the positive real axis, the set A is dense in \(\tilde{J}\) and \(\bar{A}=\tilde{J}.\) Also the Julia set of a semigroup is closed, so to prove the claim it is enough to prove that A is contained in J(G).
Now pick \((z_0,w_0) \in A\). Then \(|z_0^{p}w_0^{2^{q}}|=1\) for some \(p \ge 1\) and \(q \ge 0.\) The sequence
$$\begin{aligned} f_n(z,w)&=F_2^{p(2^{qn-q})} \circ F_1^{qn}(z,w)\\&=\big (z,wz^{p(2^{qn-q})}\big ) \circ \big (z,w^{2^{qn}}\big ) =\Big (z, \{z^p w^{2^q}\}^{2^{q(n-1)}}\Big ) \end{aligned}$$
for \(n \ge 1\). On every neighbourhood of \((z_0,w_0)\), there exists (zw) such that \(|z^pw^{2^q}|>1\) as well as (zw) such that \(|z^pw^{2^q}|<1.\) Thus, \((z_0,w_0)\) is contained in the Julia set and this completes the proof.

5 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 \(\mathbb {C}^k\). The wandering and the recurrent Fatou components for a semigroup G are defined as:

Definition 5.1

Let \(G=\langle \phi _1,\phi _2,\ldots \rangle\) where each \(\phi _i\in \mathcal {E}_k\). Given a Fatou component \(\Omega\) of G and \(\phi \in G\), let \(\Omega _{\phi }\) be the Fatou component of G containing \(\phi (\Omega \ \Sigma _\phi )\) where \(\Sigma _{\phi }\) is the set where the Jacobian of \(\phi\) vanishes. A Fatou component is wandering if the set \(\big \{ \Omega _{\phi }{:}\;\phi \in G\big \}\) contains infinitely many distinct elements.

Definition 5.2

Let \(G=\langle \phi _1,\phi _2,\ldots \rangle\) where each \(\phi _i\in \mathcal {E}_k\). A Fatou component \(\Omega\) of G is recurrent if for any sequence \(\{g_j\}_{j\ge 1}\subset G\), there exists a subsequence \(\{g_{j_m}\}\) and a point \(p\in \Omega\) (the point p depends on the chosen sequence) such that \(g_{j_m}(p)\rightarrow p_0 \in \Omega\).

Note that we assume here a stronger definition of recurrence than the existing definition for the case of iterations of a single holomorphic endomorphism of \(\mathbb C^k.\) The natural extension of this definition to the semigroup set up would have been the following, a Fatou component \(\Omega\) is recurrent if there is a point \(p \in \Omega\) and a sequence \(\phi _n \in \Omega\) such that \(\phi _n(p) \rightarrow p_0,\) where \(p_0 \in \Omega .\) If this definition of recurrence is adopted then it is possible that a Recurrent domain is Wandering. In particular, Theorem 5.3 in [7] gives an example of a polynomial semigroup \(G=\langle \phi _1,\phi _2, \ldots \rangle\) in \(\mathbb C\), such that there exists a Fatou component, (say \(\mathcal {B}\), which is conformally equivalent to a disc), that is wandering, but returns to the same component infinitely often. This means that there exists sequences say \(\phi _n^+ \in G\) and \(\phi _n^- \in G\) such that \(\phi _n^-(\mathcal {B}) \subset \mathcal {B}\) or \(\phi _n^ +(\mathcal {B})\) are contained in distinct Fatou components of G. This example can be easily adapted in higher dimensions.

Example 5.3

Consider the semigroup \(\mathcal {G}=\langle \Phi _1, \Phi _2, \ldots , \rangle\) generated by the maps
$$\begin{aligned} \Phi _i(z,w)=\left( \phi _i(z), w^2\right) \end{aligned}$$
where \(\phi _i\) are the polynomial maps as in Theorem 5.3 of [7]. Let \(\{\Phi _n^-\}_{n \ge 1} \subset G\) be the sequence that maps \(\mathcal {B} \times \mathbb {D}\) into itself and \(\{\Phi _n^+\}_{n \ge 1} \subset G\) be the sequence such that
$$\begin{aligned} \Phi ^+_i(\mathcal {B}\times \mathbb D) \cap \Phi ^+_j( \mathcal {B}\times \mathbb D) = \emptyset \end{aligned}$$
for every \(i \ne j.\) Also \(\mathcal {B}\times \mathbb D\) is a Fatou component of \(\mathcal {G}\) as any point on the boundary of \(\mathcal {B}\times \mathbb D\), is either in the Julia set of G or in the Julia set of the map \(z \rightarrow z^2\). Hence \(\mathcal {B}\times \mathbb D\) is a Fatou component which is wandering, but may be recurring as well if we adapt the classical definition of recurrence.

Hence, we work with a stronger definition of recurrence than the classical one. Next, we provide an alternative description for recurrent Fatou components of G.

Lemma 5.4

A Fatou component \(\Omega\) is recurrent if and only if for any sequence \(\{\phi _j\}\subset G\) , there exists a compact set \(K\subset \Omega\) and a subsequence \(\{\phi _{j_m}\}\) such that \(\phi _{j_m}(p_{j_m})\rightarrow p_0 \in \Omega\) for a sequence \(\{p_{j_m}\}\subset K\).

Proof

Take any sequence \(\{\phi _j\}\subset G\). Then, there exists a subsequence \(\{\phi _{j_m}\}\) and points \(\{p_{j_m}\}\subset K\) with K compact in \(\Omega\) such that
$$\begin{aligned} \phi _{j_m}(p_{j_m})\rightarrow p_0 \in \Omega . \end{aligned}$$
Without loss of generality we assume \(p_{j_m}\rightarrow q_0 \in K\). It follows that \(\phi _{j_m}(q_0)\rightarrow p_0 \in \Omega\) using the fact that any sequence of G is normal on the Fatou set of G. \(\square\)

Proposition 5.5

Let \(G=\langle \phi _1,\phi _2,\ldots ,\phi _m\rangle\) where each \(\phi _i \in \mathcal {E}_k\) for every \(1 \le i \le m\) . If \(\Omega\) is a recurrent Fatou component of G, then G is locally bounded on \(\Omega\) . Moreover, \(\Omega\) is pseudoconvex and Runge.

Proof

Assume G is not locally bounded on \(\Omega\). Then, there exists a compact set \(K\subset \Omega\) and \(\{g_r\} \subseteq G\) such that \(|g_r(z_r) |> r\) with \(z_r \in K\) for every \(r \ge 1\). Clearly, this cannot be the case since \(\Omega\) is a recurrent Fatou component, so we can always get a subsequence \(\{g_{r_k}\}\) from the sequence \(\{g_r\} \in G\) such that it converges to a holomorphic function uniformly on compact set in \(\Omega\) and in particular on K. From the proof of Proposition 2.2, it follows that local boundedness of G on \(\Omega\) implies that \(\Omega\) is polynomially convex. Hence \(\Omega\) is pseudoconvex.

Theorem 5.6

Let \(G= \langle \phi _1,\phi _2, \ldots \rangle\) where each \(\phi _i \in \mathcal {E}_k.\) Assume that \(\Omega\) is a recurrent Fatou component of G. If there exists a \(\phi \in G\) such that \(\phi (\Omega )\) is contained in the Fatou set of G,  i.e., \(\phi (\Omega ) \subset F(G)\) then one of the following is true
  1. (i)

    There exists an attracting fixed point (say \(p_0\) ) in \(\Omega\) for the map \(\phi .\)

     
  2. (ii)
    There exists a closed connected submanifold \(M_\phi \subset \Omega\) of dimension \(r_\phi\) with \(1 \le r_\phi \le k-1\) and an integer \(l_\phi >0\) such that
    1. (a)

      \(\phi ^{l_\phi }\) is an automorphism of \(M_{\phi }\) and \(\overline{\{\phi ^{nl_\phi }\}_{n \ge 1}}\) is a compact subgroup of \(\mathrm{Aut}(M_\phi ).\)

       
    2. (b)

      If \(f \in \overline{\{\phi ^n\}}\) , then f has maximal generic rank \(r_\phi\) in \(\Omega .\)

       
     
  3. (iii)

    \(\phi\) is an automorphism of \(\Omega\) and \(\overline{\{\phi ^n\}}\) is a compact subgroup of \(\mathrm{Aut}(\Omega ).\)

     

Proof

Since \(\Omega \subset F(G)\), there exists a recurrent Fatou component of the map \(\phi\) (say \(\Omega _{\phi }\)) such that \(\Omega \subset \Omega _{\phi },\) i.e., there exists an integer \(l\ge 1\) such that
$$\begin{aligned} \phi ^l(\Omega _\phi ) \cap \Omega _{\phi }\ne \emptyset \;\; \quad \text {and} \quad \;\; \phi ^{m}(\Omega _\phi ) \cap \Omega _{\phi }= \emptyset \end{aligned}$$
for \(0 \le m < l.\) So, if \(l>1\) then there do not exist any \(p \in \Omega\) such that any subsequence of \(\{\phi ^{lk+1}(p)\}_{k\ge 1}\) converges to a point in \(\Omega\). Hence \(l=1\) and by assumption it follows that \(\phi (\Omega ) \subset \Omega .\)
Let h be a limit function of \(\{\phi ^n\}\) of maximal rank (say \(r_\phi\)), i.e.,
$$\begin{aligned} h(p)=\lim _{j \rightarrow \infty } \phi ^{n_j}(p) \;\; \text {for every}\;\; p \in \Omega , \end{aligned}$$
where \(\{n_j\}\) is an increasing subsequence of natural numbers.
Case 1 If \(r_{\phi }=0.\) Then \(h(\Omega )=p_0\) for some \(p_0 \in \Omega\) since by recurrence there exists a point \(p \in \Omega\), such that \(\phi ^{n_j}(p) \rightarrow p_0\) and \(p_0 \in \Omega .\) Also \(h(p_0)=p_0.\) Then
$$\begin{aligned} \phi (p_0)= \phi (h(p_0))=h(\phi (p_0))=p_0, \end{aligned}$$
i.e., \(p_0\) is a fixed point of \(\phi .\) As some sequence of iterates of \(\phi\) converge to a constant function, \(p_0\) is an attracting fixed point for \(\phi .\)
Case 2 If \(r_{\phi } \ge 1.\) Then there exists an increasing subsequence \(\{m_j\}\) such that
$$\begin{aligned} p_j=m_{j+1}-m_j \end{aligned}$$
are increasing positive integers and the sequences \(\{\phi ^{m_j}\}\) and \(\{\phi ^{p_j}\}\) converge uniformly to the limit functions h and \(\tilde{h}\) respectively on the Fatou component \(\Omega .\) Since by recurrence \(h(\Omega ) \cap \Omega \ne \emptyset\), if \(p \in \Omega\) be such that \(p =h(q)\) for some \(q \in \Omega\) then
$$\begin{aligned} \tilde{h}(p)= \lim _{j \rightarrow \infty } \phi ^{m_{j+1}-m_j}(p)=\lim _{j \rightarrow \infty }\phi ^{m_{j+1}-m_j}\left( \phi ^{m_j}(q)\right) =p \end{aligned}$$
Define
$$\begin{aligned} M=\{x \in \Omega {:}\; \tilde{h}(x)=x \}. \end{aligned}$$
Claim M is a closed complex submanifold of \(\Omega .\)
Since \(h(\Omega ) \cap \Omega \subset M\), M is a variety of dimension \(\ge r_\phi\). But by the choice of h, the generic rank of \(\tilde{h} \le r_{\phi }\) and \(M \subset \tilde{h}(\Omega ) \cap \Omega .\) So the dimension of M is \(r_{\phi }.\) Now for any point in M,  the rank of the derivative matrix of \(\mathrm{Id}-\tilde{h}\) is greater than or equal to \(k-r_\phi\). Suppose for some \(x \in M\) the rank of \(D(\mathrm{Id}-\tilde{h})(x)>k-r_\phi ,\) then there exists a small neighbourhood of x, say \(V_x\) such that \(V_x \subset \Omega\) and
$$\begin{aligned} \text {rank of }\mathrm{Id}-\tilde{h}>k-r_\phi \;\; \text {for every }\;\; x \in V_x. \end{aligned}$$
Then \(\{\mathrm{Id}-\tilde{h}\}^{-1}(0) \cap V_x\) is a variety of dimension at most \(r_\phi -1,\) i.e., the dimension of M is strictly less than \(r_\phi ,\) which is a contradiction. Thus, the rank of \(\mathrm{Id}-\tilde{h}\) is \(k-r_\phi\) for every point in M and hence M is a closed submanifold of \(\Omega .\)

Step 1: Suppose that \(r_{\phi }=k.\)

Then clearly \(M=\Omega\) and \(\tilde{h}\) on \(\Omega\) is the identity map. Let \(h_2= \lim \phi ^{p_j-1}.\) Then
$$\begin{aligned} \tilde{h}(x)=h_2 \circ \phi (x)=x, \; \; \text {for every}\;\; x \in \Omega , \end{aligned}$$
i.e., \(\phi\) is injective on \(\Omega\) and \(\phi (\Omega )\) is an open subset of \(\Omega .\) Suppose there exists an \(x \in \Omega \ \phi (\Omega )\) then for a sufficiently small ball of radius \(r>0\) with \(B_r(x) \subset \Omega\)
$$\begin{aligned} \phi ^l(\Omega ) \cap B_r(x)=\emptyset \;\;\text {for every}\;\; l \ge 1. \end{aligned}$$
This contradicts that \(\phi ^{p_j} (x)\rightarrow x.\) Hence \(\phi\) is surjective on \(\Omega\) and hence an automorphism of \(\Omega .\)
Step 2: Suppose that \(1 \le r_{\phi } \le k-1.\) Let \(M_\phi\) denote an irreducible component of M. For every \(q \in M_{\phi }\), it follows that \(\phi ^{p_j}(q) \rightarrow q\) as \(j \rightarrow \infty .\) Since \(\phi (\Omega ) \subset \Omega\), we get \(\phi ^n(q) \in \Omega\) for every \(n \ge 1\) and
$$\begin{aligned} \tilde{h} \circ \phi ^n(q)=\phi ^n \circ \tilde{h}(q)=\phi ^n(q) \; \text {for every}\; q \in M_{\phi }, \end{aligned}$$
i.e., \(\phi ^n(M_\phi ) \subset M\) for every \(n \ge 1.\)

Claim There exists a positive integer \(l_\phi\) such that \(\phi ^{l_{\phi }}(M_{\phi }) \subset M_{\phi } .\)

Let \(p_0 \in M_\phi\) and \(\Delta \subset \Omega\) be a polydisk at \(p_0\) such that \(\Delta\) does not intersect the other components of \(M_{\phi } .\) Now choose \(\Delta ' \subset \Delta\), a sufficiently small polydisk such that \(\tilde{h}(\Delta ') \subset \Delta .\) Then \(\omega =\tilde{h}(\Delta ') \subset M_\phi\) is a \(r_\phi\)-dimensional manifold. Let \(\Delta ''\) be a \(r_\phi\)-dimensional polydisk inside \(\omega\) and \(\{w_l\}_{l \ge 1}\) be a sequence in \(\Delta ''\) such that it converges to some \(w_0 \in \Delta ''.\) But \(\phi ^{p_j}(w_{p_j}) \rightarrow w_0\) as \(j \rightarrow \infty\) hence
$$\begin{aligned} \phi ^{p_j}(M_\phi ) \cap \Delta \ne \emptyset ,\; \text {i.e.,}\; \phi ^{p_j} (M_\phi )\subset (M_\phi ) \end{aligned}$$
for j sufficiently large. Let \(l_\phi\) be the minimum value such that \(M_\phi\) is invariant under \(\phi ^{l_{\phi }}.\)

Claim \(\phi ^{l_\phi }\) is an automorphism of \(M_\phi .\)

Without loss of generality there exists a sequence \(\{k_j\}\) such that \(p_j=i_0+k_jl_\phi\) for some \(0 \le i_0 \le l_\phi -1,\) i.e.,
$$\begin{aligned} \phi ^{i_0} \circ \phi ^{k_jl_\phi }(x) \rightarrow x \;\; \text {for every} \;\; x \in M_{\phi }. \end{aligned}$$
As \(M_{\phi }\) is invariant under \(\phi ^{l_\phi }\), the sequence \(x_j=\phi ^{k_jl_\phi }(x)\) lies in \(M_{\phi }.\) Again as before let \(\Delta _x\) be a sufficiently small neighbourhood such that \(\Delta _x \subset \Omega\) and \(\Delta _x\) does not intersect the other components of M. Since \(\phi ^{i_0}(x_j) \in \Delta _x \cap M_\phi\) for large j, \(\phi ^{i_0}(M_\phi ) \subset M_\phi .\) But \(0 \le i_0 \le l_\phi -1\), i.e., \(i_0=0\) and \(\{\phi ^{k_jl_\phi }\}\) converges uniformly to the identity on \(M_{\phi }.\) Let \(\psi =\lim \phi ^{(k_j-1)l_\phi }\) then
$$\begin{aligned} \phi ^{l_\phi } \circ \psi (x)=\psi \circ \phi ^{l_\phi }(x)=x \;\; \quad \text {for every}\;\; x \in M_\phi . \end{aligned}$$
Hence \(\phi ^{l_\phi }\) is injective on \(M_\phi\) and \(\phi ^{l_\phi }(M_\phi )\) is an open subset in the manifold \(M_\phi\). Now as in Step 1 observe that \(\phi ^{k_jl_\phi }\) converges to the identity on \(M_\phi\) for an unbounded sequence \(\{k_j\}\), so \(\phi ^{l_\phi }\) is also surjective on \(M_{\phi }\). Thus the claim.

Let \(Y=\{\phi ^{nl_\phi }\}_{n \ge 1} \subset \mathrm{Aut}(M_\phi ).\)

Claim \(\bar{Y}\) is a locally compact subgroup of \(\mathrm{Aut}(M_\phi ).\)

For some \(\Psi \in Y\) and for a compact set \(K \subset M_\phi\) consider the neighbourhood of \(\Psi\) given by
$$\begin{aligned} V_\Psi (K, \epsilon )= \{\psi \in \mathrm{Aut}(M_\phi ){:}\; \Vert \psi (z)-\Psi (z)\Vert _{K} < \epsilon \}. \end{aligned}$$
One can choose \(\epsilon\) and K sufficiently small such that for every sequence \(\psi _j \in V_\Psi (K, \epsilon )\) there exists an open set \(U \subset \Omega\) such that \(\psi _j(U \cap M_\phi ) \subset \bar{V} \cap M_\phi \subset \Omega\), where V is some open subset of \(\Omega .\)
Since \(\psi _j=\phi ^{n_j l_\phi }\) for a sequence \(\{n_k\}\) and \(\Omega\) is a Fatou component, \(\psi _j\) has a convergent subsequence in \(\Omega .\) We choose appropriate subsequences such that the limit maps
$$\begin{aligned} \Psi _1=\lim _{j \rightarrow \infty } \phi ^{n_jl_{\phi }} \;\; \quad \text {and} \quad \;\; \Psi _2=\lim _{j \rightarrow \infty } \phi ^{(k_j-n_j)l_\phi } \end{aligned}$$
are defined on \(\Omega .\) Also as \(M_\phi\) is closed in \(\Omega\), \(\Psi _i(M_\phi ) \subset \overline{M_\phi }\) for every \(i=1,2\) where \(\overline{M_\phi }\) denote the closure of \(M_\phi\) in \(\mathbb C^k.\) Then \(\Psi _1(U) \subset \Omega\) and
$$\begin{aligned} \Psi _2 \circ \Psi _1(x)=x \;\; \quad \text {for every}\;\; x \in U \cap M_\phi . \end{aligned}$$
(5.1)
Since \(\Psi _1\) on \(M_\phi\) is a limit of automorphisms of \(M_\phi\), the Jacobian of \(\Psi _1\) on the manifold \(M_\phi\) is either non-zero at every point of \(M_\phi\) or vanishes identically. But by (5.1), \(\Psi _1\) restricted to \(U \cap M_{\phi }\) is injective, which is open in the manifold \(M_\phi ,\) i.e., \(\Psi _1\) is an open map of \(M_\phi\) and \(\Psi _1(M_\phi ) \subset M_{\phi }.\) So (5.1) is true for every \(x \in M_\phi .\) Now by the same arguments it follows that \(\Psi _2\) is an injective map from \(M_\phi\) such that \(\Psi _2(M_\phi ) \subset M_{\phi }.\) Hence
$$\begin{aligned} \Psi _2 \circ \Psi _1(x)=\Psi _1 \circ \Psi _2(x)=x \;\; \quad \text {for every}\;\; x \in M_\phi , \end{aligned}$$
i.e., \(\Psi _1\) is an automorphism of \(M_\phi .\) This proves that \(\bar{Y}\) is a locally compact subgroup of \(\mathrm{Aut(M_\phi )}.\)

Now since \(M_\phi\) is a complex manifold and \(\bar{Y}\) is a locally abelian subgroup of automorphisms of \(M_\phi\), by Theorem A in [2], it follows that \(\bar{Y}\) is a Lie group. Hence the component of \(\bar{Y}\) containing the identity is isomorphic to \(\mathbb T^l \times \mathbb R^m.\) Suppose \(\Psi\) is the isomorphism, then for some \(n >0\), \(\Psi (a,b)=\phi ^{nl_\phi }\). Now if \(b \ne 0\), then there does not exist an increasing sequence of \(k_j\) such that \(\phi ^{k_j l_\phi }\) converges to identity. This proves that the component of \(\bar{Y}\) containing the identity is compact and hence any component of \(\bar{Y}\) is compact by the same arguments. Also as \(M_\phi\) is contained in the Fatou set, the number of components of \(\bar{Y}\) is finite, thus \(\bar{Y}\) is a compact subgroup of \(\mathrm{Aut}(M_\phi ).\)

If \(r_\phi =k\), then \(M_\phi\) is \(\Omega\), then one can apply the same technique as discussed above to conclude that \(\overline{\{\phi ^n\}}\) is a closed compact subgroup of \(\mathrm{Aut}(\Omega ).\)

Finally, let f be a limit of \(\{\phi ^n\}_{n \ge 1},\) i.e.,
$$\begin{aligned} f(p)=\lim _{j \rightarrow \infty } \phi ^{n_j}(p) \;\; \quad \text {for every} \;\; p \in \Omega . \end{aligned}$$
Claim The generic rank of f is \(r_{\phi }.\)

By the definition of recurrence it follows that \(\Omega \subset \Omega _{\phi }\), where \(\Omega _\phi\) is a periodic Fatou component for \(\phi\) with period 1. Hence by Theorem 3.3 in [5] it follows that the limit maps of the set \(\{\phi ^n\}\) in \(\Omega _\phi\) have the same generic rank (say r). But \(\Omega\) is an open subset of the Fatou component \(\Omega _\phi\), so the rank of limit maps restricted to \(\Omega\) should be same, i.e., \(r=r_\phi\) and each limit map of \(\{\phi ^n\}\) has rank \(r_\phi .\) \(\square\)

By Proposition 5.5 a semigroup G is always locally uniformly bounded on a recurrent Fatou component semigroup G. If G is finitely generated by holomorphic endomorphisms of maximal rank k in \(\mathbb C^k\), then by Proposition 2.2 it follows that a recurrent Fatou component is mapped in the Fatou set by any element of G. Hence we have the following corollary.

Corollary 5.7

Let \(G= \langle \phi _1,\phi _2, \ldots ,\phi _m \rangle\) where each \(\phi _i \in \mathcal {E}_k\) for every \(1 \le i \le m.\) Assume that \(\Omega\) is a recurrent Fatou component of G then for every \(\phi \in G\) one of the following is true
  1. (i)

    There exists an attracting fixed point (say \(p_0\) ) in \(\Omega\) for the map \(\phi .\)

     
  2. (ii)
    There exists a closed connected submanifold \(M_\phi \subset \Omega\) of dimension \(r_\phi\) with \(1 \le r_\phi \le k-1\) and an integer \(l_\phi >0\) such that
    1. (a)

      \(\phi ^{l_\phi }\) is an automorphism of \(M_{\phi }\) and \(\overline{\{\phi ^{nl_\phi }\}_{n \ge 1}}\) is a compact subgroup of \(\mathrm{Aut}(M_\phi ).\)

       
    2. (b)

      If \(f \in \overline{\{\phi ^n\}}\) , then f has maximal generic rank \(r_\phi\) in \(\Omega .\)

       
     
  3. (iii)

    \(\phi\) is an automorphism of \(\Omega\) and \(\overline{\{\phi ^n\}}\) is a compact subgroup of \(\mathrm{Aut}(\Omega ).\)

     

Example 5.8

Let \(G=\langle \phi _1,\phi _2 \rangle\) be a semigroup of entire maps in \(\mathbb C^2\) generated by
$$\begin{aligned} \phi _1(z,w)=(w, \alpha z-w^2) \; \; \text{ and } \;\; \phi _2(z,w)=(zw,w) \end{aligned}$$
where \(0<\alpha < 1.\) Then G is locally uniformly bounded on a sufficiently small neighbourhood around the origin, and \(\phi (0)=0\) for every \(\phi \in G.\) So the Fatou component of G containing 0 (say \(\Omega _0\)) is recurrent. Now note that for \(\phi _2\)
$$\begin{aligned} r_{{\phi _2}}=1\;\; \text{ and } \;\;M_{\phi _2}=\{(0,w){:}\; w \in \mathbb C\} \cap \Omega _0, \end{aligned}$$
whereas for \(\phi _1\) the origin is an attracting fixed point. This illustrates the different behaviour of the sequences \(\{\phi _1^n\}\) and \(\{\phi _2^n\}\) (both of which are in G) on \(\Omega _0.\)
Note that for any other \(\phi \in G\) which is not of the form \(\phi _1^k,~k \ge 2\), contains a factor of \(\phi _2\) at least once. Since for a small enough ball (say B) around origin, \(\phi _2\) is contracting, and \(\phi _1(B) \subset B\) so there exists a constant \(0<a_\phi <1\) such that
$$\begin{aligned} |\phi (z)| \le a_{\phi } |z| \;\; \quad \text {for every} \;\; z \in B, \end{aligned}$$
i.e., the origin is an attracting fixed point.

Proposition 5.9

Let \(G=\langle \phi _1,\phi _2,\ldots ,\phi _m\rangle\) where each \(\phi _i\in \mathcal {V}_k\) for every \(1 \le i \le m\) and let \(\Omega\) be an invariant Fatou component of G. Then either \(\Omega\) is recurrent or there exists a sequence \(\{\phi _n\}\subset G\) converging to infinity.

Proof

If \(\Omega\) is not recurrent, then there exists a sequence \(\{\phi _n\}\subset G\) such that \(\{\phi _n\} \rightarrow \partial \Omega \cup \{\infty \}\) uniformly on compact sets of \(\Omega\). Assume \(\{\phi _{n_k}\}\) converges to a holomorphic function f on \(\Omega\). This implies that \(f(\Omega )\subset \partial \Omega\) contradicting the assumption that each \(\phi _{n_k}\) is volume preserving. Hence, \(\{\phi _{n_k}\}\) diverges to infinity uniformly on compact subsets of \(\Omega\). \(\square\)

Proposition 5.10

Let \(G=\langle \phi _1,\phi _2,\ldots ,\phi _m \rangle\) where each \(\phi _i\in \mathcal {V}_k\) for every \(1 \le i \le m\) and let \(\Omega\) be a wandering Fatou component of G. Then, there exists a sequence \(\{\phi _n\}\subset G\) converging to infinity.

Proof

Since \(\Omega\) is wandering, one can choose a sequence \(\{\phi _n \}\subset G\) so that
$$\begin{aligned} \Omega _{\phi _n}\cap {\Omega }_{\phi _m}=\emptyset \end{aligned}$$
(5.2)
for \(n\ne m\). If this sequence \(\{\phi _n\}\) does not diverge to infinity uniformly on compact subsets, some subsequence \(\{\phi _{n_k}\}\) will converge to a holomorphic function h on \(\Omega\). By abuse of notation, we denote \(\{\phi _{n_k}\}\) still by \(\{\phi _n \}\). Fix \(z_0\in \Omega\). Then for any given \(\epsilon\), there exists \(\delta\) such that
$$\begin{aligned} \left| \phi _{n_0}(z)-\phi _n(z)\right| <\epsilon \end{aligned}$$
(5.3)
for all \(n\ge n_0\) and for all \(z\in B(z_0,\delta )\). From (5.3) it follows that vol\((\cup _{n\ge n_o}\phi _n(B(z_0,\delta )))\) is finite. On the other hand, since each \({\phi _n}\) is volume preserving and (5.2) holds, we get
$$\begin{aligned} \text{ Vol }\Big (\bigcup _{n\ge n_o}\phi _n\big (B(z_0,\delta )\big )\Big ) =+\infty . \end{aligned}$$
Hence, we have proved the existence of a sequence in G converging to infinity. \(\square\)

6 Concluding remarks

As mentioned in the introduction, the classification of recurrent Fatou components for iterations of holomorphic endomorphisms of complex projective spaces has been studied in [4] and [3]. It would be interesting to explore the same question for semigroups of holomorphic endomorphisms of complex projective spaces. The main theorem in [4] and [3] is proved under the assumption that the given recurrent Fatou component is also forward invariant. The analogue of such a condition in the case of semigroups is not clear to us since we are then dealing with a family of maps none of which is distinguishable from the other.

Declarations

Authors’ contribution

This is a joint work and both authors read and approved the final manuscript.

Acknowledgements

We would like to thank Kaushal Verma for valuable discussions and comments. We would also like to thank the referee for useful suggestions.

Competing interests

The authors declare that they have no competing interests.

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Harish-Chandra Research Institute
(2)
Department of Mathematics, Indian Institute of Science

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