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

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.

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 $\phi \in G$

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

• (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

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

• (ii) $\phi$ is invertible at its fixed points.

• (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 (z, w). 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 (z, w) such that $|z^pw^{2^q}|>1$ as well as (z, w) 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

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

• (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

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

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

• (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

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

• (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

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

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

• (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.