����������� ������ 56 (2004), 85–90

UDK 517.548

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RENORMALIZING ITERATED REPELLING GERMS OF C^{2}

Claudio Meneghini

Abstract. We find bounded-degree renormalizing polynomial families for iterated repelling germs of (C^{2}, 0). These families consist in contracting mappings and yield germs whose differentials have rank two.

1. Foreword

In this paper we deal with renormalizing families of iterates of holomorphic mappings with a repelling fixed point at 0. The process of renormalization could be described as composing on the right the elements of a holomorphic family F with a family of mappings of a fixed type (depending on the nature of the problem: linear, affine, polynomial ones, etc) and then extracting a normally convergent subsequence from the new family. This is a useful tool in complex analysis: in one variable, for instance, it allows, as shown in [5], Chap. 8, to get a ’linearizing’ change of coordinates (known as Königs coordinates) in the neighbourhood of a repelling (or attractive) fixed point p of a holomorphic endomorphism φ of ℙ^{1}: at a repelling fixed point, this coordinates could be in fact achieved by composing on the right the iterates φ^{n} with multiplication by [φ′(p)]^{−n} and letting n diverge.

Moreover, by a slightly different point of view, one could consider ’Zalcman’s renormalization lemma’ (see [7], [1], p. 9 ), which amounts to getting a normal family from a nonnormal one by means of composition on the right with a family of contracting affine functions: this yields an entire limit function and allows quite direct proofs of both great Picard’s and Montel’s theorems (see [1], pp. 10/11): this topics will not be discussed in this paper.

The situation is different in higher dimension: the following example, adapted from [6] (9.2) shows, for instance, the existence of nonnormal families of iterates of an automorphisms of ℂ^{2}, with a repelling fixed point in 0, which are by no means linearly renormalizable.

Let F ∈ Aut(ℂ^{2}) be defined by F (z, w) = (αz, βw + z^{2}), with |α| > 1, |β| > 1 and |β| ≥ |α|^{2}; F admits a repelling fixed point in 0, hence {F^{o k}} cannot be normal at 0.

AMS Subject Classification: 32 H 50

Now F^{o k}(z, w) = (α^{k}z, β^{k}w + β^{k−1}[1 + c + ⋯ c^{k−1}]z^{2}), where c = α^{2}/β; then F^{o k} o[F_{∗}(0)]^{−k}(z, w) = (z, w + β^{−1}[c^{−k} + c^{−k+1} + ⋯ + c^{−1}]z^{2}); since c ≤ 1, the coefficient of z^{2} diverges as k → ∞.

Thus, composing with ’division’ by differentials in 0 allows in general no kind of renormalization; however, see [3] for a deep analysis of renormalization by differentials in connection with ’Lattes examples’ in ℙ^{N}.

By contrast, we shall show that the family of the iterates of a repelling germ of (ℂ^{2}, 0) admits a renormalizing family consisting of polynomial mappings fixing 0, with uniformly bounded degrees and converging unformly on compacta to 0, in such a way that a holomorphic germ H tangent to the identity is yielded after renormalization. This fact implies, in particular, that the dimension of the image of any representative of H will be maximal.

2. Some definitions and lemmata

2.1. Elementary mappings

We recall that G = (g_{1}, g_{2}) is called an elementary mapping of ℂ^{2} if g_{1}(z) = c_{1}z_{1}, g_{2}(z) = c_{2}z_{2} + h(z_{1}), where the c_{k}’s are complex constants and h is a holomorphic function of z_{1}; G is an automorphism if and only if each c_{k} is nonzero.

Lemma 1. If H: ℂ^{2} → ℂ^{2} is defined by setting H(x, y) = (αx, βy + h(x)), where h(x) = ∑_{l=0}^{∞} η_{l}x^{l} and α, β are complex constants, then, for n ≥ 0, H^{∘n}(x, y) = (α^{n}x, [∑_{k=0}^{n−1} β^{k}h(α^{n−1−k}x)] + β^{n}y); moreover, if α ≠ 0, β ≠ 0, then

H is invertible and there holds H^{−n}(x, y) = (α^{−n}x, [− ∑_{k=1}^{n} β^{−k}h(α^{−n−1+k}x)] + β^{−n}y).

Proof : by induction on n.

Let now p_{N}(x) = ∑_{l=0}^{N} η_{l}x^{l} be the N-th degree truncation of the development of h and

H_{N}^{−n}(x, y) = (α^{−n}x, [− ∑_{k=1}^{n} β^{−k}p_{N}(α^{−n−1+k}x)] + β^{−n}y)

the corresponding truncation of H^{−n}; note that, if |α| > 1 and |β| > 1, then H_{N}^{−n} is a polynomial contracting mapping for every n ≥ 0 and {H^{−n}} → 0 uniformly on compacta in ℂ^{2}.

Theorem 2. If |α| > 1, |β| > 1 and |β| < |α|^{N}, then {H^{∘n} ∘ H_{N}^{−n}} converges uniformly on compacta to a lower triangular automorphism of the form H(x, y) = (x, h(x) + y) for a suitable entire function ψ.

Proof. Trivially (H^{∘n} ∘ H_{N}^{−n})_{1} (x, y) ≡ x and

(H^{∘n} ∘ H_{N}^{−n})_{2} (x, y) = ∑_{k=0}^{n−1} β^{k}h(α^{−1−k}x) − ∑_{k=1}^{n} β^{n−k}p_{N}(α−n − 1 + kx) + y

= ∑_{k=0}^{n−1} β^{k}h(α^{−1−k}x) − ∑_{k=0}^{n−1} β^{k}p_{N}(α−1 − kx) + y

Renormalizing iterated repelling germs of C^{2} 87

= ∑_{k=0}^{n−1} β^{k}R_{N}(α−1 − kx) + y := ψ_{n}(x) + y,

where R_{N} is the N-th remainder in the development of h.

Now h_{n}(x) = ∑_{k=0}^{n−1} β^{k} ∑_{l=N}^{∞} η_{l}(α^{k+1})^{−l}x^{l}; since, for N ≥ l, we have

|β^{k}α^{−(k+1)l}| ≤ |βα^{−N}|^{k},

we can let n diverge and exchange the order of summation, getting, uniformly on compact sets:

h(x) := lim_{n→∞} ψ_{n}(x) = ∑_{l=N}^{∞}( ∑_{k=0}^{∞} β^{k}(α^{k+1})^{−l})η_{l}x^{l}
= [ ∑_{k=0}^{∞}(βα^{−l})^{k}]α^{−l}η_{l}x^{l} = ∑_{l=N}^{∞}(α^{l} − β)^{−1}η_{l}x^{l};  (1)

by comparison with h = ∑_{l=0}^{∞} η_{l}x^{l}, we see that the last series in (1) represents an entire function ψ: this ends the proof.

2.2. A lemma by Rosay and Rudin

We recall the lemma from the Appendix of [6], specialized to ℂ^{2}.

Lemma 3. Let V be a neighbourhood of 0 in ℂ^{2}, F : V → ℂ^{2} a holomorphic mapping with F (0) = 0 and F_{∗}(0) lower triangular ; suppose that all eigenvalues λ_{i} of F_{∗}|_{0} := A satisfy |λ_{i}| < 1. Then there exist: (i) an elementary polynomial automorphism G of ℂ^{2} such that G(0) = 0 and G_{∗}|_{0} = A (thus c_{i} = λ_{i} for every i), (ii) polynomial applications T_{m}: ℂ^{2} → ℂ^{2}, with T_{m}(0) = 0, T_{m}_{∗}|_{0} = id such that G^{−1} o T_{m} o F − T_{m} = O(|z|^{m}), (m = 2, 3, …).

2.3. A lemma on attractive germs

The following lemma shows that a holomorphic mapping has a contracting behaviour near an attractive fixed point.

Lemma 4. Let V be a neighbourhood of 0 in ℂ^{N}, F : V → ℂ^{N} a holomorphic mapping admitting an attractive fixed point at 0: then there exists A < 1 and a neighbourhood ℝ ⊂ V of 0 such that F^{n}(ℝ) ⊂ A^{n}ℝ.

Proof. By Schur’s lemma, we may suppose, without loss of generality, that

F_{∗}(0) = ( \begin{matrix}λ_{1} & … & … & … & 0 \\ a_{21} & λ_{2} & … & … & 0 \\ ⋮ & ⋮ & ... & & ⋮ \\ ⋮ & ⋮ & & ... & ⋮ \\ a_{N1} & … & … & a_{N N−1} & λ_{N} \end{matrix} ) ,

where the λ_{k}’s (with |λ_{1}| ≤ |λ_{2}| ≤ ⋯ ≤ |λ_{N}|) are the eigenvalues of F_{∗}(0) and the a_{jk}’s complex constants.

Set

E_{ε} = ( \begin{matrix}ε^{N} & 0 & … & 0 \\ 0 & ε^{N−1} & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & … & … & ε \end{matrix} ) :

if ε is small enough, there exists A < 1 such that

∥E_{ε}^{−1}F_{∗}(0)E_{ε}∥(= ∥E_{ε}^{−1}F_{∗}E_{ε}(0)∥) < A;

thus there exists ϱ > 0 such that

E_{ε}^{−1} ∘ F ∘ E_{ε}(B(0, ϱ)) ⊂ B(0, ϱ),

hence, if p ∈ B(0, ϱ), ∥E_{ε}^{−1}F_{∗}^{n}E_{ε}(p)∥ < A^{n} and ∥E_{ε}^{−1}F^{n}E_{ε}(p)∥ < A^{n}∥p∥ ≤ A^{n}ϱ, i.e. E_{ε}^{−1} ∘ F^{n} ∘ E_{ε}(B(0, ϱ)) ⊂ B(0, A^{n}ϱ).

This fact will eventually imply

F^{n}(E_{ε}(B(0, ϱ))) ⊂ E_{ε}(B(0, A^{n}ϱ)) = A^{n}E_{ε}(B(0, ϱ));

we conclude by setting ℝ = E_{ε}(B(0, ϱ)).

3. The main result

Lemma 5. Let f be a holomorphic mapping in a neighbourhood of 0 ∈ ℂ^{2}, with f(0) = 0 and f_{∗}(0) attractive. There exist: a neighbourhood ℝ of 0, a biholomorphic mapping Ψ : ℝ → ℂ^{2} with ψ(0) = 0 and ψ_{∗}(0) = id and an elementary polynomial automorphism G of ℂ^{2} such that G^{n}oΨ = Ψo f^{n} for each n ≥ 0.

Proof: By Lemma 4 there exists a neighbourhood ℝ of 0 such that f^{∘n}(ℝ) ⊂ A^{n}ℝ for a suitable real constant A < 1 and every n > 0.

We may suppose that f_{∗}(0) is lower triangular (necessarily attractive) at 0: Lemma 3 gives us a lower triangular polynomial automorphism G : ℂ^{2} → ℂ^{2} such that G(0) = 0 and G_{∗}(0) = f_{∗}(0); by Lemma 1, there exists a complex constant γ such that |G^{−k}(w) − G^{−k}(w′) ≤ γ^{k}|w − w′| for each (w, w′) ∈ [𝔻 (0, 1/2)]. Take an integer m such that A^{m} ≤ 1/γ: Lemma 3 yields a polynomial application T := T_{m} correspondingly.

Let us proceed exactly like in the proof if the Theorem in the appendix of [6] (see from line 29 of page 84 up to line 4 of page 85), whose notations we have kept; this shows that the limit lim_{k→∞} G^{−k} ∘ T ∘ f^{k} exists uniformly on each compact set of ℝ so it canonically defines a holomorphic application Ψ : ℝ → ℂ^{2} such that Ψ(0) = 0, Ψ_{∗}(0) = id and G^{n}oΨ = Ψo f^{n}.

Theorem 6. Let h be a repelling holomorphic germ of (ℂ^{2}, 0), with h_{∗} admitting the eigenvalues α, β such that β ≥ α > 1 and |β| < |α|^{N}; then there exists a sequence {Q_{n}} of polynomial mappings which

• are contracting in a neighbourhood of 0;

• converge uniformly on compacta to 0 in ℂ^{2};

• have uniformly bounded degrees such that {h∘Q_{n}} converges to a holomorphic germ H of (ℂ^{2}, 0), with H_{∗}(0) = id.

Proof. Let (U, h) be a representative of h with inverse (V, f); maybe shrinking V we may suppose, by Lemma 4, that there exists A ∈ ℝ such that f^{n} ⊂ A^{n}V (i.e. V = ℝ with respect to the notation of Lemma 4.

By lemma 5 there exist: a biholomorphic mapping ψ : V → ℂ^{2} tangent to the identity in 0 and an elementary polynomial automorphism G of ℂ^{2} such that f^{n}(z) = ψ^{−1}G^{∘n}ψ(z), for each n > 0 and z ∈ V: thus h^{n}(z) = ψ^{−1}G^{−n}ψ(z), for each n > 0 and z ∈ f^{n}(V).

Set H := G^{−1}: we have H_{∗}(0) = G_{∗}^{−1}(0) = f_{∗}^{−1}(0) = h_{∗}(0) hence H will be defined by setting H(x, y) = (αx, βy + h(x)), with h entire and h(0) = 0.

For each M ≥ 0, let Θ_{M} be a polynomial mapping such that

ψ ∘ Θ_{M} = id + O(|z|^{M}).

By Theorem 2, {H^{∘n}} admits a family of contracting polynomial mappings {P_{n}} converging uniformly on compacta to 0, with P_{n}(0) = 0 and uniformly bounded degrees such that {H^{∘n}∘P_{n}} converges uniformly on compacta to an entire mapping S on ℂ^{2} (note that P_{n} := H_{N}^{−n} with respect to the notation of Theorem 2).

Since |α| ≤ |β| and h(0) = 0 , we have:

| ∑_{k=1}^{n} β^{−k}h(α^{−n−1+k}u)| = O(n|α|^{−n}|u|);

thus P_{n}(z) = O ((n + 1)|α|^{−n}|z|), where z = (u, v); moreover

ψ ∘ Θ_{M} ∘ P_{n} = P_{n} + O ([(n + 1)|α|^{−n}|z|]^{M}) .

Now P_{n}^{−1} ∘ f^{n} is tangent to the identity at 0 for all n, hence we may assume that there exists a neighbourhood A of 0 such that P_{n}^{−1} ∘ f^{n}(V) ⊃ A.

This eventually implies that, if z ∈ A,

ψ ∘ h^{n} ∘ Θ_{M} ∘ P_{n} = H^{−n} ∘ ψ ∘ Θ_{M} ∘ P_{n}
= H^{−n} ∘ ψ ∘ Θ_{M} ∘ (P_{n} + O ([(n + 1)|α|^{−n}|z|]^{M}))
= S + O ((n + 1)|β|^{n} [(n + 1)|α|^{−n}|z|]^{M})
= S + O ((n + 1)^{M+1}|α|^{n(N−M)}|z|^{M}) .

By choosing M > N and passing to germs at 0, we see that Ψ ∘ h^{n} ∘ Θ_{M} ∘ P_{n} converges to S, hence h^{n} ∘ Θ_{M} ∘ P_{n} converges to Ψ^{−1} ∘ S.

Now the Θ_{M} ∘P_{n}’s are polynomial mappings, contracting in a neighbourhood of 0, with uniformly bounded degrees, since so are the P_{n}’s; they converge uniformly on compacta to 0 by the corresponding property of the P_{n}’s; moreover, Ψ^{−1} is tangent to the identity at 0 and rk(S_{∗}(0)) = 2 by Theorem 2: setting H = Ψ^{−1} ∘S ends the proof.

REFERENCES

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[3] Ch. Dupont, Propriétés extrémales et caractéristiques des exemples de Lattès, Thèse de doctorat, université Paul Sabatier, Toulouse, le 29/11/2002.

[4] M. Gromov, Foliated plateau problem: part II: harmonic maps of foliations, GAFA, 1, 3 (1991), 253–320.

[5] J. Milnor, Dynamics in One Complex Variable, Friedr. Vieweg und Sohn Verlagsgesellshaft mbH, Braunschweig/Wiesbaden, 1999.

[6] J.-P. Rosay, W. Rudin, Holomorphic maps from C^{n} to C^{n}, Trans. of the A.M.S. 310, 1, (1988).

[7] L. Zalcman, Normal families: new perspectives, Bull. Amer. Math. Soc. 35 (1998).

(received 06.05.2003)

Dept. of Mathematics, University of Parma, 85, Strada M.D’Azeglio, 43100 Parma (I), Italy

 E-mail: clamengh@bluemail.ch