Canonical models for the forward and backward iteration of holomorphic maps
aa r X i v : . [ m a t h . C V ] A p r CANONICAL MODELS FOR THE FORWARD AND BACKWARDITERATION OF HOLOMORPHIC MAPS
LEANDRO AROSIO
Abstract.
We prove the existence and the essential uniqueness of canonical models for theforward (resp. backward) iteration of a holomorphic self-map f of a cocompact Kobayashihyperbolic complex manifold, such as the ball B q or the polydisc ∆ q . This is done perform-ing a time-dependent conjugacy of the dynamical system ( f n ), obtaining in this way a non-autonomous dynamical system admitting a relatively compact forward (resp. backward) orbit,and then proving the existence of a natural complex structure on a suitable quotient of the directlimit (resp. subset of the inverse limit). As a corollary we prove the existence of a holomorphicsolution with values in the upper half-plane of the Valiron equation for a holomorphic self-mapof the unit ball. Contents
Introduction 1
Part 1. Forward iteration
71. Preliminaries 72. Non-autonomous iteration 83. Autonomous iteration 124. The unit ball 15
Part 2. Backward iteration
Introduction
In order to study the forward or backward iteration of a holomorphic self-map f : X → X of acomplex manifold, it is natural to search for a semi-conjugacy of f with some automorphism of acomplex manifold. The first example of this approach is old as complex dynamics itself: if D ⊂ C Mathematics Subject Classification.
Primary 32H50; Secondary 37F99.
Key words and phrases.
Canonical models; holomorphic iteration.Supported by the ERC grant “HEVO - Holomorphic Evolution Equations” n. 277691. is the unit disc and f : D → D is a holomorphic self-map such that f (0) = 0 and 0 < | f ′ (0) | < h : D → C solving the Schr¨oder equation h ◦ f = f ′ (0) h, and satisfying h ′ (0) = 1. Clearly h gives a semi-conjugacy between f and the automorphism z f ′ (0) z of C . Notice that ∪ n ≥ f ′ (0) − n h ( D ) = C .We call semi-model for f a triple (Λ , h, ϕ ), where Λ is a complex manifold called the basespace , h : X → Λ is a holomorphic mapping called the intertwining mapping and ϕ : Λ → Λ isan automorphism, such that the following diagram commutes: X f / / h (cid:15) (cid:15) X h (cid:15) (cid:15) Λ ϕ / / Λ , and Λ = S n ≥ ϕ − n ( h ( X )). A model for f is a semi-model (Λ , h, ϕ ) such that the intertwiningmapping h is univalent on an f -absorbing domain, that is, a domain A such that f ( A ) ⊂ A andsuch that every orbit of f eventually lies in A .There is a “dual” way of semi-conjugating f with an automorphism: we call pre-model for f a triple (Λ , h, ϕ ), where Λ is a complex manifold called the base space , h : Λ → X is aholomorphic mapping called the intertwining mapping and ϕ : Λ → Λ is an automorphism, suchthat the following diagram commutes: Λ ϕ / / h (cid:15) (cid:15) Λ h (cid:15) (cid:15) X f / / X. We refer to [4, 3, 2] for a brief history and recent developments in the theories of semi-modelsand pre-models. We recall that semi-models and pre-models, besides giving informations on theiteration of the self-map f , can also be fruitfully applied to the study of composition operators[7, 15, 21, 24] and of commuting self-maps [14, 8, 12].We now need to recall some definitions and results for a holomorphic self-map f of the unitdisc D ⊂ C . A point ζ ∈ ∂ D is a boundary regular fixed point if ∠ lim z → ζ f ( z ) = ζ, where ∠ limdenotes the non-tangential limit, and if λ := lim inf z → ζ − | f ( z ) | − | z | < + ∞ . The number λ ∈ (0 , + ∞ ) is called the dilation of f at ζ . The point ζ is repelling if λ >
1. Theclassical Denjoy–Wolff theorem states that if f admits no fixed point z ∈ D , then there existsa boundary regular fixed point p ∈ ∂ D with dilation λ ≤ f n ) converges to theconstant map p uniformly on compact subsets. The self-map f is called hyperbolic if λ <
1. Wedenote by H ⊂ C the upper half-plane. ANONICAL MODELS 3
We are interested in the following examples of semi-models and pre-models in D , given respec-tively by Valiron [29] and by Poggi-Corradini [25]. Both examples can be seen as the solutionof a generalized Schr¨oder equation at the boundary of the disc. Theorem 0.1 (Valiron) . Let f : D → D be a hyperbolic holomorphic self-map with dilation λ < at its Denjoy–Wolff point. Then there exists a model ( H , h, z λ z ) for f . Theorem 0.2 (Poggi-Corradini) . Let f : D → D be a holomorphic self-map and let ζ be aboundary repelling fixed point with dilation λ > . Then there exists a pre-model ( H , h, z λ z ) for f . A proof of the essential uniqueness of the intertwining mapping in Theorem 0.1 was given byBracci–Poggi-Corradini [11], and Poggi-Corradini [25] proved that the intertwining mapping inTheorem 0.2 is essentially unique.These two results were generalized to the unit ball B q ⊂ C q (for a definition of dilation,hyperbolic self-maps, Denjoy–Wolff point and boundary repelling points in the ball, see Sections4 and 8). Bracci–Gentili–Poggi-Corradini [10] studied the case of a hyperbolic holomorphic self-map f : B q → B q with dilation λ < p ∈ ∂ B q , and, assuming someregularity at p , they proved the existence of a one-dimensional semi-model ( H , h, z λ z ) for f (for other results about semi-models for hyperbolic self-maps, see [9, 21, 6]).Ostapyuk [23] studied the case of a holomorphic self-map f : B q → B q with a boundaryrepelling fixed point ζ ∈ ∂ B q with dilation λ >
1, and, assuming that ζ is isolated from otherboundary repelling fixed points with dilation less or equal than λ , she proved the existence of aone-dimensional pre-model ( H , h, z λ z ) for f .Theorems 0.1 and 0.2 were generalized respectively by Bracci and the author [4] and by theauthor [3] to the case of a univalent self-map f : X → X of a cocompact Kobayashi hyperboliccomplex manifold (such as the unit ball B q or the unit polydisc ∆ q ). The approach used isgeometric, much in the spirit of the work of Cowen [13] for the forward iteration in the unit disc.We first consider the forward iteration case. Let k denote the Kobayashi distance. Noticethat if ( z n := f n ( z )) is a forward orbit, then for all fixed m ≥ k X ( z n , z n + m )) n ≥ is non-increasing. The limit s m ( z ) := lim n →∞ k X ( z n , z n + m ) is called the forward m -step at z . The divergence rate of a self-map is a generalization introduced in [4] of the dilation at theDenjoy–Wolff point of a holomorphic self-map of B q . Theorem 0.3 (A.–Bracci) . Let X be Kobayashi hyperbolic and cocompact and let f : X → X bea univalent self-map. Then there exists an essentially unique model (Ω , σ, ψ ) . Moreover, thereexists a holomorphic retract Z of X , a surjective holomorphic submersion r : Ω → Z , and anautomorphism τ : Z → Z with divergence rate c ( τ ) = c ( f ) = lim m →∞ s m ( x ) m , x ∈ X, (0.1) such that ( Z, r ◦ σ, τ ) is a semi-model for f , called a canonical Kobayashi hyperbolic semi-model .Moreover, the semi-model ( Z, r ◦ σ, τ ) satisfies the following universal property. If (Λ , h, ϕ ) is asemi-model for f such that Λ is Kobayashi hyperbolic, then there exists a surjective holomorphicmapping η : Z → Λ such that the following diagram commutes: L. AROSIO X h / / r ◦ σ & & ◆◆◆◆◆◆◆◆◆◆◆◆◆ f (cid:15) (cid:15) Λ ϕ (cid:15) (cid:15) Z η > > ⑦⑦⑦⑦⑦⑦⑦⑦ τ (cid:15) (cid:15) X h / / r ◦ σ & & ◆◆◆◆◆◆◆◆◆◆◆◆◆ Λ Z. η > > ⑦⑦⑦⑦⑦⑦⑦ In particular, if X = B q and f is hyperbolic with dilation λ < Z is biholomorphic to a ball B k with 1 ≤ k ≤ q , and the automorphism τ is hyperbolicwith dilation λ at its Denjoy-Wolf point. As a corollary Theorem 0.3 yields the existence ofa semi-model ( H , ϑ, z λ z ) for f , hence ϑ : B q → H is a holomorphic solution of the Valironequation ϑ ◦ f = 1 λ ϑ. (0.2)Now we recall the backward iteration case. A backward orbit is a sequence β := ( y n ) in X suchthat f ( y n +1 ) = y n for all n ≥
0. Notice that if ( y n ) is a backward orbit, then for all fixed m ≥ k X ( y n , y n + m )) n ≥ is non-decreasing. The limit σ m ( β ) := lim n →∞ k X ( y n , y n + m ) iscalled the backward m -step of β . A backward orbit β has bounded step if σ ( β ) < + ∞ . Theorem 0.4 (A.) . Let X be Kobayashi hyperbolic and cocompact and let f : X → X be aunivalent self-map. Let β := ( y n ) be a backward orbit for f with bounded step. Then thereexists a holomorphic retract Z of X , an injective holomorphic immersion ℓ : Z → X , and anautomorphism τ : Z → Z with divergence rate c ( τ ) = lim m →∞ σ m ( β ) m , (0.3) such that ( Z, ℓ, τ ) is a pre-model for f , called a canonical pre-model associated with [ y n ] .Moreover ( Z, ℓ, τ ) satisfies the following universal property. If (Λ , h, ϕ ) is a pre-model for f such that for some (and hence for any) w ∈ Λ , the non-decreasing sequence ( k X ( h ( ϕ − n ( w )) , y n )) n ≥ is bounded, then there exists an injective holomorphic mapping η : Λ → Z such that the followingdiagram commutes: Λ h / / η ❅❅❅❅❅❅❅❅ ϕ (cid:15) (cid:15) X f (cid:15) (cid:15) Z ℓ ♣♣♣♣♣♣♣♣♣♣♣♣♣ τ (cid:15) (cid:15) Λ h / / η ❅❅❅❅❅❅❅ XZ. ℓ ♣♣♣♣♣♣♣♣♣♣♣♣♣ ANONICAL MODELS 5
In particular, if X = B q and the backward orbit ( y n ) converges to a boundary repelling fixedpoint ζ ∈ ∂ B q with dilation λ >
1, then Z is biholomorphic to a ball B k with 1 ≤ k ≤ q , and theautomorphism τ is hyperbolic with dilation µ ≥ λ at its unique boundary repelling fixed point.In this paper we generalize Theorems 0.3 and 0.4 to non-necessarily univalent holomorphicself-maps f : X → X , and then we apply our results to the case of the unit ball B q . Our proofsunderline the strong duality between the forward case and the backward case.In the first part of the paper we prove Theorem 3.6, which generalizes Theorem 0.3. Let(Ω , Λ n : X → Ω) be the direct limit of the sequence ( f n : X → X ). Consider the equivalencerelation ∼ on Ω, where [( x, n )] , [( y, u )] ∈ Ω are equivalent by ∼ if and only if k X ( f m − n ( x ) , f m − u ( y )) m →∞ −→ . The bijective self-map ψ : Ω → Ω defined by [( x, n )] [( f ( x ) , n )] satisfies ψ ◦ Λ = Λ ◦ f andpasses to the quotient inducing a bijective self-map ˆ ψ : Ω / ∼ → Ω / ∼ satisfying X f / / ˆΛ (cid:15) (cid:15) X ˆΛ (cid:15) (cid:15) Ω / ∼ ˆ ψ / / Ω / ∼ , where ˆΛ := π ∼ ◦ Λ . A natural candidate for a canonical Kobayashi hyperbolic semi-modelfor f would be the triple (Ω / ∼ , Λ , ˆ ψ ). Indeed, by the universal property of the direct limit, if(Λ , h, ϕ ) is a semi-model for f such that Λ is Kobayashi hyperbolic, then there exists a mapping η : Ω / ∼ → Λ which makes the following diagram commute: X h / / ˆΛ ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖ f (cid:15) (cid:15) Λ ϕ (cid:15) (cid:15) Ω / ∼ η = = ④④④④④④④④ ˆ ψ (cid:15) (cid:15) X h / / ˆΛ ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖ ΛΩ / ∼ . η = = ④④④④④④④④ We have to show that Ω / ∼ can be endowed with a suitable complex structure. If f is univalent,then it follows from the proof of Theorem 0.3 that the direct limit Ω admits a natural complexstructure which passes to the quotient to a complex structure on Ω / ∼ (see [4]). The problem inthe non-univalent case is that Ω may not admit a natural complex structure. Rather surprisingly,even if Ω does not, the quotient set Ω / ∼ can always be endowed with a complex structure whichmakes it biholomorphic to a holomorphic retract of X . We prove this by conjugating ( f n ) toa non-autonomous holomorphic forward dynamical system ( ˜ f n,m : X → X ) m ≥ n ≥ which admitsa relatively compact forward orbit. This orbit is used to prove the existence of a holomorphic L. AROSIO retract Z of X and a family of holomorphic mappings ( α n : X → Z ) satisfying α m ◦ ˜ f n,m = α n , ∀ ≤ n ≤ m. By the universal property of the direct limit there exists a mapping Φ : Ω → Z which induces abijection ˆΦ : Ω / ∼ → Z , which pulls back the desired complex structure to Ω / ∼ . Formula (0.1)for the divergence rate of τ is a consequence of the fact that the Kobayashi distance on Ω / ∼ admits a description in terms of the forward iteration of f .In the second part of the paper, we consider the backward iteration of f : X → X and weprove Theorem 7.5, which generalizes Theorem 0.4. Let (Θ , V n : Θ → X ) be the inverse limit ofthe sequence ( f n : X → X ). Let ( y n ) be a backward orbit with bounded step and let [ y n ] ⊂ Θbe the subset consisting of the backward orbits ( z n ) ∈ Θ such that the non-decreasing sequence( k X ( z n , y n )) n ≥ is bounded. The bijective self-map ψ : Θ → Θ defined by ( z , z , z , . . . ) [( f ( z ) , z , z , . . . )] satisfies ψ ([ y n ]) = [ y n ], and the following diagram commutes:[ y n ] ψ | [ yn ] / / V (cid:15) (cid:15) [ y n ] V (cid:15) (cid:15) X f / / X. A natural candidate for a canonical pre-model for f associated with [ y n ] would be the triple([ y n ] , V , ψ | [ y n ] ). Indeed, by the universal property of the inverse limit, if (Λ , h, ϕ ) is a pre-model for f such that for some (and hence for any) w ∈ Λ the non-decreasing sequence( k X ( h ( ϕ − n ( w )) , y n )) n ≥ is bounded, then there exists a mapping η : Λ → [ y n ] which makesthe following diagram commute: Λ h / / η ! ! ❇❇❇❇❇❇❇❇ ϕ (cid:15) (cid:15) X f (cid:15) (cid:15) [ y n ] V ♣♣♣♣♣♣♣♣♣♣♣♣♣ ψ | [ yn ] (cid:15) (cid:15) Λ h / / η ! ! ❇❇❇❇❇❇❇❇ X [ y n ] . V ♣♣♣♣♣♣♣♣♣♣♣♣♣ We have to show that [ y n ] can be endowed with a suitable complex structure. If f is univalent,then V : Θ → X is injective, and it follows from the proof of Theorem 0.4 that the image V ([ y n ]) is an injectively immersed complex submanifold of X which is biholomorphic to aholomorphic retract of X . In the non-univalent case the mapping V : Θ → X is no longerinjective, but the subset [ y n ] can however be endowed with a natural complex structure whichmakes it biholomorphic to a holomorphic retract of X . We prove this by conjugating ( f n ) to anon-autonomous holomorphic backward dynamical system ( ˜ f n,m : X → X ) m ≥ n ≥ which admitsa relatively compact backward orbit. This orbit is used to prove the existence of a holomorphic ANONICAL MODELS 7 retract Z of X and a family of holomorphic mappings ( α n : Z → X ) satisfying˜ f n,m ◦ α m = α n , ∀ ≤ n ≤ m. By the universal property of the inverse limit there exists an injective mapping Φ : Z → Θ,which pushes forward the desired complex structure to its image Φ( Z ) = [ y n ]. Formula (0.3) forthe divergence rate of τ is a consequence of the fact that the Kobayashi distance of [ y n ] admitsa description in terms of the backward iteration of f . Part Forward iteration Preliminaries
Definition 1.1.
Let X be a complex manifold. We call forward (non-autonomous) holomorphicdynamical system on X any family ( f n,m : X → X ) m ≥ n ≥ of holomorphic self-maps such thatfor all m ≥ u ≥ n ≥
0, we have f u,m ◦ f n,u = f n,m . For all n ≥ f n,n +1 also by f n . A forward holomorphic dynamical system ( f n,m : X → X ) m ≥ n ≥ is called autonomous if f n = f for all n ≥
0. Clearly in this case f n,m = f m − n . Remark 1.2.
Any family of holomorphic self-maps ( f n : X → X ) n ≥ determines a forwardholomorphic dynamical system ( f n,m : X → X ) in the following way: for all n ≥
0, set f n,n = id ,and for all m > n ≥
0, set f n,m = f m − ◦ · · · ◦ f n . Definition 1.3.
Let X be a complex manifold, and let ( f n,m : X → X ) be a forward holomorphicdynamical system. A direct limit for ( f n,m ) is a pair (Ω , Λ n ) where Ω is a set and (Λ n : X → Ω) n ≥ is a family of mappings such thatΛ m ◦ f n,m = Λ n , ∀ m ≥ n ≥ , satisfying the following universal property: if Q is a set and if ( g n : X → Q ) is a family ofmappings satisfying g m ◦ f n,m = g n , ∀ m ≥ n ≥ , then there exists a unique mapping Γ : Ω → Q such that g n = Γ ◦ Λ n , ∀ n ≥ . Remark 1.4.
The direct limit is essentially unique, in the following sense. Let (Ω , Λ n ) and( Q, g n ) be two direct limits for ( f n,m ). Then there exists a bijective mapping Γ : Ω → Q suchthat g n = Γ ◦ Λ n , ∀ n ≥ . Remark 1.5.
A direct limit for ( f n,m ) is easily constructed. We define an equivalence relationon the set X × N in the following way: ( x, n ) ≃ ( y, m ) if and only if there exists u ≥ max { n, m } such that f n,u ( x ) = f m,u ( y ). We denote the equivalence class of ( x, n ) by [( x, n )], and we setΩ := X × N / ≃ . We define a family of mappings (Λ n : X → Ω) n ≥ in the following way: for all x ∈ X and n ≥
0, set Λ n ( x ) = [( x, n )]. It is easy to see that (Ω , Λ n ) is a direct limit for ( f n,m ). L. AROSIO
Definition 1.6.
In what follows we will need the following equivalence relation on Ω:[( x, n )] ∼ [( y, u )] iff k X ( f n,m ( x ) , f u,m ( y )) m →∞ −→ . It is easy to see that this is well-defined. We denote by π ∼ : Ω → Ω / ∼ the projection to thequotient.We now introduce a modified version of the direct limit for ( f n,m ) which is more suited forour needs. Definition 1.7.
Let X be a complex manifold and let ( f n,m : X → X ) be a forward holomorphicdynamical system. We call canonical Kobayashi hyperbolic direct limit for ( f n,m ) a pair ( Z, α n )where Z is a Kobayashi hyperbolic complex manifold and ( α n : X → Z ) n ≥ is a family ofholomorphic mappings such that α m ◦ f n,m = α n , ∀ m ≥ n ≥ , which satisfies the following universal property: if Q is a Kobayashi hyperbolic complex manifoldand if ( g n : X → Q ) is a family of holomorphic mappings satisfying g m ◦ f n,m = g n , ∀ m ≥ n ≥ , then there exists a unique holomorphic mapping Γ : Z → Q such that g n = Γ ◦ α n , ∀ n ≥ . Proposition 1.8.
The canonical Kobayashi hyperbolic direct limit is essentially unique, in thefollowing sense. Let ( Z, α n ) and ( Q, g n ) be two canonical Kobayashi hyperbolic direct limits for ( f n,m ) . Then there exists a biholomorphism Γ : Z → Q such that g n = Γ ◦ α n , ∀ n ≥ . Proof.
There exist holomorphic mappings Γ : Z → Q and Ξ : Q → Z such that for all n ≥
0, wehave g n = Γ ◦ α n and α n = Ξ ◦ g n . Thus the holomorphic mapping Ξ ◦ Γ : Z → Z satisfiesΞ ◦ Γ ◦ α n = α n , ∀ n ≥ , By the universal property of the canonical Kobayashi hyperbolic direct limit, this implies thatΞ ◦ Γ = id Z . Similarly, we obtain Γ ◦ Ξ = id Q . (cid:3) Non-autonomous iteration
Let X be a taut complex manifold. Let ( f n,m : X → X ) m ≥ n ≥ be a forward holomorphicdynamical system, and assume that it admits a relatively compact forward orbit ( f ,m ( x )) m ≥ . Remark 2.1.
Let K ⊂ X be a compact subset such that { f ,m ( x ) } m ≥ ⊂ K. It follows that,for all fixed n ≥ f n,m ( K ) ∩ K = ∅ ∀ m ≥ n. (2.1)The sequence of holomorphic self-maps ( f ,m : X → X ) m ≥ is not compactly divergent by(2.1), and since X is taut, there exists a subsequence ( f ,m k ) k ≥ converging uniformly oncompact subsets to a holomorphic self-map α : X → X . The sequence of holomorphic self-maps ( f ,m k : X → X ) k ≥ is not compactly divergent by (2.1), and since X is taut, thereexists a subsequence ( f ,m k ) k ≥ converging to a holomorphic self-map α : X → X . Iterating ANONICAL MODELS 9 this procedure we obtain a family of holomorphic self-maps ( α n : X → X ) satisfying for all m ≥ n ≥ α m ◦ f n,m = α n . (2.2)Notice that for all n ≥ α n ( K ) ∩ K = ∅ . (2.3)Let now ( m k ) k ≥ be a sequence of integers which for all j ≥ m k j ) k j ≥ (such a sequence exists by a classical diagonal argument).The sequence of holomorphic self-maps ( α m k : X → X ) k ≥ is not compactly divergent by(2.3), and since X is taut, there exists a subsequence ( α m h ) h ≥ converging uniformly on compactsubsets to a holomorphic self-map α : X → X . Proposition 2.2.
The holomorphic self-map α : X → X is a holomorphic retraction, and forall n ≥ , α ◦ α n = α n . (2.4) Proof.
Fix n ≥ x ∈ X . Then for all h ≥ m h ≥ n , we have α n ( x ) = α m h ( f n,m h ( x )) h →∞ −→ α ( α n ( x )) . Thus we have, for all h ≥ α ( α m h ( x )) = α m h ( x ) . When h → ∞ , the left-hand side converges to α ( α ( x )), while the right-hand side converges to α ( x ). (cid:3) Remark 2.3.
The image α ( X ) is a closed complex submanifold of X (see e.g. [1, Lemma2.1.28]). Definition 2.4.
We denote α ( X ) by Z . Remark 2.5.
By (2.4), we have α n ( X ) ⊂ Z for all n ≥
0, and by (6.2) we have that α n ( X ) ⊂ α m ( X )for all m ≥ n ≥ , Λ n ) be the direct limit of the directed system ( X, f n,m ). By the universal property ofthe direct limit, there exists a mapping Ψ : Ω → Z such that for all n ≥ α n = Ψ ◦ Λ n . The mapping Ψ is defined in the following way: if [( x, n )] ∈ Ω, then Ψ([( x, n )]) = α n ( x ). Proposition 2.6.
The mapping
Ψ : Ω → Z is surjective, and Ψ([( x, n )]) = Ψ([( y, u )]) if andonly if [( x, n )] ∼ [( y, u )] .Proof. Since α is a retraction, we have α ( z ) = z for all z ∈ Z , that is, α m h ( z ) h →∞ −→ z forall z ∈ Z . Consider the sequence of holomorphic mappings ( α m h | Z : Z → Z ). This sequenceconverges uniformly on compact subsets to id Z , and thus it is eventually injective on compactsubsets of Z . Fix z ∈ Z and let U be a neighborhood of z in Z such that ( α m h | U : U → Z ) iseventually injective. Then the image α m h | U eventually contains z (see e.g. [5, Corollary 3.2]).Hence we obtain that Ψ : Ω → Z is surjective. Assume now that [( x, n )] ∼ [( y, u )]. For all m ≥ max { n, u } , we have that Ψ([( x, n )]) = α m ( f n,m ( x )), and Ψ([( y, u )]) = α m ( f u,m ( y )). We have k X (Ψ([( x, n )]) , Ψ([( y, u )])) ≤ k X ( f n,m ( x ) , f u,m ( y )) m →∞ −→ , which implies Ψ([( x, n )]) = Ψ([( y, u )]).Conversely, assume that Ψ([( x, n )]) = Ψ([( y, u )]). It follows thatlim h →∞ f n,m h ( x ) = lim h →∞ f u,m h ( y ) , and thus lim h →∞ k X ( f n,m h ( x ) , f u,m h ( y )) = 0 . Since the sequence ( k X ( f n,m ( x ) , f u,m ( y ))) m ≥ max { n,u } is non-increasing, we have [( x, n )] ∼ [( y, u )]. (cid:3) Remark 2.7.
It follows from Proposition 2.6 that S n ≥ α n ( X ) = Z , and that Ψ induces abijection ˆΨ : Ω / ∼ → Z . Proposition 2.8.
The pair ( Z, α n ) is a canonical Kobayashi hyperbolic direct limit for ( f n,m ) .Proof. First of all, Z is Kobayashi hyperbolic since it is a submanifold of X . Let Q be aKobayashi hyperbolic complex manifold and let ( g n : X → Q ) be a family of holomorphic map-pings satisfying g m ◦ f n,m = g n , ∀ m ≥ n ≥ . By the universal property of the direct limit, there exists a unique mapping Φ : Ω → Q suchthat g n = Φ ◦ Λ n , ∀ n ≥ . The mapping Φ is defined in the following way: if [( x, n )] ∈ Ω, then Φ([( x, n )]) = g n ( x ). Weclaim that [( x, n )] ∼ [( y, u )] = ⇒ Φ([( x, n )]) = Φ[( y, u )] . Indeed, if [( x, n )] ∼ [( y, u )], then for all m ≥ max { n, u } , we have that Φ([( x, n )]) = g m ( f n,m ( x )),and Φ([( y, u )]) = g m ( f u,m ( y )). We have k X (Φ([( x, n )]) , Φ([( y, u )])) ≤ k X ( f n,m ( x ) , f u,m ( y )) m →∞ −→ . Thus there exists a unique mapping ˆΦ : Ω / ∼ → Q such that ˆΦ ◦ π ∼ = Φ.Set Γ := ˆΦ ◦ ˆΨ − : Z → Q. For all n ≥
0, Γ ◦ α n = Γ ◦ Ψ ◦ Λ n = ˆΦ ◦ π ∼ ◦ Λ n = Φ ◦ Λ n = g n . The uniqueness of the mapping Γ follows easily from the uniqueness of the mappings Φ and ˆΦ.The mapping Γ acts in the following way: if z ∈ Z , then there exists x ∈ X and n ≥ α n ( x ) = z , and then Γ( z ) = g n ( x ) . We now prove that Γ is holomorphic. Let z ∈ Z , and let x ∈ X and n ≥ α n ( x ) = z . Since α has maximal rank at z , there exists a neighborhood V of z in X suchthat, for m large enough, α m has maximal rank at every point y ∈ V . Since the sequence( f n,m kn ( x )) k n ≥ converges to α n ( x ) = z as k n → ∞ , it is eventually contained in V . Hence there ANONICAL MODELS 11 exists m ′ ≥ w := f n,m ′ ( x ) ∈ V and α m ′ has maximal rank at w . Thus there existsan open neighborhood U ⊂ Z of z and a holomorphic function σ : U → X such that α m ′ ◦ σ = id U . Then, for all y ∈ U , Γ( y ) = Γ( α m ′ ( σ ( y ))) = g m ′ ( σ ( y )) , which means that Γ is holomorphic in U . (cid:3) We denote by κ the Kobayashi–Royden metric. Proposition 2.9.
For all n ≥ , lim m →∞ f ∗ n,m k X = α ∗ n k Z , (2.5) and lim m →∞ f ∗ n,m κ X = α ∗ n κ Z . (2.6) Proof.
Let x, y ∈ X , and fix n ≥
0. We have thatlim k n →∞ k X ( f n,m kn ( x ) , f n,m kn ( y )) = k X ( α n ( x ) , α n ( y )) = k Z ( α n ( x ) , α n ( y )) , where the last identity follows from the fact that α n ( x ) , α n ( y ) ∈ Z and Z is a holomorphicretract. Then (2.5) follows since the sequence ( k X ( f n,m ( x ) , f n,m ( y ))) m ≥ n is non-increasing.The proof of (2.6) is similar. (cid:3) Definition 2.10.
Let X be a Kobayashi hyperbolic complex manifold. We say that X is cocompact if X/ aut( X ) is compact.Notice that this implies that X is complete Kobayashi hyperbolic [18, Lemma 2.1]. Theorem 2.11.
Let X be a cocompact Kobayashi hyperbolic complex manifold, and let ( f n,m : X → X ) m ≥ n ≥ be a forward holomorphic dynamical system. Then there exists a canonical Kobayashihyperbolic direct limit ( Z, α n ) for ( f n,m ) , where Z is a holomorphic retract of X . Moreover, Z = [ n ≥ α n ( X ) , (2.7) and lim m →∞ f ∗ n,m k X = α ∗ n k Z , lim m →∞ f ∗ n,m κ X = α ∗ n κ Z . (2.8) Proof.
Let K ⊂ X be a compact subset such that X = Aut( X ) · K . Let x ∈ X . For all n ≥ h n ∈ Aut( X ) be such that h n ( f ,n ( x )) ∈ K . For all m ≥ n ≥ f n,m := h m ◦ f n,m ◦ h − n .It is easy to see that ( ˜ f n,m : X → X ) is a forward holomorphic dynamical system such that { ˜ f ,m ( h ( x )) } m ≥ ⊂ K. (2.9)We can now apply Proposition 2.8 to ( ˜ f n,m : X → X ), obtaining a canonical Kobayashi hyper-bolic direct limit ( Z, ˜ α n ) for ( ˜ f n,m ), where Z is a holomorphic retract of X . For all n ≥ α n := ˜ α n ◦ h n . Clearly α m ◦ f n,m = α n , ∀ m ≥ n ≥ . Let Q be a Kobayashi hyperbolic manifold and let ( g n : X → Q ) be a family of holomorphicmappings satisfying g m ◦ f n,m = g n , ∀ m ≥ n ≥ . For all n ≥ g n := g n ◦ h − n . Then for all m ≥ n ≥ g m ◦ ˜ f n,m = g m ◦ h − m ◦ ˜ f n,m = g m ◦ f n,m ◦ h − n = g n ◦ h − n = ˜ g n . By the universal property of the canonical Kobayashi hyperbolic direct limit applied to ( Z, ˜ α n )we obtain a holomorphic mapping Γ : Z → Q such that˜ g n = Γ ◦ ˜ α n , ∀ n ≥ . Hence g n = Γ ◦ α n for all n ≥ n ≥ h n : X → X is an isometry for k X and κ X . (cid:3) Remark 2.12.
Let (Ω , Λ n ) be the direct limit of the directed system ( X, f n,m ). Let (
Z, α n )be the canonical Kobayashi hyperbolic direct limit given by Theorem 2.11. By the universalproperty of the direct limit, there exists a mapping Ψ : Ω → Z such that α n = Ψ ◦ Λ n for all n ≥
0. It is easy to see that Ψ is surjective and induces a bijection ˆΨ : Ω / ∼ → Z such that α n = ˆΨ ◦ π ∼ ◦ Λ n , ∀ n ≥ . Autonomous iteration
Definition 3.1.
Let X be a complex manifold and let f : X → X be a holomorphic self-map.Let x ∈ X , and let m ≥
0. The m -step s m ( x ) of f at x is the limit s m ( x ) = lim n →∞ k X ( f n ( x ) , f n + m ( x )) . Such a limit exists since the sequence ( k X ( f n ( x ) , f n + m ( x )) n ≥ is non-increasing. The divergencerate c ( f ) of f is the limit c ( f ) = lim m →∞ k X ( f m ( x ) , x ) m . It is shown in [4] that such a limit exists, does not depend on x ∈ X and equals inf m ∈ N k X ( f m ( x ) ,x ) m . Definition 3.2.
Let X be a complex manifold and let f : X → X be a holomorphic self-map. A semi-model for f is a triple (Λ , h, ϕ ) where Λ is a complex manifold, h : X → Λ is a holomorphicmapping, and ϕ : Ω → Ω is an automorphism such that h ◦ f = ϕ ◦ h, (3.1)and [ n ≥ ϕ − n ( h ( X )) = Λ . (3.2)We call the manifold Λ the base space and the mapping h the intertwining mapping . ANONICAL MODELS 13
Let (
Z, ℓ, τ ) and (Λ , h, ϕ ) be two semi-models for f . A morphism of semi-models ˆ η : ( Z, ℓ, τ ) → (Λ , h, ϕ ) is given by a holomorphic map η : Z → Λ such that the following diagram commutes: X h / / ℓ & & ◆◆◆◆◆◆◆◆◆◆◆◆◆ f (cid:15) (cid:15) Λ ϕ (cid:15) (cid:15) Z η > > ⑦⑦⑦⑦⑦⑦⑦⑦ τ (cid:15) (cid:15) X h / / ℓ & & ◆◆◆◆◆◆◆◆◆◆◆◆◆ Λ Z. η > > ⑦⑦⑦⑦⑦⑦⑦ If the mapping η : Z → Λ is a biholomorphism, then we say that ˆ η : ( Z, ℓ, τ ) → (Λ , h, ϕ ) is an isomorphism of semi-models . Notice that then η − : Λ → Z induces a morphism ˆ η − : (Λ , h, ϕ ) → ( Z, ℓ, τ ) . Remark 3.3.
It is shown in [4, Lemmas 3.6 and 3.7] that if (
Z, ℓ, τ ) , (Λ , h, ϕ ) are semi-modelsfor f , then there exists at most one morphism ˆ η : ( Z, ℓ, τ ) → (Λ , h, ϕ ), and that the holomorphicmap η : Z → Λ is surjective.
Definition 3.4.
Let X be a complex manifold and let f : X → X be a holomorphic self-map.Let ( Z, ℓ, τ ) be a semi-model for f whose base space Z is Kobayashi hyperbolic. We say that( Z, ℓ, τ ) is a canonical Kobayashi hyperbolic semi-model for f if for any semi-model (Λ , h, ϕ ) for f such that the base space Λ is Kobayashi hyperbolic, there exists a morphism of semi-modelsˆ η : ( Z, ℓ, τ ) → (Λ , h, ϕ ) (which is necessarily unique by Remark 3.3). Remark 3.5.
If (
Z, ℓ, τ ) and (Λ , h, ϕ ) are two canonical Kobayashi hyperbolic semi-models for f , then they are isomorphic. Theorem 3.6.
Let X be a cocompact Kobayashi hyperbolic complex manifold, and let f : X → X be a holomorphic self-map. Then there exists a canonical Kobayashi hyperbolic semi-model ( Z, ℓ, τ ) for f , where Z is a holomorphic retract of X . Moreover, the following holds: (1) if α n := τ − n ◦ ℓ for all n ≥ , then lim m →∞ ( f m ) ∗ k X = α ∗ n k Z , lim m →∞ ( f m ) ∗ κ X = α ∗ n κ Z , (2) the divergence rate of τ satisfies c ( τ ) = c ( f ) = lim m →∞ s m ( x ) m = inf m ∈ N s m ( x ) m . Proof.
Let ( f n,m : X → X ) be the autonomous dynamical system defined by f n,m = f m − n . ByTheorem 2.11, there exist a holomorphic retract Z of X and a family of holomorphic mappings( α n : X → Z ) such that the pair ( Z, α n ) is a canonical Kobayashi hyperbolic direct limit for( f n,m ). The sequence of holomorphic mappings ( β n := α n ◦ f : X → Z ) satisfies, for all m ≥ n ≥ β m ◦ f n,m = α m ◦ f ◦ f m − n = α n ◦ f = β n . By the universal property of the canonical Kobayashi hyperbolic direct limit there exists aholomorphic self-map τ : Z → Z such that for all n ≥ τ ◦ α n = α n ◦ f. We claim that τ is a holomorphic automorphism. For all n ≥
0, set γ n := α n +1 . For all m ≥ n ≥ γ m ◦ f n,m = α m +1 ◦ f m − n = α n +1 = γ n . Thus there exists a holomorphic self-map δ : Z → Z such that δ ◦ α n = α n +1 for all n ≥
0. Forall n ≥ τ ◦ δ ◦ α n = τ ◦ α n +1 = α n , and δ ◦ τ ◦ α n = δ ◦ α n ◦ f = α n +1 ◦ f = α n . By the universal property of the canonical Kobayashi hyperbolic direct limit we have that τ isa holomorphic automorphism and δ = τ − . Since for all n ≥ τ n ◦ α n = α n ◦ f n = α , it follows that α n = τ − n ◦ α .Set ℓ := α . We claim that the triple ( Z, ℓ, τ ) is a canonical Kobayashi hyperbolic semi-modelfor f . Indeed, let (Λ , h, ϕ ) be a semi-model for f such that the base space Λ is Kobayashihyperbolic. For all n ≥
0, let λ n := ϕ − n ◦ h . Then by the universal property of the canonicalKobayashi hyperbolic direct limit there exists a holomorphic mapping η : Z → Λ such that forall n ≥ η ◦ α n = λ n , that is η ◦ τ − n ◦ ℓ = ϕ − n ◦ h. Notice that this implies η ◦ ℓ = h , and if n ≥ ϕ ◦ η ◦ τ − ◦ α n = ϕ ◦ ϕ − n − ◦ h = λ n = η ◦ α n . Thus by the universal property of the canonical Kobayashi hyperbolic direct limit, η = ϕ ◦ η ◦ τ − .Hence the mapping η : Z → Λ gives a morphism of semi-models ˆ η : ( Z, ℓ, τ ) → (Λ , h, ϕ ).Property (1) follows clearly from Theorem 2.11. Property (1) implies in particular that forall m ≥ x ∈ X , the m -step s m ( x ) satisfies s m ( x ) = k Z ( ℓ ( z ) , τ m ( ℓ ( z ))) . By [4, Proposition 2.7] c ( τ ) = lim m →∞ k Z ( ℓ ( z ) , τ m ( ℓ ( z ))) m = lim m →∞ s m ( x ) m = lim m →∞ k X ( f m ( x ) , x ) m = c ( f ) , and c ( τ ) = inf m ∈ N k Z ( ℓ ( z ) , τ m ( ℓ ( z ))) m = inf m ∈ N s m ( x ) m , which proves Property (2). (cid:3) ANONICAL MODELS 15
Remark 3.7.
Actually, the proof shows that the semi-model (
Z, ℓ, τ ) satisfies the followingstronger universal property. If Λ is a Kobayashi hyperbolic complex manifold, if ϕ : Λ → Λ is anautomorphism and if h : X → Λ is a holomorphic mapping such that h ◦ f = ϕ ◦ h (notice thatwe do not assume (3.2)), then there exists a holomorphic mapping η : Z → Λ such that η ◦ ℓ = h and η ◦ τ = ϕ ◦ η . Clearly, η ( Z ) = ∪ n ≥ ϕ − n h ( X ).4. The unit ball
Definition 4.1.
The Siegel upper half-space H q is defined by H q = (cid:8) ( z, w ) ∈ C × C q − , Im ( z ) > k w k (cid:9) . Recall that H q is biholomorphic to the ball B q via the Cayley transform
Ψ : B q → H q defined asΨ( z, w ) = (cid:18) i z − z , w − z (cid:19) , ( z, w ) ∈ C × C q − . Let h· , ·i denote the standard Hermitian product in C q . In several complex variables, thenatural generalization of the non-tangential limit at the boundary is the following. If ζ ∈ ∂ B q ,then the set K ( ζ, R ) := { z ∈ B q : | − h z, ζ i| < R (1 − k z k ) } is a Kor´anyi region of vertex ζ and amplitude R >
1. Let f : B q → C m be a holomorphic map.We say that f has K -limit L ∈ C m at ζ (we write K - lim z → ζ f ( z ) = L ) if for each sequence ( z n )converging to ζ such that ( z n ) belongs eventually to some Kor´anyi region of vertex ζ , we havethat f ( z n ) → L . The Kor´anyi regions can also be easily described in the Siegel upper half-space H q , see e.g. [10].Let ζ ∈ ∂ B q . A sequence ( z n ) ⊂ B q converging to ζ ∈ ∂ B q is said to be restricted at ζ if h z n , ζ i → D , while it is said to be special at ζ iflim n →∞ k B q ( z n , h z n , ζ i ζ ) = 0 . We say that f has restricted K -limit L at ζ (we write ∠ K lim z → ζ f ( z ) = L ) if for every specialand restricted sequence ( z n ) converging to ζ we have that f ( z n ) → L .One can show that K - lim z → ζ f ( z ) = L = ⇒ ∠ K lim z → ζ f ( z ) = L, but the converse implication is not true in general. Definition 4.2.
A point ζ ∈ ∂ B q such that K - lim z → ζ f ( z ) = ζ andlim inf z → ζ − k f ( z ) k − k z k = λ < + ∞ is called a boundary regular fixed point , and λ is called its dilation .The following result [20] generalizes the Denjoy–Wolff theorem in the unit disc. Theorem 4.3.
Let f : B q → B q be holomorphic. Assume that f admits no fixed points in B q .Then there exists a point p ∈ ∂ B q , called the Denjoy–Wolff point of f , such that ( f n ) convergesuniformly on compact subsets to the constant map z p . The Denjoy–Wolff point of f is aboundary regular fixed point and its dilation λ is smaller than or equal to . Remark 4.4.
Let f : B q → B q be a holomorphic self-map without fixed points, and let λ be thedilation at its Denjoy–Wolff fixed point. Then by [4, Proposition 5.8] the divergence rate of f satisfyies c ( f ) = − log λ. Definition 4.5.
A holomorphic self-map f : B q → B q is called(1) elliptic if it admits a fixed point z ∈ B q ,(2) parabolic if it admits no fixed points z ∈ B q , and its dilation at the Denjoy–Wolff pointis equal to 1,(3) hyperbolic if it admits no fixed points z ∈ B q , and its dilation at the Denjoy–Wolff pointis strictly smaller than 1.If s ( z ) > z ∈ B q , then we say that f is nonzero-step .The next result generalizes Theorem 0.1 to the unit ball. Theorem 4.6.
Let f : B q → B q be a hyperbolic holomorphic self-map, with dilation λ at itsDenjoy–Wolff point p ∈ ∂ B q . Then there exist (1) an integer k such that ≤ k ≤ q , (2) a hyperbolic automorphism ϕ : H k → H k of the form ϕ ( z, w ) = (cid:18) λ z, e it √ λ w , . . . , e it k − √ λ w k − (cid:19) , (4.1) where t j ∈ R for ≤ j ≤ k − , (3) a holomorphic mapping h : B q → H k with K - lim x → p h ( x ) = ∞ , such that the triple ( H k , h, ϕ ) is a canonical Kobayashi hyperbolic model for f .Proof. Since B q is cocompact and Kobayashi hyperbolic, by Theorem 3.6 there exists a canonicalKobayashi hyperbolic semi-model ( Z, ℓ, τ ) for f . Since Z is a holomorphic retract of B q , it isbiholomorphic to B k for some 0 ≤ k ≤ q (see e.g. [1, Corollary 2.2.16]). By Remark 4.4 and by (2)of Theorem 3.6, we have c ( τ ) = c ( f ) = − log λ , hence k ≥ τ is a hyperbolic automorphismwith dilation λ at its Denjoy–Wolff point. Thus there exists (see e.g. [1, Proposition 2.2.11]) abiholomorphism γ : Z → H k such that ϕ := γ ◦ τ ◦ γ − is of the form (4.1). Setting h := γ ◦ ℓ we have that ( H k , h, ϕ ) is also a canonical Kobayashi hyperbolic semi-model for f . By [4,Proposition 5.11], we have K - lim x → p h ( x ) = ∞ . (cid:3) Corollary 4.7.
Let f : B q → B q be a hyperbolic holomorphic self-map, with dilation λ at itsDenjoy–Wolff point p ∈ ∂ B q . Then there exists a holomorphic mapping ϑ : B q → H solving theValiron equation (0.2).Proof. Let ( H k , h, ϕ ) be the canonical Kobayashi hyperbolic semi-model given by Theorem 4.6.Let π : H k → H be the projection π ( z, w ) = z. Then (cid:0) H , ϑ := π ◦ h, x λ x (cid:1) is a semi-modelfor f , and thus ϑ solves the Valiron equation (0.2). (cid:3) ANONICAL MODELS 17
Remark 4.8. If q = 1, then the following uniqueness result holds [11]: any holomorphic solutionof the Valiron equation (0.2) is a positive multiple of a given solution ϑ : H → H .If q ≥
2, the situation is quite different. It is easy to see that the solutions of (0.2) are all theholomorphic mappings of the form Γ ◦ h, where ( H k , h, ϕ ) is the canonical Kobayashi hyperbolicsemi-model given by Theorem 4.6, and Γ : H k → H is a holomorphic function such thatΓ ◦ ϕ = 1 λ Γ . (4.2)Notice that for all z ∈ H , Γ (cid:18) λ z, (cid:19) = 1 λ Γ( z, , which by a result of Heins [19] implies that Γ( z,
0) = az for some a > H k ) = H ).Thus if k = 1 we obtain again a uniqueness result: any holomorphic solution of (0.2) is a positivemultiple of a given solution ϑ : H q → H .Assume now that k ≥
2. The function Γ is unique up to positive multiples on the slice { w = 0 } of H k , but is far from being unique on H k r { w = 0 } . This can be seen, for example, in thefollowing way. If γ : H k → H k is a holomorphic self-map which commutes with the hyperbolicautomorphism ϕ , then clearly Γ := π ◦ γ satisfies (4.2). The family of holomorphic mappingsof the form π ◦ γ is large, as shown (and made precise) in [16, Theorem 2.5].Recall the following result on the Abel equation in the unit disc. Theorem 4.9 (Pommerenke [28]) . Let f : D → D be a parabolic nonzero-step holomorphic self-map. Then there exists a model ( H , h, z z ± for f . The essential uniqueness of the intertwining mapping in the previous theorem is proved in[27]. The next result gives a generalization of this result to the unit ball.
Theorem 4.10.
Let f : B q → B q be a parabolic nonzero-step holomorphic self-map with Denjoy–Wolff point p ∈ ∂ B q . Then there exist (1) an integer k such that ≤ k ≤ q , (2) a parabolic automorphism ϕ : H k → H k of the form ϕ ( z, w ) = ( z ± , e it w , . . . e it k − w k − ) , (4.3) where t j ∈ R for ≤ j ≤ k − , or of the form ϕ ( z, w ) = ( z − w + i, w − i, e it w , . . . e it k − w k − ) , (4.4) where where t j ∈ R for ≤ j ≤ k − , (3) a holomorphic mapping h : B q → H k with ∠ K - lim z → h ( z ) = ∞ , such that the triple ( H k , h, ϕ ) is a canonical Kobayashi hyperbolic model for f .Proof. Since B q is cocompact and Kobayashi hyperbolic, by Theorem 3.6 there exists a canonicalKobayashi hyperbolic semi-model ( Z, ℓ, τ ) for f . Since Z is a holomorphic retract of B q , it isbiholomorphic to B k for some 0 ≤ k ≤ q . Let z ∈ Z , x ∈ B q , and n ≥ τ − n ( ℓ ( x )) = z .Then, by (1) of Theorem 3.6, k Z ( z, τ ( z )) = s ( z ) > . Hence k ≥
1, and τ is not elliptic. By Remark 4.4 and by (2) of Theorem 3.6, we have c ( τ ) = c ( f ) = 0. Hence τ is parabolic. There exists (see e.g. [17]) a biholomorphism γ : Z → H k such that ϕ := γ ◦ τ ◦ γ − is of the form (4.3) or of the form (4.4). Setting h := γ ◦ ℓ we havethat ( H k , h, ϕ ) is also a canonical Kobayashi hyperbolic semi-model for f . By [4, Proposition5.11], we have ∠ K - lim x → p h ( x ) = ∞ . (cid:3) Part Backward iteration Preliminaries
Definition 5.1.
Let X be a complex manifold. We call backward (non-autonomous) holomor-phic dynamical system on X any family ( f n,m : X → X ) m ≥ n ≥ of holomorphic self-maps suchthat for all m ≥ u ≥ n ≥
0, we have f n,u ◦ f u,m = f n,m . For all n ≥ f n,n +1 also by f n . A backward holomorphic dynamical system( f n,m : X → X ) m ≥ n ≥ is called autonomous if f n = f for all n ≥
0. Clearly in this case f n,m = f m − n . Remark 5.2.
Any family of holomorphic self-maps ( f n : X → X ) n ≥ determines a backwardholomorphic dynamical system ( f n,m : X → X ) in the following way: for all n ≥
0, set f n,n = id ,and for all m > n ≥
0, set f n,m = f n ◦ · · · ◦ f m − . Definition 5.3.
Let X be a complex manifold, and let ( f n,m : X → X ) be a backward holo-morphic dynamical system. An inverse limit for ( f n,m ) is a pair (Θ , V n ) where Θ is a set and( V n : Θ → X ) n ≥ is a family of mappings such that f n,m ◦ V m = V n , ∀ m ≥ n ≥ , satisfying the following universal property: if Q is a set and if ( g n : Q → X ) is a family ofmappings satisfying f n,m ◦ g m = g n , ∀ m ≥ n ≥ , then there exists a unique mapping Γ : Q → Θ such that g n = V n ◦ Γ , ∀ n ≥ . Remark 5.4.
The inverse limit is essentially unique, in the following sense. Let (Θ , V n ) and( Q, g n ) be two inverse limits for ( f n,m ). Then there exists a bijective mapping Γ : Q → Θ suchthat g n = V n ◦ Γ , ∀ n ≥ . Definition 5.5.
Let X be a complex manifold, and let ( f n,m : X → X ) be a backward holo-morphic dynamical system. A backward orbit for ( f n,m ) is a sequence ( x n ) n ≥ in X such that,for all m ≥ n ≥ f n,m ( x m ) = x n . ANONICAL MODELS 19
Remark 5.6.
An inverse limit for ( f n,m ) is easily constructed. We define Θ as the set of allbackward orbits for ( f n,m ). We define a family of mappings ( V n : Θ → X ) n ≥ in the followingway. Let β = ( x m ) m ≥ be a backward orbit. Then for all n ≥ V n ( β ) = x n . It is easy to see that (Θ , V n ) is an inverse limit for ( f n,m ). Definition 5.7.
Let X be a complex manifold and let ( f n,m : X → X ) m ≥ n ≥ be a backwardholomorphic dynamical system. Let (Θ , V n ) be the inverse limit of the inverse system ( X, f n,m ).We define an equivalence relation ∼ on Θ in the following way. The backward orbits ( z n ) and( w n ) are equivalent if and only if the non-decreasing sequence ( k X ( z n , w n )) n ≥ is bounded. Theclass of the backward orbit ( z n ) will be denoted by [ z n ]. Lemma 5.8.
Let X be a complex manifold, and let ( f n,m : X → X ) be a backward holomorphicdynamical system. Let Z be a complex manifold and let ( α n : Z → X ) be a sequence of holo-morphic mappings such that f n,m ◦ α m = α n for all m ≥ n ≥ . Then ( α n ( z )) ∼ ( α n ( w )) for all z, w ∈ Z .Proof. It follows since k X ( α n ( z ) , α n ( w )) ≤ k Z ( z, w ) for all n ≥ . (cid:3) We now introduce a modified version of the inverse limit for ( f n,m ) which is more suited forour needs. Definition 5.9.
Let X be a complex manifold. Let ( f n,m : X → X ) be a backward holomorphicdynamical system. We call canonical inverse limit associated with the class [ y n ] ∈ Θ / ∼ for ( f n,m )a pair ( Z, α n ) where Z is a complex manifold and ( α n : Z → X ) is a sequence of holomorphicmappings such that(1) f n,m ◦ α m = α n , for all m ≥ n ≥ , (2) ( α n ( z )) ∈ [ y n ] for some (and hence for any) z ∈ Z ,which satisfies the following universal property: if Q is a complex manifold and if ( g n : Q → X )is a family of holomorphic mappings satisfying(1’) f n,m ◦ g m = g n , for all m ≥ n ≥ , (2’) ( g n ( q )) ∈ [ y n ] for some (and hence for any) q ∈ Q ,then there exists a unique holomorphic mapping Γ : Q → Z such that g n = α n ◦ Γ , ∀ n ≥ . Proposition 5.10.
The canonical inverse limit for ( f n,m ) associated with the class [ y n ] ∈ Θ / ∼ is unique in the following sense. Let ( Z, α n ) and ( Q, g n ) be two canonical inverse limit for ( f n,m ) associated with the same class [ y n ] . Then there exists a biholomorphism Γ : Q → Z such that g n = α n ◦ Γ , ∀ n ≥ . Proof.
There exist holomorphic mappings Γ : Q → Z and Ξ : Z → Q such that for all n ≥
0, wehave g n = α n ◦ Γ and α n = g n ◦ Ξ. Thus the holomorphic mapping Γ ◦ Ξ : Z → Z satisfies α n ◦ Γ ◦ Ξ = α n , ∀ n ≥ , By the universal property of the canonical inverse limit associated with the class [ y n ] ∈ Θ / ∼ ,this implies that Γ ◦ Ξ = id Z . Similarly, we obtain Ξ ◦ Γ = id Q . (cid:3) Non-autonomous iteration
Let X be a complete Kobayashi hyperbolic complex manifold. Let ( f n,m : X → X ) m ≥ n ≥ be a backward holomorphic dynamical system, and assume that it admits a relatively compactbackward orbit ( y m ) m ≥ . Remark 6.1.
The class [ y n ] ∈ Θ / ∼ coincides with the subset of Θ defined by all relativelycompact backward orbits of ( f n,m ). Remark 6.2.
Let K ⊂ X be a compact subset such that { y m } m ≥ ⊂ K. It follows that, for allfixed n ≥ f n,m ( K ) ∩ K = ∅ ∀ m ≥ n. (6.1)The sequence ( f ,m : X → X ) m ≥ is not compactly divergent by (6.1), and since X is taut,there exists a subsequence ( f ,m k ) k ≥ converging to a holomorphic self-map α : X → X . Thesequence ( f ,m k : X → X ) k ≥ is not compactly divergent by (6.1), and since X is taut, thereexists a subsequence ( f ,m k ) k ≥ converging to a holomorphic self-map α : X → X . Iteratingthis procedure we obtain a family of holomorphic self-maps ( α n : X → X ) satisfying for all m ≥ n ≥ f n,m ◦ α m = α n . (6.2)Notice that for all n ≥ α n ( K ) ∩ K = ∅ . (6.3)Let now ( m k ) k ≥ be a sequence which for all j ∈ N is eventually a subsequence of ( m k j ) k j ≥ (such a sequence exists by a diagonal argument). The sequence of holomorphic self-maps( α m k : X → X ) k ≥ is not compactly divergent by (6.3), and since X is taut, there exists asubsequence ( α m h ) h ≥ converging to a holomorphic self-map α : X → X . Lemma 6.3.
The holomorphic self-map α : X → X is a holomorphic retraction, and for all n ≥ , α n ◦ α = α n . (6.4) Proof.
Fix n ≥ x ∈ X . Then for all h ≥ m h ≥ n , we have α n ( x ) = f n,m h ( α m h ( x )) h →∞ −→ α n ( α ( x )) . Thus we have, for all h ≥ α m h ( α ( x )) = α m h ( x ) . When h → ∞ , the left-hand side converges to α ( α ( x )), while the right-hand side converges to α ( x ). (cid:3) Definition 6.4.
We denote the closed complex submanifold α ( X ) by Z .In what follows we denote the restriction α n | Z simply by α n . Let (Θ , V n ) be the inverse limitof the inverse system ( X, f n,m ). By the universal property of the inverse limit, there exists amapping Ψ : Z → Θ such that for all n ≥ α n = V n ◦ Ψ . The mapping Ψ is defined in the following way: if z ∈ Z , then Ψ( z ) is the backward orbit( α m ( z )) m ≥ . ANONICAL MODELS 21
Proposition 6.5.
The mapping
Ψ : Z → Θ is injective and its image is [ y n ] .Proof. Let z, w ∈ Z and assume that Ψ( z ) = Ψ( w ). It follows that α m ( z ) = α m ( w ) for all m ≥
0, that is α ( z ) = α ( w ). Since α is a retraction, we obtain z = w . Hence Ψ : Z → Θ isinjective.We now show that Ψ( Z ) ⊂ [ y n ]. If z ∈ Z , we have to show that the sequence ( k X ( α m ( z ) , y m ))is bounded. Since y m ∈ K for all m ≥ α m h ( z ) → α ( z ), we have that the subsequence( k X ( α m h ( z ) , y m h )) is bounded. Since the sequence ( k X ( α m ( z ) , y m )) is non-decreasing, it isbounded too.Finally, we show that for all ( z m ) ∈ [ y n ], there exists z ∈ Z such that α m ( z ) = z m for all m ≥
0. Let thus ( z m ) be a backward orbit such that the sequence ( k X ( y m , z m )) is bounded.Clearly, the subsequence ( k X ( y m h , z m h )) is also bounded, and thus there exists a subsequence( z m u ) of ( z m h ) converging to a point z ∈ X . It follows that for all n ≥ z n = f n,m u ( z m u ) u →∞ → α n ( z ) . We claim that z ∈ Z . Indeed, letting u → ∞ in the identity α m u ( z ) = z m u we obtain α ( z ) = z . (cid:3) Proposition 6.6.
The pair ( Z, α n ) is a canonical inverse limit for ( f n,m ) associated with [ y n ] .Proof. Let Q be a complex manifold and let ( g n : Q → X ) be a family of holomorphic mappingssatisfying(1) f n,m ◦ g m = g n , for all m ≥ n ≥ , (2) ( g n ( q )) ∈ [ y n ] for some (and hence for any) q ∈ Q .By the universal property of the inverse limit, there exists a unique mapping Φ : Q → Θ suchthat g n = V n ◦ Φ , ∀ n ≥ . The mapping Φ is defined in the following way: if q ∈ Q , then Φ( q ) is the backward orbit( g m ( q )) m ≥ . Property (2) implies that Φ( Q ) ⊂ [ y n ]. SetΓ := Ψ − ◦ Φ : Q → Z. For all n ≥ α n ◦ Γ = V n ◦ Ψ ◦ Γ = V n ◦ Φ = g n . (6.5)The uniqueness of the mapping Γ follows easily from the uniqueness of the mapping Φ. Themapping Γ acts in the following way: if q ∈ Q , then Γ( q ) ∈ Z is uniquely defined by α m (Γ( q )) = g m ( q ) , ∀ m ≥ . (6.6)We now prove that Γ is holomorphic. Recall that the sequence ( α m h : Z → X ) h ≥ convergesuniformly on compact subsets to id Z . By Remark 6.1, the sequence ( g m : Q → X ) is notcompactly divergent. Since X is taut, the sequence ( g m h : Q → X ) admits a subsequence( g m u : Q → X ) converging uniformly on compact subsets to a holomorphic mapping g : Q → X .Thus taking the limit in both sides of α m u ◦ Γ = g m u , as m u → ∞ , we have Γ = g , which implies that Γ is holomorphic. (cid:3) Proposition 6.7.
We have lim m →∞ α ∗ m k X = k Z , and lim n →∞ α ∗ m κ X = κ Z . Proof.
Let z, w ∈ Z . We havelim m h →∞ k X ( α m h ( z ) , α m h ( w )) = k X ( α ( z ) , α ( w )) = k X ( z, w ) = k Z ( z, w ) . where the last identity follows from the fact that Z is a holomorphic retract of X . The firststatement follows since the sequence ( k X ( α m ( z ) , α m ( w ))) m ≥ is non-decreasing. The proof ofthe second statement is similar. (cid:3) Theorem 6.8.
Let X a cocompact Kobayashi hyperbolic complex manifold, and let ( f n,m : X → X ) m ≥ n ≥ be a backward dynamical system. Let ( y n ) be a backward orbit. Then there exists acanonical inverse limit ( Z, α n ) for ( f n,m ) associated with [ y n ] , where Z is a holomorphic retractof X . Moreover, lim m →∞ α ∗ m k X = k Z , and lim m →∞ α ∗ m κ X = κ Z . (6.7) Proof.
Let K ⊂ X be a compact subset such that X = Aut( X ) · K . For all n ≥
0, let h n ∈ Aut( X ) be such that h − n ( y n ) ∈ K . For all m ≥ n ≥ f n,m = h − n ◦ f n,m ◦ h m . It is easyto see that ( ˜ f n,m : X → X ) is a forward holomorphic dynamical system with a relatively compactbackward orbit (˜ y n := h − n ( y n )) . We can now apply Proposition 6.6 to ( ˜ f n,m : X → X ), obtaininga canonical inverse limit ( Z, ˜ α n ) for ( ˜ f n,m ) associated with [˜ y n ], where Z is a holomorphic retractof X . For all n ≥ α n := h n ◦ ˜ α n . Clearly f n,m ◦ α m = α n , ∀ m ≥ n ≥ . Let Q be a complex manifold and let ( g n : Q → X ) be a family of holomorphic mappingssatisfying f n,m ◦ g m = g n , ∀ m ≥ n ≥ . For all n ≥ g n := h − n ◦ g n . Then for all m ≥ n ≥ f n,m ◦ ˜ g m = ˜ f n,m ◦ h − m ◦ g m = h − n ◦ f n,m ◦ g m = ˜ g n . By the universal property of the canonical inverse limit ( Z, ˜ α n ) we obtain a holomorphic mappingΓ : Q → Z such that ˜ g n = ˜ α n ◦ Γ , ∀ n ≥ . Hence g n = α n ◦ Γ for all n ≥ n ≥ h n : X → X is an isometry for k X and κ X . (cid:3) Remark 6.9.
Let (Θ , V n ) be the inverse limit of the inverse system ( X, f n,m ). Let ( y n ) bea backward orbit and let ( Z, α n ) be the canonical inverse limit associated with ( y n ) given byTheorem 6.8. By the universal property of the inverse limit, there exists a mapping Ψ : Z → Θsuch that α n = V n ◦ Ψ , ∀ n ≥ . ANONICAL MODELS 23
It is easy to see that Ψ is injective and that Ψ( Z ) = [ y n ]. In particular, for all n ≥ α n ( Z ) = V n ([ y n ]) . Autonomous iteration
Definition 7.1.
Let X be a complex manifold and let f : X → X be a holomorphic self-map.A pre-model for f is a triple (Λ , h, ϕ ) such that Λ is a complex manifold, h : Λ → X is aholomorphic mapping and ϕ : Λ → Λ is an automorphism such that f ◦ h = h ◦ ϕ. The mapping h is called the intertwining mapping .Let (Λ , h, ϕ ) and ( Z, ℓ, τ ) be two pre-models for f . A morphism of pre-models ˆ η : (Λ , h, ϕ ) → ( Z, ℓ, τ ) is given by a holomorphic mapping η : Λ → Z such that the following diagram commutes:Λ h / / η ❅❅❅❅❅❅❅❅ ϕ (cid:15) (cid:15) X f (cid:15) (cid:15) Z ℓ ♣♣♣♣♣♣♣♣♣♣♣♣♣ τ (cid:15) (cid:15) Λ h / / η ❅❅❅❅❅❅❅ XZ. ℓ ♣♣♣♣♣♣♣♣♣♣♣♣♣ If the mapping η : Λ → Z is a biholomorphism, then we say that ˆ η : (Λ , h, ϕ ) → ( Z, ℓ, τ ) is an isomorphism of pre-models . Notice that then η − : Z → Λ induces a morphism ˆ η − : ( Z, ℓ, τ ) → (Λ , h, ϕ ) . Definition 7.2.
Let X be a complex manifold and let f : X → X be a holomorphic self-map.Let ( y n ) be a backward orbit for f . Let ( Z, ℓ, τ ) be a semi-model for f such that for some(and hence for any) z ∈ Z we have ( ℓ ( τ − n ( z ))) ∈ [ y n ]. We say that ( Z, ℓ, τ ) is a canonicalpre-model associated with [ y n ] for f if for any pre-model (Λ , h, ϕ ) for f such that for some (andhence for any) x ∈ Λ we have ( h ( ϕ − n ( x ))) ∈ [ y n ], there exists a unique morphism of pre-modelsˆ η : (Λ , h, ϕ ) → ( Z, ℓ, τ ). Remark 7.3.
If (
Z, ℓ, τ ) and (Λ , h, ϕ ) are two canonical pre-models for f associated with thesame class [ y n ], then they are isomorphic. Lemma 7.4.
Let X be a complex manifold and let f : X → X be a holomorphic self-map. Let ( y n ) be a backward orbit. If there exists a canonical pre-model ( Z, ℓ, τ ) for f associated with [ y n ] ,then every backward orbit ( w n ) ∈ [ y n ] has bounded step.Proof. Let z ∈ Z . The backward orbit ( ℓ ( τ − n ( z ))) has bounded step since for all n ≥ k X ( ℓ ( τ − n ( z )) , ℓ ( τ − n − ( z ))) ≤ k Z ( τ − n ( z ) , τ − n − ( z )) = k Z ( z, τ ( z )) . Let ( w n ) ∈ [ y n ]. Since for all n ≥ k X ( w n , w n +1 ) ≤ k X ( w n , ℓ ( τ − n ( z ))) + k X ( ℓ ( τ − n ( z )) , ℓ ( τ − n − ( z ))) + k X ( ℓ ( τ − n − ( z )) , w n +1 ) , it follows that ( w n ) has also bounded step. (cid:3) Theorem 7.5.
Let X be a cocompact Kobayashi hyperbolic complex manifold, and let f : X → X be a holomorphic self-map. Let ( y n ) be a backward orbit with bounded step. Then there exists acanonical pre-model ( Z, ℓ, τ ) for f associated with [ y n ] , where Z is a holomorphic retract of X .Moreover, the following holds: (1) ℓ ( Z ) = V ([ y n ]) , (2) if α m := ℓ ◦ τ − m for all m ≥ , then lim m →∞ α ∗ m k X = k Z , lim m →∞ α ∗ m κ X = κ Z , (3) if β is a backward orbit in the class [ y n ] , c ( τ ) = lim m →∞ σ m ( β ) m = inf m ∈ N σ m ( β ) m . Proof.
Let ( f n,m : X → X ) be the autonomous dynamical system defined by f n,m = f m − n . ByTheorem 6.8, there exist a holomorphic retract Z of X and a family of holomorphic mappings( α n : Z → X ) such that the pair ( Z, α n ) is a canonical inverse limit associated with [ y n ]. Thesequence of holomorphic mappings ( β n := f ◦ α n : Z → X ) satisfies, for all m ≥ n ≥ f n,m ◦ β m = f m − n ◦ f ◦ α m = f ◦ α n = β n . Let z ∈ Z be the unique point such that α m ( z ) = y m for all m ≥
0. Then for all m ≥ k X ( β m ( z ) , y m ) = k X ( α m − ( z ) , y m ) = k X ( y m − , y m ) , which is bounded since by assumption the backward orbit ( y n ) has bounded step. By theuniversal property of the canonical inverse limit associated with [ y n ] there exists a holomorphicself-map τ : Z → Z such that for all n ≥ α n ◦ τ = f ◦ α n . We claim that τ is a holomorphic automorphism. Set for all n ≥ γ n := α n +1 . For all m ≥ n ≥ f n,m ◦ γ m = f m − n ◦ α m +1 = α n +1 = γ n . Let z ∈ Z be the unique point such that α m ( z ) = y m for all m ≥
0. For all m ≥ k X ( γ m ( z ) , y m ) = k X ( α m +1 ( z ) , y m ) = k X ( y m +1 , y m ) , which is bounded since by assumption the backward orbit ( y n ) has bounded step. Thus thereexists a holomorphic self-map δ : Z → Z such that α n ◦ δ = α n +1 for all n ≥
0. For all n ≥ α n ◦ τ ◦ δ = f ◦ α n ◦ δ = f ◦ α n +1 = α n , and α n ◦ δ ◦ τ = α n +1 ◦ τ = α n . By the universal property of the canonical inverse limit associated with [ y n ] we have that τ is aholomorphic automorphism and δ = τ − . Since for all n ≥ α n ◦ τ n = f n ◦ α n = α , ANONICAL MODELS 25 it follows that α n = α ◦ τ − n . Set ℓ := α . We claim that the triple ( Z, ℓ, τ ) is a canonical pre-model for f associated with[ y n ]. Indeed, let (Λ , h, ϕ ) be a pre-model for f such that for some (and hence for any) x ∈ Λ wehave h ( ϕ − n ( x )) ∈ [ y n ]. For all n ≥
0, let λ n := h ◦ ϕ − n . Then by the universal property of thecanonical inverse limit associated with [ y n ] there exists a holomorphic mapping η : Λ → Z suchthat for all n ≥ α n ◦ η = λ n , that is ℓ ◦ τ − n ◦ η = h ◦ ϕ − n . Notice that this implies ℓ ◦ η = h , and if n ≥ α n ◦ τ − ◦ η ◦ ϕ = h ◦ ϕ − n − ◦ ϕ = λ n . Thus by the universal property of the canonical Kobayashi hyperbolic direct limit, η = τ − ◦ η ◦ ϕ. Hence the mapping η : Λ → Z gives a morphism of pre-models ˆ η : (Λ , h, ϕ ) → ( Z, ℓ, τ ).Property (1) follows from Remark 6.9. Property (2) follows from (6.7). We now proveProperty (3). Let β := ( w n ) be a backward orbit [ y n ], and let z ∈ Z be the unique point suchthat α n ( z ) = w n for all n ≥
0. Then by Property (2) the backward m -step σ m ( β ) satisfies σ m ( β ) = lim n →∞ k X ( α n ( z ) , α n + m ( z )) = lim n →∞ k X ( α n ( z ) , α n ( τ − m ( z ))) = k Z ( z, τ − m ( z )) . Notice that k Z ( z, τ − m ( z )) = k Z ( z, τ m ( z )). We have c ( τ ) = lim m →∞ k Z ( z, τ m ( z )) m = lim m →∞ σ m ( β ) m , and c ( τ ) = inf m ∈ N k Z ( z, τ m ( z )) m = inf m ∈ N σ m ( β ) m . (cid:3) The unit ball
Definition 8.1.
Let f : B q → B q be a holomorphic self-map. Let ζ ∈ ∂ B q be a boundary regularfixed point. The stable subset of f at ζ is defined as the subset consisting of all z ∈ B q such thatthere exists a backward orbit with bounded step starting at z and converging to ζ . We denoteit by S ( ζ ).Clearly S ( ζ ) coincides with the union of all backward orbits in B q with bounded step con-verging to ζ . Definition 8.2.
Let f : B q → B q be a holomorphic self-map. A boundary repelling fixed point ζ ∈ ∂ B q is a boundary regular fixed point with dilation λ > Theorem 8.3.
Let f : B q → B q be a holomorphic self-map and let ζ ∈ ∂ B q be a boundaryrepelling fixed point with dilation < λ < ∞ . Let ( y n ) be a backward orbit with bounded stepwhich converges to ζ . Define µ by µ := lim m →∞ e σm ( β ) m ≥ λ, where β ∈ [ y n ] . Then µ does not depend on β ∈ [ y n ] and there exist (1) an integer k such that ≤ k ≤ q , (2) a hyperbolic automorphism ϕ : H k → H k with dilation µ at its unique repelling point ∞ ,of the form ϕ ( z, w ) = (cid:18) µ z, e it √ µ w , . . . , e it k − √ µ w k − (cid:19) , (8.1) where t j ∈ R for ≤ j ≤ k − , (3) a holomorphic mapping h : H k → B q with K - lim z →∞ h ( z ) = ζ, such that ( H k , h, ϕ ) is a canonical pre-model for f associated with [ y n ] , and h ( H k ) = V ([ y n ]) ⊂ S ( ζ ) . If [ y n ] contains backward orbit whose convergence to ζ is special and restricted, then µ = λ .Proof. Since B q is cocompact and Kobayashi hyperbolic, by Theorem 7.5 there exists a canonicalpre-model ( Z, ℓ, τ ) for f associated with [ y n ]. Since Z is a holomorphic retract of B q , it isbiholomorphic to B k for some 0 ≤ k ≤ q . By (3) of Theorem 7.5, if β is a backward orbit in theclass [ y n ], µ = lim m →∞ e σm ( β ) m = e c ( τ ) . In particular, µ does not depend on β ∈ [ y n ].We claim that µ ≥ λ . Let n ≥
0. Since λ n is the dilation at ζ of the mapping f n , we have,for any w ∈ B q (see e.g. [1]), n log λ = lim inf z → ζ ( k B q ( w, z ) − k B q ( w, f n ( z ))) . Since k B q ( w, z ) − k B q ( w, f n ( z )) ≤ k B q ( z, f n ( z )) , we have that n log λ ≤ σ n ( β ), that is, λ ≤ e σn ( β ) n . Thus µ ≥ λ .The automorphism τ is hyperbolic since the dilation at its Denjoy–Wolff point is equal to e − c ( τ ) and e − c ( τ ) = 1 µ ≤ λ < . There exists (see e.g. [1, Proposition 2.2.11]) a biholomorphism γ : Z → H k such that ϕ := γ ◦ τ ◦ γ − is of the form (8.1). Setting h := ℓ ◦ γ − we have that ( H k , h, ϕ ) is also a canonicalpre-model for f associated with [ y n ].We now address the regularity at ∞ of the intertwining mapping h . Let ( z n , w n ) be a backwardorbit in H k for τ . Then ( z n , w n ) converges to ∞ and there exists C > k H k (( z n , w n ) , ( z n +1 , w n +1 )) ≤ C, and k H k (( z n , w n ) , ( z n , ≤ C. Clearly g ( z n , w n ) is a backward orbit for f which converges to ζ ∈ ∂ B q . Then [4, Theorem 5.6]yields the result.Theorem 7.5 yields that h ( H k ) = V ([ y n ]). Let x ∈ V ([ y n ]). Then there exists a backwardorbit ( w n ) ∈ [ y n ] starting at x , which clearly converges to ζ . By Lemma 7.4 the backward orbit( w n ) has bounded step, and thus V ([ y n ]) ⊂ S ( ζ ). ANONICAL MODELS 27
Let β := ( w n ) be a special and restricted backward orbit in [ y n ] converging to ζ . Then thesame proof as in [3, Proposition 4.12] shows thatlog µ = lim m →∞ σ m ( β ) m = log λ. (cid:3) We leave the following open questions.
Question 8.4.
With notations from the previous theorem, does the identity λ = µ always hold? Question 8.5.
Let f : B q → B q be a holomorphic self-map and let ζ ∈ ∂ B q be a boundaryrepelling fixed point with dilation 1 < λ < ∞ . By [23, Lemma 3.1], if ζ is isolated from otherboundary repelling fixed points with dilation less or equal than λ , then S ( ζ ) = ∅ . Is the sametrue if the point ζ is not isolated? Question 8.6.
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