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Dynamics of semigroups of entire maps of \(\mathbb C^k\)
 Sayani Bera^{1}Email author and
 Ratna Pal^{2}
https://doi.org/10.1186/s406270160006x
© The Author(s) 2016
 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.

\(\mathcal {E}_k{:}\;\) 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 \(\mathbb C^k.\)
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 nonrecurrent, nonwandering 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
 (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 \(\phi .\) \(\square\)
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
Proposition 2.3
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 \(\phi (\Omega _G) \subset \bar{\Omega }_{\phi }.\)
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
Definition 2.5
Proposition 2.6
Let G be a semigroup generated by proper holomorphic endomorphisms of \(\mathbb C^k\) and H be a subsemigroup of G which has a finite (or cofinite) index in G. Then, \(F(G)=F(H)\) and \(J(G)=J(H).\)
Proof
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
Example 2.9
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
Claim There exists a sequence \(\phi _n \in G\) such that \(\phi _n\) diverges to infinity on A.
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 \(0 \in B(0,\epsilon )\subset \psi (B(0,\epsilon ))\), 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
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
 (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 \(\Omega =\{zp{:}\; z \in U \}.\) Then, \(\psi\) is the required map with the properties \(\Omega \subset \subset \psi (\Omega )\) and 0 is a fixed point for \(\psi .\)
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 \(\phi _i \in \mathcal P_k.\)
Case 2 Suppose that \(G=\langle \phi _1,\phi _2, \ldots \rangle\) where each \(\phi _i \in \mathcal E_k.\)
Claim \(\psi ^{1}(p) \cap \omega =p\)
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
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
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
Case 2 Suppose \(z>1\) and \(w>1.\)
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
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
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
 (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 \(1 \le r_\phi \le k1\) 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 \(\Omega .\)
 (a)
 (iii)
\(\phi\) is an automorphism of \(\Omega\) and \(\overline{\{\phi ^n\}}\) is a compact subgroup of \(\mathrm{Aut}(\Omega ).\)
Proof
Step 1: Suppose that \(r_{\phi }=k.\)
Claim There exists a positive integer \(l_\phi\) such that \(\phi ^{l_{\phi }}(M_{\phi }) \subset M_{\phi } .\)
Claim \(\phi ^{l_\phi }\) is an automorphism of \(M_\phi .\)
Let \(Y=\{\phi ^{nl_\phi }\}_{n \ge 1} \subset \mathrm{Aut}(M_\phi ).\)
Claim \(\bar{Y}\) is a locally compact subgroup of \(\mathrm{Aut}(M_\phi ).\)
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 \(\mathrm{Aut}(\Omega ).\)
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
 (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 \(1 \le r_\phi \le k1\) 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 \(\Omega .\)
 (a)
 (iii)
\(\phi\) is an automorphism of \(\Omega\) and \(\overline{\{\phi ^n\}}\) is a compact subgroup of \(\mathrm{Aut}(\Omega ).\)
Example 5.8
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
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
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