aa r X i v : . [ m a t h . C V ] A p r AUTOMORPHISMS OF C m WITH BOUNDED WANDERING DOMAINS
LUKA BOC THALER
Abstract.
We prove that the Euclidean ball can be realized as a Fatou component of aholomorphic automorphism of C m , in particular as the escaping and the oscillating wanderingdomain. Moreover, the same is true for a large class of bounded domains, namely for allbounded simply connected regular open sets Ω ⊂ C m whose closure is polynomially convex.Our result gives in particular the first example of a bounded Fatou component with a smoothboundary in the category of holomorphic automorphisms. Introduction
With the emergence of new techniques, the study of wandering domains has flourished inrecent years. Many strong results have been established, in particular about the existence andthe geometry of wandering domains in the category of transcendental holomorphic functions[Ba76, Bi15, EL92, BEGRS19] and in the category of holomorphic endomorphisms of CP [ABDPR16, AsBTP19]. On the other hand there are only few known results in the categoryof holomorphic automorphisms of C m for m ≥ Theorem 1.
For every m ≥ there exists an automorphism of C m with an escaping wan-dering domain equal to the Euclidean ball. Theorem 2.
For every m ≥ there exists an automorphism of C m with an oscillatingwandering domain equal to the Euclidean ball. As we will argue in the last section, the proofs of these theorems can easily be modified sothat the same statements hold for any bounded simply connected regular open sets Ω ⊂ C m whose closure is polynomially convex, in particular for all bounded convex domains.Let F be a holomorphic automorphism of C m , and recall that the Fatou set F is thelargest open subset of C m on which the family of iterates ( F n ) n ≥ is locally equicontinuous.Connected component Ω of the Fatou set is called the Fatou component and we say that suchcomponent is wandering if and only if F n (Ω) ∩ Ω = ∅ for all n ≥
1. We will call a wanderingFatou component a wandering domain . There are three types of wandering domains:(1) escaping; if all orbits converge to the line at infinity,(2) oscillating; if there exists an unbounded orbit and an orbit with a bounded subse-quence,(3) orbitally bounded; if every orbit is bounded.
Mathematics Subject Classification.
Key words and phrases. holomorphic dynamics, automorphisms, Fatou components, wandering domains,polynomially convex, Anders´en-Lempert theory,Supported by the research program P1-0291 from ARRS, Republic of Slovenia. he first construction of a holomorphic automorphism of C with a wandering domain isdue to Fornæss-Sibony [FS98] and their wandering domain is of the oscillating type. Morerecently Arosio-Benini-Fornæss-Peters [ABTP19] constructed transcendental H´enon maps,i.e. a holomorphic automorphism of C of the form F ( z, w ) = ( f ( z ) + aw, az ) with f : C → C a transcendental function, that admit wandering domains. In particular they constructexamples of wandering domains which are of the escaping and of the oscillating type andthey are biholomorphic to C . The first example of a polynomial automorphism (of C )with a wandering domain was given by Hahn-Peters [HP18] and their example was of theorbitally bounded type. In [ABTP19] we have shown that oscillating wandering domains oftranscendental H´enon maps can also have different complex structures. In particular we haveconstructed a wandering domain that supports a non-constant bounded plurisubharmonicfunction and therefore it can not be biholomorphic to C . The most recent result is dueto Berger-Biebler [BB20] who have solved a long standing problem by proving existence ofpolynomial H´enon maps which admit wandering domains and note that those can only be ofthe orbitally bounded type.Our Theorem 1 and Theorem 2 give new examples of wandering domains in terms of theirgeometry. In particular, these are also the first examples of bounded Fatou components witha smooth boundary.Note that the only other known example of an automorphism of C m that has a Fatoucomponent (non-wandering) with a smooth boundary, is a shear automorphism of C , con-structed recently by [BC20], for which the invariant Fatou component is the product of thecomplex line with the Euclidean unit disc. Also, the only known examples of bounded Fatoucomponents are the Siegel balls for polynomial H´enon maps, and their boundary is very farfrom being smooth. It is known that such a domain must be biholomorphic to one of thefollowing three the polydisc, the unit ball and a Thullen domain, but which of them can berealised as the Siegel ball for polynomial H´enon maps it is presently unknown.We believe that the construction behind Theorem 1 will open the way for the studyof intrinsic dynamics in wandering domains, as has recently been done for transcendentalfunctions in dimension one [BEGRS19].Finally let us mention that tools Anders´en–Lempert theory, on which or our proof rely,apply to the large class of Stein manifolds with the density property, see [For17, Section 4].Therefore we strongly believe that, in many of those cases, the constructions presented in thispaper could easily be modified and used to produce automorphisms with different types ofwandering domains.Our paper is organized as follows:In Section 2 we introduce the notation and recall the basic ingredients that will be usedin the paper.In Section 3 we modify the constructions from [ABFP18, ABTP19] and reprove the exis-tence of wandering domains biholomorphic to C m and to Short C m using tools of Anders´en–Lempert theory. This modification will serve as the basis for the construction of the oscillatingwandering ball in Section 5.In Section 4 we use tools of Anders´en–Lempert theory to inductively construct a sequenceof automorphisms ( F k ), that converge uniformly on compacts, to an automorphism F withthe following properties: (1) The Euclidean diameter of F k ( B ( P , k ≥
0, (2) F k ( B ( P , → ∞ as k → ∞ and (3) there is a sequence of points ( T j ) that ccumulate densely on the b B ( P ,
1) and each of them is contained in some attracting basin.This implies that the Fatou component is exactly the ball B ( P ,
1) which settles Theorem 1.In Section 5 we prove Theorem 2 by carefully combining two constructions previouslyintroduced in Section 3 and Section 4.2.
Preliminaries
In this section, we introduce the notation and recall the basic ingredients that will be usedin the paper. Throughout this paper we will always assume that m ≥ polynomially-convex hull of a compact set K ⊂ C m is defined as b K = { z ∈ C m : | p ( z ) | ≤ sup K | p | for all holomorphic polynomials p } . We say that K is polynomially convex if b K = K .Given a point z ∈ C m , we denote by B ( z , r ) ⊂ C m to denote the open m -dimensionalEuclidean ball of radius r centered at z . We will write B = B (0 , Lemma 3.
Assume that K ⊂ C n is a compact polynomially convex set. For any finiteset p , . . . , p k ∈ C m \ K and for all sufficiently small numbers r > , . . . , r k > , the set S kj =1 B ( p j , r j ) ∪ K is polynomially convex. We will frequently use the fact that the union of any two disjoint closed Euclidean ballsis polynomially convex.Recall that a domain D ⊂ C m is called starshaped (in some literature star-like ) if thereexists a point p ∈ D such that the line segment between p and any other point q ∈ D iscontained in D . We will say that a domain D ⊂ C m is starshapelike if there exists Φ anautomorphism of C m and a starshaped domain D ′ so that D = Φ( D ′ ). For example, anyimage of the Euclidean ball under an automorphism of C m is a starshapelike domain. It isa well known result that the closure of any bounded starshapelike domian is polynomiallyconvex.The key ingredient in our proofs will be the following result of the Anders´en–Lemperttheory, which is a combination of Theorem 4.9.2 and Corollary 4.12.4 from [For17]. Theorem 4.
Let A , A , . . . , A n be pairwise disjoint compact sets in C m such that all but oneare starshapelike. Let q j ∈ Aut ( C m ) ( j = 1 , . . . , n ) be such that the images B j = q j ( A j ) arepairwise disjoint. If the sets K = ∪ nj =1 A j and K ′ = ∪ nj =1 B j are polynomially convex, then forevery ε > there exists g ∈ Aut ( C m ) such that k g ( z ) − q j ( z ) k < ε for all z ∈ A j , j = 1 , . . . , m .In particular the automorphism of g can be chosen so that its finite order jets agree with thecorresponding jets of q j at any given finite set of points in A j , for ≤ j ≤ m . Note that in the above theorem the compact set A j can also be a point, since by Lemma3 we can always find a small closed ball around A j , such that its union with all the other setsis polynomially convex.If ( H n ) n ≥ is a sequence of automorphisms of C m , then for all 0 ≤ n ≤ k we denote H k,n := H k ◦ · · · ◦ H n +1 . Notice that with these notations we have for all n ≥ H n +1 ,n = H n +1 , H n,n = id . f for all n ≥ H n ( B ) ⊂ B then we define the basin of the sequence ( H n ) as thedomain Ω H := [ n ≥ H − n, ( B ) . The following lemma was established in [ABTP19] and will be used in the following sectionto determine the complex structure of a wandering domain.
Lemma 5.
To every finite family ( F , . . . , F n ) of holomorphic automorphisms of C m satisfying F j ( B ) ⊂⊂ B for all ≤ j ≤ n we can associate ε ( F , . . . , F n ) > such that the following holds:Given any two sequences ( H n ) n ≥ and ( G n ) n ≥ of holomorphic automorphisms of C m satisfying H n ( B ) ⊂⊂ B and G n ( B ) ⊂ B for all n ≥ , and moreover satisfying k H n − G n k B ≤ ε ( H , . . . , H n ) , ∀ n ≥ , the basins Ω G and Ω H are biholomorphically equivalent. Oscillating wandering domains
The existence holomorphic automorphisms of C m , that admit an oscillating wanderingdomain, has been proven by Fornæss and Sibony [FS98]. Recently constructed examplesshow that such wandering domains can be biholomorphic to C m [ABFP18] and to a Short C m [ABTP19]. In this section we slightly modify these two constructions and reprove themusing tools of Anders´en–Lempert theory. This modification will serve as the basis for theconstruction of the oscillating wandering ball, which will be presented in the last section ofthis paper. Proposition 6.
Let ( H k ) k ≥ be a sequence of holomorphic automorphisms of C m satisfying H k ( B ) ⊂⊂ B for all k ≥ and let ε k = ε k ( H , . . . , H k ) be as in Lemma . There exists a se-quence ( F k ) k ≥ of holomorphic automorphims of C m , a sequence of points ( P n ) n ≥ , sequencespositive real numbers ( β n ) n ≥ ց , ( R k ) k ≥ ր ∞ , ( r k ) k ≥ ր ∞ , strictly increasing sequencesof integers ( n k ) k ≥ and ( N k ) k ≥ satisfying n = 0 and N k − ≤ n k ≤ N k , and such that thefollowing properties are satisfied:(a) B (0 , r k − ) ⊂⊂ F k ( B (0 , R k )) for all k ≥ ,(b) k F k − F k − k B (0 ,R k − ) ≤ − k for all k ≥ ,(c) F k ( P n ) = P n +1 for all ≤ n < N k ,(d) k P n k k ≤ k for all k ≥ ,(e) k P N k k > R k for all k ≥ ,(f ) for all k ≥ we have β j < k +1 for N k < j ≤ N k +1 .(g) for all ≤ s ≤ k , F jk ( B ( P n s , β n s )) ⊂⊂ B ( P n s + j , β n s + j ) , ∀ ≤ j ≤ N k − n s , (h) for all ≤ s ≤ k , k Φ − n s ◦ F n s − n s − k ◦ Φ n s − − H s k B < ε s , where Φ n ( z ) := P n + β n z . In the above proposition, the properties ( a ) and ( b ) imply that the sequence F k convergesuniformly on compacts to an automorphism F . The properties ( c )–( g ) ensure the existence ofan oscillating wandering domain for F . Furthermore, the property ( h ) determines the intrinsicdynamics and the geometry of such a domain. roof. We prove this proposition by induction on k . We start the induction by letting F ( z , . . . , z m ) = ( z , . . . , z k , z k +1 , . . . , z m ) for some 1 ≤ k < m . Let r = 1 and R > B (0 , r ) ⊂⊂ F ( B (0 , R )) and set K = B (0 , R ). Moreover we let n = N = 0, β = 1, and choose any P with k P k > R + 1 such that all conditions are satisfied for k = 0.Let us suppose that conditions ( a )—( h ) hold for certain k , and proceed with the construc-tions satisfying the conditions for k + 1.First let R k +1 > k P N k k + 1 such that K k ⊂ B (0 , R k +1 ), where K k ∪ B ( P N k , β N k ) ispolynomially convex and B (0 , R k ) ⊆ K k . By the λ -Lemma (see [PdM82, Lemma 7.1]) thereexist a finite F k orbit ( Q j ) ≤ j ≤ M , i.e. F jk ( Q ) = Q j for all 0 ≤ j ≤ M , such that:(1) k Q j k < R k +1 for all 0 ≤ j < M ,(2) k Q M k > R k +1 ,(3) k Q ℓ k < k +1 for some 0 < ℓ < M .We can choose small enough 0 < θ < k +1 so that:(i) the ball B ( Q M , θ ) is disjoint from B (0 , R k +1 ),(ii) the balls B ( P N k , β N k ) , B ( Q , θ ) B ( Q M , θ ) (1)are pairwise disjoint, and disjoint from the set L := K k ∪ [
Note that the map g k +1 already satisfies all the properties ( b )—( h ) of theProposition 6. Next we need to post-compose our map g k +1 with an appropriate automor-phism to make sure that also property ( a ) is satisfied. Recall that this property is needed toensure that the limit map of the sequence ( F k ) is surjective.Let us continue with induction by choosing r k +1 > r k + 1 such that g k +1 ( B (0 , R k +1 )) ⊂⊂ B (0 , r k +1 ) . Since compact sets B ( Q M , θ ) and B (0 , R k +1 ) are disjoint starshapelike domains whose unionis polynomially convex the same holds for U := g k +1 ( B ( Q M , θ )) , V := g k +1 ( B (0 , R k +1 )) . Choose any point Q ′ ∈ C m such that the ball B ( Q ′ , θ ) lies in the complement of B (0 , r k +1 )and let ψ be a linear map satisfying ( ψ ◦ g k +1 )( Q M ) = Q ′ and ψ ( U ) ⊂ B ( Q ′ , θ ) . By Theorem 4 there exists an automorphism h such that(1) k h − id k V ≤ δ ′ k ,(2) k h − ψ k U ≤ δ ′ k ,(3) h ( Q j ) = Q j for all 0 < j ≤ M ,(4) ( h ◦ g k +1 )( Q M ) = Q ′ (5) h (0) = 0, d h = id , h ( P j ) = P j for all 1 ≤ j ≤ P N k ,where we have chosen δ ′ k ≤ k small enough such that(i) ( h ◦ g k +1 ) j ( W ) ⊂ B ( Q j , θ ) , for all 0 < j < ℓ, (ii) ( h ◦ g k +1 ) ℓ ( W ) ⊂ B ( Q ℓ , s ℓ ) , (iii) for all 0 ≤ s ≤ k ( h ◦ g k +1 ) j ( B ( P n s , β n s )) ⊂⊂ B ( P n s + j , β n s + j ) , ∀ ≤ j ≤ N k − n s . (iv) ( h ◦ g k +1 ) j ( B ( Q ℓ , s ℓ )) ⊂⊂ B ( Q ℓ + j , θ ) , for all 0 ≤ j ≤ M − ℓ, (v) k Φ − n s +1 ◦ ( h ◦ g k +1 ) n s +1 − n s ◦ Φ n s − H s +1 k B < ε s for all 0 ≤ s ≤ k (vi) ( h ◦ g k +1 )( U ) ⊂ B ( Q ′ , θ ).We define F k +1 := h ◦ g k +1 , so that the sequences of points( P j ) ≤ j ≤ N k , ( Q j ) Note that in the k th step of the induction we chose the radius β n k , i.e. s ℓ ,before we make an ε k -approximation to H k , i.e. before introducing ϕ . This means that wecan choose H k freely, in particular we can choose it so that it satisfies k H k ( z ) k ≤ β kn k k z k .Let us now show that this proposition implies the existence of an oscillating wanderingFatou component. The arguments in the following two paragraphs are similar to those in theproof of [ABTP19, Theorem 1] but we choose to present them here, so that the paper remainsself-contained.Let ( F k ) be a sequence of automorphisms of C m satisfying conditions ( a ) − ( h ) of Propo-sition 6. The sequence ( F k ) converges uniformly on compact subsets to an automorphism F of C m with an isolated fixed point at the origin and with d F being a diagonal matrix witheigenvalues equal to and 2. There is an unbounded orbit ( P n ), a sequence β n → n k ) such that the following properties are satisfied:(i) P n k → k ≥ F j ( B ( P n k , β n k )) ⊂ B ( P n k + j , β n k + j ) , ∀ j ≥ . (4)(iii) if for all k ≥ G k := Φ − n k ◦ F n k − n k − ◦ Φ n k − ∈ Aut( C m ) , then by combining conditions (g) and (h) it follows that G k ( B ) ⊂ B for all k , and k G k − H k k B ≤ ε ( H , . . . , H k ) , ∀ k ≥ . (5)Next show that Ω F := ∞ [ k =0 F − n k ( B ( P n k , β n k ))is contained in an oscillating Fatou component F . It suffices to prove that for all k ≥ 0, the ball B ( n k , β n k ) is contained in the Fatou set. But this follows from (4) since theEuclidean diameter of F j ( B ( P n k , β n k )) is bounded for all j ≥ 0. For all j ≥ F n j the atou component containing B ( P n j , β n j ). Since Ω F is connected, it is contained in the Fatoucomponent F .Since β n → 0, by (4) it follows that all limit functions on each F n j are constants. Weclaim that F n i = F n j iff j = i , which implies that they are all oscillating wandering domains.Assume by contradiction that F n j = F n i , and set k := n i − n j . Since the origin a fixed pointand d F is diagonal matrix with eigenvalues equal to and 2, there exists a neighborhood U ofthe origin that contains no periodic points of order less than or equal to k . Since the sequence( P n ) oscillates, there exists a subsequence ( P m ℓ ) of ( P n ) such that P m ℓ → z ∈ U \ { } . Butthen F m ℓ − n j ( P n i ) = F n i − n j ( P m ℓ ) → F k ( z ) = z, which contradicts F m ℓ − n j ( P n j ) = P m ℓ → z .This completes the proof of the existence of an oscillating wandering domain.Next we will see how we can use the sequence ( H k ) to determine the complex structure ofthe Fatou component F . Theorem 7. Let F and ( β n k ) k ≥ be as above. If the sequence of automorphisms ( H k ) k ≥ satisfies k H k ( z ) k ≤ β kn k k z k on B for all k ≥ then the oscillating wandering Fatou component F is biholomorphic to Ω H .Proof. Since we have (5) the Lemma 5 implies that the basins Ω H = S k ≥ H − k, ( B ) andΩ G = S k ≥ G − k, ( B ) are biholomorphic. Next observe that Ω F = Φ (Ω G ), hence Ω F is alsobiholomorphic to Ω H . It remains to prove that Ω F = F .This will be done using the plurisubharmonic method which was introduced in [ABFP18]and further developed in [ABTP19, Theorem 7].First observe that by taking smaller ε k ( H , . . . , H k ) if necessary we may assume that k G k ( z ) k ≤ k H k ( z ) k for all z ∈ B and all k ≥ k ( z ) := log k F n k ( z ) − P n k k− k log β n k . Let Ψ = lim sup j →∞ Ψ j on F and let Ψ ⋆ be its upper semi-continuous regularization (see[Kl91]), hence Ψ ⋆ is a plurisubharmonic function on F . Since the sequence ( P n k ) is bounded,it follows that for all compact subsets K ⊂ F , we have k F n k ( z ) − P n k k → 0. This impliesthat Ψ ≤ F , and hence Ψ ⋆ ≤ F .Recall that Ω F = Φ (Ω G ) and observe thatΨ k ( z ) = log k Φ n k ◦ G k ◦ . . . ◦ G ◦ Φ − ( z ) − P n k k− k log β n k = log k G k ◦ . . . ◦ G ◦ Φ − ( z ) k + log β n k − k log β n k = log k G k ◦ . . . ◦ G ◦ Φ − ( z ) k− k log β n k − k Given any z ∈ F we know that the sequence G k ◦ . . . ◦ G ◦ Φ − ( z ) is bounded. Inparticular G k ◦ . . . ◦ G ◦ Φ − ( z ) ∈ B (0 , 1) for some k ≥ z ∈ Ω F . This impliesthat Ψ( z ) = Ψ ⋆ ( z ) = 0 on F \ Ω F . n the other hand, let z ∈ Ω F , and let k ≥ z k := G k − ◦ . . . ◦ G ◦ Φ − ( z ) ∈ B (0 , k ( z ) = log k G k ( z j ) k− k log β n k − k ≤ log 2 k H k ( z j ) k− k log β n k − k ≤ log 2 + k log β n k − k log β n k − k ≤ − − k log β n k − k . It follows that Ψ( z ) ≤ − z ∈ Ω F , which implies that Ψ ⋆ ( z ) ≤ − z ∈ Ω F .Since F is open and connected, it follows from the maximum principle for plurisubharmonicfunctions that F \ Ω F must be empty, which completes the proof. (cid:3) Example 1. For H k ( z ) = β n k z it is easy to see that Ω H = C m and therefore by Theorem7 the oscillating wandering Fatou component F is biholomorphic to C m . Example 2. Let H k ( z , . . . , z m ) = (( z ) d k + 2 − d k ··· d z m , − d k ··· d z , . . . , − d k ··· d z m − )where integers d k > k H k ( z ) k ≤ β kn k k z k on B for all k ≥ 1. By [ABTP19,Proposition 3], which is a slightly modified version of [Fo04, Theorem 1.4.] we know that Ω H is a Short C m , and therefore by Theorem 7 the oscillating wandering Fatou component F isalso a Short C m . Since such a domain supports a non-constant bounded plurisubharmonicfunction this implies that F is not biholomorphic to C m .4. Escaping wandering Ball In this section we prove Theorem 1 using the following proposition. Proposition 8. There exists a sequence ( F k ) k ≥ of holomorphic automorphisms of C m , dis-joint sequences of points ( P n ) n ≥ , ( T jn ) j ≥ ,n ≥ , ( S j ) j ≥ and sequences positive real numbers ( R k ) k ≥ ր ∞ and ( r k ) k ≥ ր ∞ , such that the following properties are satisfied:(a) B (0 , r k − ) ⊂⊂ F k ( B (0 , R k )) for all k ≥ ,(b) k F k − F k − k B (0 ,R k − ) ≤ − k for all k ≥ ,(c) k P k k > R k for all k ≥ ,(d) F jk ( B ( P , ⊂⊂ B ( P j , , for all ≤ j ≤ k and all k ≥ .(e) points T j accumulate densely on the b B ( P , .(f ) F k ( T jn ) = T jn +1 for all ≤ j ≤ k and all ≤ n ≤ j − (g) T jj ∈ B ( S j , for all j ≥ .(h) F k ( S j ) = S j for all ≤ j ≤ k .(i) k F k ( z ) − S j k ≤ k k +1) k z − S j k on B ( S j , for all ≤ j ≤ k . Before proving this proposition, let us show that it implies the existence of an escapingwandering Fatou component which is the unit ball.4.1. Proof of Theorem 1. Let ( F k ) k ≥ be a sequence of holomorphic automorphisms of C m given by Proposition 8. This sequence converges uniformly on compacts to a holomo-prhic automorphism F of C m . Moreover there exist disjoint sequences of points ( P n ) n ≥ ,( T j ) j ≥ , ( S j ) j ≥ and strictly increasing sequence of positive real numbers ( R k ) k ≥ such thatthe following holds:(1) k P k k > R k for all k ≥ F k ( B ( P , ⊂⊂ B ( P k , 2) for all k ≥ F ( S j ) = S j and k F ( z ) − S j k ≤ k z − S j k on B ( S j , 1) for all j ≥ T j accumulate densely on the b B ( P , 1) and F k ( T j ) → S j as k → ∞ ,Since the Euclidean diameter of F j ( B ( P , j ≥ B ( P , F . Assume that F = B ( P , F is an openset there exists j > T j ∈ F . Since F contains B ( P , 1) whose orbit eventuallyleaves every compact set, the same must hold for any point in F . But we know that for T j we have F k ( T j ) → S j as k → ∞ which brings us to the contradiction. (cid:3) Proof of Proposition 8. We prove this proposition by induction on k . We start theinduction by letting F = id, r = 1 and R = 2. We define K = B (0 , R ). Finally wechoose a point P satisfying k P k > R + 3 and a sequence ( T j ) j ≥ ⊂ B ( P , \ B ( P , 1) whichaccumulates densely on b B ( P , 1) and for which the sequence of distances k T j − P k → a )—( i ) are satisfied for k = 0.Let us suppose that conditions ( a )—( i ) hold for certain k , and proceed with the construc-tions satisfying the conditions for k + 1.The sets K k and B ( P k , 2) are disjoint and their union is polynomially convex. Observethat B (0 , R k ) ⊆ K k . Next choose a point S k +1 so that the ball B ( S k +1 , 1) is disjoint from thesets K k , F k ( K k ) and B ( P k , 2) and so that both sets B ( S k +1 , ∪ K k ∪ B ( P k , 2) and B ( S k +1 , ∪ F k ( K k ) are polynomially convex. Choose R k +1 > R k + 1 so that B ( S k +1 , ∪ K k ∪ F k ( K k ) ∪ B ( P k , ⊂ B (0 , R k +1 )and point P k +1 such that B ( P k +1 , 2) and B (0 , R k +1 ) are disjoint. Finally choose 1 < ρ k +1 < k T k +20 − P k < ρ k +1 < k T k +10 − P k and define a starshapelike compact set W := F kk ( B ( P , ρ k +1 )) ⊂ B ( P k , . In the terminology of Theorem 4 we define A := K k , A := B ( S k +1 , , A := W, A := T k +1 k . (6)It follows from our construction that all these sets are pairwise disjoint and that their unionis polynomially convex. Also note that all these sets are all starshapelike. Next we define q ( z ) := F k ( z ) on A , q ( z ) := ( k +1)2( k +2)+1 ( z − S k +1 ) + S k +1 on A , q ( z ) = z − P k + P k +1 on A and q ( z ) = z − T k +1 k + Q + on A . Observe that their images B j := q j ( A j ) where1 ≤ j ≤ B ⊂ B ( P k +1 , B ( S j , 1) for 1 ≤ j ≤ k are contained inthe compact set K k . By Theorem 4 there exists an automorphism g k +1 such that(1) k F k − g k +1 k ≤ δ k on K k (2) k q − g k +1 k ≤ δ k on W .(3) g k +1 ( S j ) = S j for all 1 ≤ j ≤ k + 1(4) k g k +1 ( z ) − S k +1 k ≤ k +12( k +2)+ k +1 k z − S k +1 k on B ( S k +1 , g k +1 ( T jn ) = F k ( T jn ) for all 1 ≤ j ≤ k and 0 ≤ n ≤ j − g k +1 ( T k +1 n ) = T k +1 n +1 for all 0 ≤ n < k where T k +1 n := F nk ( T k +10 )(7) T k +1 k +1 := g k +1 ( T k +1 k ) ∈ B ( S k +1 , \ B ( S k +1 , )where we have chosen δ k ≤ k small enough such that:(a) g jk +1 ( B ( P , ρ k +1 )) ⊂⊂ B ( P j , ≤ j ≤ k + 1 b) k g k +1 ( z ) − S j k ≤ k +12( k +2)+ k +1 k z − S j k on B ( S j , 1) for all 1 ≤ j ≤ k ,At this point the automorphism g k +1 already satisfies properties ( b )–( i ) and we continuesimilarly as in the proof of Proposition 6. We choose r k +1 > r k + 1 so that g k +1 ( B (0 , R k +1 )) ⊂⊂ B (0 , r k +1 ) . Since compact sets B ( P k +1 , 2) and B (0 , R k +1 ) are disjoint starshapelike domains whose unionis polynomially convex the same holds for their images U := g k +1 ( B ( P k +1 , , V := g k +1 ( B (0 , R k +1 )) . Let Q ′ ∈ C be a point such that the ball B ( Q ′ , θ ) lies in the complement of B (0 , r k +1 ). Alsolet ψ be a linear map satisfying ψ ( U ) ⊂ B ( Q ′ , θ ) . By Theorem 4 there exists an automorphism h such that(1) k h − id k V ≤ δ ′ k ,(2) k h − ψ k U ≤ δ ′ k ,(3) h ( S j ) = S j for all 0 < j ≤ k + 1,(4) h ( T jn ) = T jn for all 1 ≤ j ≤ k + 1 and 1 ≤ n ≤ j where we have chosen δ ′ k ≤ k small enough such that(i) ( h ◦ g k +1 ) j ( B ( P , ρ k +1 )) ⊂⊂ B ( P j , ≤ j ≤ k + 1(ii) k ( h ◦ g k +1 )( z ) − S j k ≤ k +12( k +2) k z − S j k on B ( S j , 1) for all 1 ≤ j ≤ k + 1,Finally define K k +1 := F − k +1 ( B (0 , r k +1 )) and observe that B (0 , R k +1 ) ⊂ K k +1 . Since F k +1 ( B ( P k +1 , ⊂⊂ B ( Q ′ , , and since the set B (0 , r k +1 ) ∪ B ( Q ′ , θ ) is polynomially convex it follows that K k +1 and B ( P k +1 , 2) are disjoint and their union is polynomially convex.It is immediate that properties ( c )—( i ) are satisfied for the ( k + 1)-th step. The properties( a ) and ( b ) can be verified by following the last paragraph of the proof of Proposition 6verbatim. This concludes the inductive step. (cid:3) Remark: We believe that by a slight modification of the above proof, in particular bychoosing different map q , one can construct examples of wandering balls with different interiordynamics, as it was recently done for transcendental functions in dimension one [BEGRS19].5. Oscillating wandering Ball In the previous two sections we have seen how the tools of Anders´en–Lepert theory can beused to construct various examples of oscillating wandering domains and also of the escapingwandering ball. In this section we will see that by combining these two constructions we canconstruct an oscillating wandering ball. The poof of Theorem 2 is based on the followingproposition which is a hybrid between Proposition 8 and Proposition 6. Proposition 9. There exists a sequence ( F k ) k ≥ of holomorphic automorphisms of C m , dis-joint sequences of points ( P n ) n ≥ , ( T jn ) j ≥ ,n ≥ , ( S j ) j ≥ with ( S j ) j being bounded away fromthe origin, sequences positive real numbers ( β n ) n ≥ ց , ( τ n ) n ≥ ց , ( R k ) k ≥ ր ∞ , ( r k ) k ≥ ր ∞ , strictly increasing sequences of integers ( n k ) k ≥ and ( N k ) k ≥ satisfying n = 0 and N k − ≤ n k ≤ N k , and such that the following properties are satisfied:(a) B (0 , r k − ) ⊂⊂ F k ( B (0 , R k )) for all k ≥ , b) k F k − F k − k B (0 ,R k − ) ≤ − k for all k ≥ ,(c) F k ( P n ) = P n +1 for all ≤ n < N k ,(d) k P n k k ≤ k for all k ≥ ,(e) k P N k k > R k for all k ≥ (f ) for all k ≥ we have β j < k +1 for N k < j ≤ N k +1 .(g) F jk ( B ( P , β n )) ⊂⊂ B ( P j , β j ) , for all ≤ j ≤ N k and all k ≥ .(h) points T j accumulate densely on the b B ( P , .(i) F k ( T jn ) = T jn +1 for all ≤ j ≤ k and all ≤ n ≤ N j − (j) T jN j − +1 ∈ B ( S j , τ j ) for all j ≥ .(k) F k ( S j ) = S j for all ≤ j ≤ k .(l) k F k ( z ) − S j k ≤ k k +1) k z − S j k on B ( S j , τ j ) for all ≤ j ≤ k . In the proposition above properties ( a ) and ( b ) imply that the sequence F k convergesuniformly on compacts to an automorphism F . Properties ( c )—( g ) ensure the existence of anoscillating wandering domain for F and properties ( h )—( l ) ensure that the F -orbit of everypoint T j converges to an attracting fixed point S j .Before proving this proposition, let us show how it is used to prove our main theorem.5.1. Proof of Theorem 2. Let ( F k ) k ≥ be a sequence of holomorphic automorphisms of C m given by Proposition 9. This sequence converges uniformly on compacts to a holomo-prhic automorphism F of C m . Moreover there exists a disjoint sequences of points ( P n ) n ≥ ,( T jn ) j ≥ ,n ≥ , ( S j ) j ≥ , sequences positive real numbers ( β n ) n ≥ ց 0, ( τ n ) n ≥ ց 0, strictlyincreasing sequences of integers ( n k ) k ≥ and ( N k ) k ≥ such that the following holds:(1) F j ( P ) = P j for all j ≥ P n k → P N k → ∞ as k → ∞ ,(3) F j ( B ( P , ⊂⊂ B ( P j , β j ) for all j ≥ F ( S j ) = S j and k F ( z ) − S j k ≤ on B ( S j , τ j ) for all j ≥ T j accumulate densely on the b B ( P , 1) and F k ( T j ) → S j as k → ∞ for all j ≥ F j ( B ( P , j ≥ B ( P , F . Assume that F = B ( P , F is an openset there exists j > T j ∈ F . Since F contains B ( P , 1) we have F n k ( z ) → F as k → ∞ but on the other hand we have F n k ( T j ) → S j = 0 as k → ∞ which brings us to acontradiction. (cid:3) Proof of Proposition 9. We prove this proposition by induction on k . We start theinduction by letting F ( z , . . . , z m ) = ( z , . . . , z k , z k +1 , . . . , z m ) for some 1 ≤ k < m . Let r = 1 and let R > B (0 , r ) ⊂⊂ F ( B (0 , R )). Define K = B (0 , R ), n = N = 0, β = 2, and choose any P with k P k > R + 3. Finally choose a sequence( T j ) j ≥ ⊂ B ( P , \ B ( P , 1) which accumulates densely on b B ( P , 1) and for which the se-quence of distances k T j − P k → a )—( l ) are satisfied for k = 0.Let us suppose that conditions ( a )—( l ) hold for certain k , and proceed with the construc-tions satisfying the conditions for k + 1. irst let R k +1 > k P N k k + 1 such that K k ⊂ B (0 , R k +1 ), where K k ∪ B ( P N k , β N k ) ispolynomially convex and B (0 , R k ) ⊆ K k . By the λ -Lemma (see [PdM82, Lemma 7.1]) thereexist a finite F k orbit ( Q j ) − ≤ j ≤ M , t.i. F k ( Q j − ) = Q j for 0 ≤ j ≤ M , such that:(1) k Q j k < R k +1 for all − ≤ j < M ,(2) k Q M k > R k +1 ,(3) k Q ℓ k < k +1 for some 0 < ℓ < M .By increasing R k +1 if necessary, we can choose 0 < θ ≤ η < k +1 so that:(i) the ball B ( Q M , θ ) is disjoint from B (0 , R k +1 ),(ii) B ( Q , θ ) ⊂⊂ F k ( B ( Q − , η ))(iii) the balls B ( P N k , β N k ) , B ( Q , θ ) , B ( Q M , θ ) , B ( Q − , η ) (7)are pairwise disjoint, and disjoint from the set L := K k ∪ [ r k + 1 such that g k +1 ( B (0 , R k +1 )) ⊂⊂ B (0 , r k +1 ) . Since the compact sets B ( Q M , θ ) and B (0 , R k +1 ) are disjoint starshapelike domains whoseunion is polynomially convex the same holds for their images U := g k +1 ( B ( Q M , θ )) , V := g k +1 ( B (0 , R k +1 )) . Let Q ′ ∈ C be a point such that the ball B ( Q ′ , θ ) lies in the complement of B (0 , r k +1 ).Moreover let ψ be a linear map satisfying ( ψ ◦ g k +1 )( Q M ) = Q ′ and ψ ( U ) ⊂ B ( Q ′ , θ ) . By Theorem 4 there exists an automorphism h such that(1) k h − id k V ≤ δ ′ k ,(2) k h − ψ k U ≤ δ ′ k ,(3) h ( Q j ) = Q j for all 0 < j ≤ M ,(4) ( h ◦ g k +1 )( Q M ) = Q ′ (5) h ( S j ) = S j for all 0 < j ≤ k + 1,(6) h (0) = 0, d h = id , h ( P j ) = P j for all 1 ≤ j ≤ P N k ,(7) h ( T jn ) = T jn for all 1 ≤ j ≤ k + 1 and 1 ≤ n ≤ N j − + 1where we have chosen δ ′ k ≤ k small enough such that(i) ( h ◦ g k +1 ) j ( B ( P , ρ k +1 )) ⊂⊂ B ( P j , β j ), for all 1 ≤ j ≤ N k (ii) ( h ◦ g k +1 ) N k + j ( B ( P , ρ k +1 )) ⊂⊂ B ( Q j , θ ) , for all 0 < j ≤ M (iii) ( h ◦ g k +1 ) N k + ℓ ( B ( P , ρ k +1 )) ⊂⊂ B ( Q ℓ , s ℓ ) , iv) k ( h ◦ g k +1 )( z ) − S j k ≤ k +12( k +2) k z − S j k on B ( S j , τ j ) for all 1 ≤ j ≤ k + 1,Similarly as in the proof of Proposition 6 we define F k +1 := h ◦ g k +1 , so that the sequencesof points ( P j ) ≤ j ≤ N k , ( Q j ) In Theorem 1 and Theorem 2 the term unit ball can be replacedby any bounded simply connected regular open set Ω ⊂ C m whose closure is polynomiallyconvex.Here we explain how can one adapt the proof of Proposition 9 to include also these domainsand note that similarly can be done for the proof of Proposition 8.Since each compact polynomially convex set admits a basis of Stein neighborhoods thatare Runge in C m , there exists a decreasing sequence of compact polynomially convex neigh-borhoods ( U k ) of Ω.In the above proof we simply replace the role of B ( P , 1) with Ω and B ( P , ρ k +1 ) with U k +1 and choose a point ˜ P ∈ Ω. Recall that the automorphism Φ defined in (9) and usedin (10) maps B ( P , ρ k +1 ) into B (0 , 1) with Φ ( P ) = 0. We replace this with a automorphism˜Φ which maps U k +1 into B (0 , 1) and satisfies ˜Φ ( ˜ P ) = 0.By choosing θ sufficiently small we may assume that for every 0 < j < M the ball B ( Q j , θ )is either contained in K k or else they are disjoint, hence set L defined in (8) is a union ofstarshapelike domains. Finally in (11) we get finitely many sets A j , so that all but one arestarshapelike. The only one that might not be starshapelike is the set W = F N k ( U k +1 ). Therest of the proof follows verbatim. References [ABDPR16] M. Astorg, X. Buff, R. Dujardin, H. Peters, J. Raissy. A two-dimensional polynomial mappingwith a wandering Fatou component . Ann. of Math. (2), 184(1): 263 313, 2016.[ABFP18] L. Arosio, Benini, J. E. Fornæss, H. Peters, Dynamics of transcendental H´enon maps , Math. Ann. (2018) no. 1-2, 853–894.[ABTP19] L. Arosio, L. Boc Thaler, H. 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Translated fromthe Portuguese by A. K. Manning. Springer-Verlag, New York-Berlin, (1982).[St07] Stout, E. L.: Polynomial convexity. Progress in Mathematics, 261. Birkh¨auser Boston, Inc., Boston,MA (2007) L. Boc Thaler: Faculty of Education, University of Ljubljana, SI–1000 Ljubljana, Slovenia. E-mail address : [email protected]@pef.uni-lj.si