Non-linearizability of Polynomials at Irrationally Indifferent Fixed Points

Yûsuke Okuyama

Department of Mathematics, Graduate School of Science

Kyoto University, Kyoto 606-8502, Japan E-mail; okuyama@kusm.kyoto-u.ac.jp

October 19, 1998

1991 Mathematics Subject Classification. Primary 58F23, 30C62, 30D05, 39B12 Keywords and phrases. linearizability, irrationally indifferent fixed point, Brjuno number, cubic-like map

Abstract In this paper, we consider the non-linearizability of polynomials with irrationally indifferent fixed points. Under the assumption that there exists a cubic polynomial which is linearlizable at an irrationally indifferent fixed point with a non-Brjuno multiplier, we show that, for every degree more than two, one can construct a holomorphic family of possible maximal dimension consisting of polynomials linearizable at the fixed point.

1 Introduction

Let f be a germ of a holomorphic map at z_{0} ∈ ℂ with f(z_{0}) = z_{0} and call λ := f′(z_{0}) the multiplier of f at z_{0}. We consider the linearization problem of f at z = z_{0}, i.e. whether there exists a holomorphic local change of coordinate z = h(w) with h(0) = z_{0} and h′(0) ≠ 0 which conjugates f to the linear map w ↦ λw. If such h exists, the germ f is said to be linearizable at z_{0} and we call h the (analytic) linearizing map of f at z_{0} or the solution of the linearization problem of f at z_{0}.

If λ = 0, Böttcher showed that f(z) = z^{n} + a_{n+1}z^{n+1} + ⋯ is always analytically conjugate to w ↦ w^{n}. In the case 0 < |λ| < 1 (resp. 1 < |λ|), Kœnigs showed that f is always linearizable at z_{0} and the fixed point z_{0} is called attracting (resp. repelling). If |λ| = 1 and λ is a root of unity, f is always non-linearizable at z_{0} and z_{0} is called parabolic (for the details, see [5]).

If |λ| = 1 and λ is not a root of unity, the fixed point z_{0} of f is said to be irrationally indifferent. In this case, some are linearizable at z_{0}, others non-linearizable at z_{0}. For example, a rational or entire function f which has an irrationally indifferent fixed point z_{0} is linearizable there if and only if the fixed point z_{0} belongs to the Fatou set of f (cf. [5] and [8]).

From now on, we always assume that a real number α is irrational. Let G_{z_{0},α} be the set of holomorphic germs at z_{0} which have an irrationally indifferent fixed point z_{0} with multiplier λ = exp(2πiα). If α is a Brjuno number, any germ f ∈ G_{z_{0},α} is linearizable at z_{0} ([2]). We define

B := {α ∈ ℝ − ℚ ; α is a Brjuno number}.

The Brjuno numbers are defined in terms of the continued fractional expansion. Since we do not need the definition of them, we omit it. For the precise definition of them, see, for example, [5].

In [9] Yoccoz showed that the Brjuno condition is optimal. In other words, if α ∉ B, we can find a germ f ∈ G_{z_{0},α} which is non-linearizable at z_{0}, indeed, the quadratic polynomial P (z) = e^{2πiα}z + z^{2} is non-linearizable at the origin. Thus it follows that a quadratic polynomial with an irrationally indifferent fixed point with multiplier e^{2πiα} is linearizable there if and only if α is a Brjuno number, since it is affinely conjugate to P (z). Can we extend this result for polynomials of degree more than two?

In this paper, we shall study the linearizability of polynomials of degree more than two at irrationally indifferent fixed points.

In [7] Pérez-Marco showed that for λ = exp(2πiα) (α ∉ B) and d > 2, the family

𝒫_{λ,d} :={P (z) = λz + a_{2}z^{2} + ⋯ + a_{d}z^{d}; (a_{2}, …, a_{d}) ∈ ℂ^{d−1}} ≅ ℂ^{d−1}

contains an open dense subset whose elements are non-linearizable at the origin. It is not known whether this subset coincides with 𝒫_{λ,d}.

The main theorem in this paper is the following.

Main Theorem. Fix λ = e^{2πiα} (α ∉ B). Suppose there exists A ∈ ℂ such that the cubic polynomial P_{λ,A}(z) = λz + Az^{2} + z^{3} is linearizable at the origin, then for any d ≥ 3, the family 𝒫_{λ,d} contains a holomorphic subfamily of complex dimension d − 2 whose elements are linearizable at the origin.

Remark. Fix λ = e^{2πiα} (α ∉ B). If 𝒫_{λ,d} has a nonlinear element P_{d} linearizable at the origin, \frac{1}{c}P_{d}(cz) is also linearizable at the origin for any c ∈ ℂ^{∗}, so 𝒫_{λ,d} always contains a holomorphic subfamily of complex dimension one whose elements are linearizable at the origin.

By Pérez-Marco’s result, if 𝒫_{λ,d} contains a holomorphic subfamily consisting of elements linearizable at the origin, the complex dimension of it is at most d − 2. In this sense, Main Theorem says that for d ≥ 3, 𝒫_{λ,d} contains a holomorphic subfamily of possible maximal dimension consisting of elements linearizable at the origin if it is impossible to extend Yoccoz’s result on quadratic polynomials to the case of cubic polynomials. Thus we have the following.

Corollary 1.1. Fix λ = e^{2πiα} (α ∉ B). The cubic polynomial P_{λ,A} is nonlinearizable at the origin for any A ∈ ℂ if and only if the α satisfies the degeneration property : For some d ≥ 3, 𝒫_{λ,d} does not contain a holomorphic subfamily of complex dimension d − 2 whose elements are linearizable at the origin.

Corollary 1.2. If every non-Brjuno number has the degeneration property, it holds that a cubic polynomial with an irrationally indifferent fixed point with multiplier e^{2πiα} is linearizable there if and only if α is a Brjuno number.

Remark. Fix λ = e^{2πiα} (α ∉ B) and d > 3. If 𝒫_{λ,d} has an element P_{d} linearizable at the origin, by computing the dimension of Teichmüller space of the polynomial P_{d} (c.f. McMullen-Sullivan [4]), it may be possible to show a result similar to the above. But in this paper, we explicitly construct a holomorphic subfamily in the proof of Main Theorem.

In the rest of this paper, we shall prove Main Theorem. We fix λ = e^{2πiα}. In Section 2, we shall show that for a univalent function f on 𝔻 which has a fixed point z = 0 with multiplier λ, the function f_{a,A,b}(z) := a^{−1}f(az) + Abz^{2}+b^{2}z^{3} can be regarded as a cubic-like map under a suitable condition. In Section 3, we shall show that the cubic-like map is quasiconformally conjugate to a cubic polynomial which has a fixed point at the origin with multiplier λ. In Section 4, we shall complete the proof of Main Theorem.

2 Cubic perturbation of univalent maps

We set

S := {f; holomorphic and univalent functions on 𝔻,
f(0) = 0, and |f′(0)| = 1},
S_{λ} := {f ∈ S; f′(0) = λ} and
𝔻_{r} := {z; |z| < r} for r > 0.

Let λ = e^{2πiα} for α ∈ ℝ − ℚ, and let f be an element of S_{λ}. We set, for a ∈ \overline{𝔻} − \{0\}, A ∈ ℂ and b ∈ ℂ,

f_{a,A,b}(z) := a^{−1}f(az) + Abz^{2} + b^{2}z^{3}.

A triplet (Ũ, U, f) is called a cubic-like map if Ũ and U are simply connected proper subdomains of ℂ, and Ũ is relatively compact in U, and f : Ũ → U is a proper holomorphic map of degree 3.

Lemma 2.1. For A ∈ ℂ and b ∈ ℂ, we define

R_{A,b} :=\frac{10}{9}|A||b| + \frac{15}{2},
B_{A,b} :=27R_{A,b} + 3|A||b| + \frac{81}{4} = 33|A||b| + \frac{891}{4},
W :=𝔻_{R_{A,b}} = {z; |z| < R_{A,b}} and
W̃ :=𝔻_{1/3} ∩ f_{a,A,b}^{−1}(W).

For f ∈ S, a ∈ \overline{𝔻} − \{0\}, A ∈ ℂ and |b|^{2} > B_{A,b}, the triplet (W̃, W, f_{a,A,b}) is a cubic-like map.

Proof. It is sufficient to prove this in the case a = 1. Since f ∈ S, it follows that \frac{|z|}{(1+|z|)^{2}} ≤ |f(z)| ≤ \frac{|z|}{(1−|z|)^{2}} for z ∈ 𝔻. In particular, if |z| = 1/3, we have 3/16 ≤ |f(z)| ≤ 3/4 and it follows that

|f_{1,A,b}(z)| ≥ |b^{2}z^{3}| − |Abz^{2} + f(z)| ≥ \frac{|b|^{2}}{27} − \frac{|A||b|}{9} − \frac{3}{4} > R_{A,b}. (1)

Thus f_{1,A,b}(𝔻_{1/3}) contains the disk \overline{W}, so f_{1,A,b} : W̃ → W is proper and W̃ is simply connected by the maximum modulus principle. And for |z| = 1/3 and z_{1} ∈ W, it follows from (1) that

|b^{2}z^{3} − z_{1}| >|b^{2}z^{3}| − R_{A,b} > |Abz^{2} + f(z)| and
\sqrt[3]{|\frac{z_{1}}{b^{2}} |} <\sqrt[3]{\frac{R_{A,b}}{|b|^{2}}} < \sqrt[3]{\frac{R_{A,b}}{27R_{A,b}}} = \frac{1}{3}

since |b|^{2} > B_{A,b} > 27R_{A,b} by definition. Thus by the theorem of Rouché, f_{1,A,b} : W̃ → W is a proper map of degree 3.

If W̃ is not connected, then the number of connected components of W̃ is three or two. First, if it is three, the connected component of W̃ containing the origin is conformally mapped to W by f_{1,A,b}. However this contradicts the Schwarz lemma because |λ| = 1. Second, if it is two, two cases occur. If f_{1,A,b} conformally maps the connected component of W̃ containing the origin onto W, we can derive a contradiction by the same argument as above. Otherwise f_{1,A,b} conformally maps the other component W′ of W̃ onto W. So there exists the only one point z_{0} ∈ W′ such that f_{1,A,b}(z_{0}) = 0. We define ϕ := (f_{1,A,b}|W′)^{−1}, and ψ(z) := ϕ(R_{A,b}z). Then ψ conformally maps 𝔻 onto W′, and ψ(0) = z_{0} (see Figure 1).

𝔻 0  0 z_{0}  ϕ  ψ \{|z| < \frac{1}{3}\} W′  < \frac{1}{6} _{\frac{1}{3}} f_{1,A,b}|W′  R_{A,b} W = \{|z| < R_{A,b}\} Figure 1:

By the Koebe one-quarter theorem, it follows that W′ contains the open disk of which the radius is \frac{1}{4}|ψ′(0)| = \frac{1}{4}R_{A,b}|ϕ′(0)|. Since 𝔻_{1/3} ⊃ W′ and W′ ∌ 0, we have \frac{1}{4}R_{A,b}|ϕ′(0)| < \frac{1}{6}. Hence

\frac{1}{|ϕ′(0)|} > \frac{3R_{A,b}}{2} .

On the other hand, we have

\frac{1}{|ϕ′(0)|} = |f′_{1,A,b}(z_{0})| ≤ |f′(z_{0})| + 2|A||b||z_{0}| + 3|b|^{2}|z_{0}|^{2}

and by the Koebe distortion theorem, |f′(z_{0})| < 9/2 for |z_{0}| < 1/3. Since f_{1,A,b}(z_{0}) = f(z_{0})+Abz_{0}^{2}+b^{2}z_{0}^{3} = 0 and |f(z_{0})| ≤ \frac{|z_{0}|}{(1−|z_{0}|)^{2}} , we have 3|b|^{2}|z_{0}|^{2}+ 2|A||b||z_{0}| < \frac{27}{4} + \frac{5}{3}|A||b| for |z_{0}| < 1/3. Hence

\frac{1}{|ϕ′(0)|} < \frac{9}{2} + \frac{27}{4} + \frac{5}{3}|A||b| = \frac{45}{4} + \frac{5}{3}|A||b| = \frac{3}{2}R_{A,b}.

This is a contradicton. So W̃ is connected, and the proof is completed.

3 Straightening of the cubic-like mapping

We define N(M) := \frac{33M+\sqrt{1089M^{2}+891}}{2} for M > 0. Then |b|^{2} > B_{A,b} if and only if |b| > N(|A|).

Let M be an arbitrary positive number and take a smooth function η : ℝ → [0, 1] which is identically 1 on (−∞, 1/3] and 0 on [R_{A,b}, +∞). And we define the round annulus A(M) := {b; N(M) < |b| < N(M) + 1}. There exists a constant R(M) > 0 which depends only on M such that R_{A,b} < R(M) for any A ∈ 𝔻_{M} and b ∈ A(M).

For f ∈ S_{λ}, a ∈ \overline{𝔻_{1/R(M)}} − \{0\}, A ∈ 𝔻_{M} and b ∈ A(M), we define

f̃_{a,A,b}(z) := η(|z|)f_{a,A,b}(z) + (1 − η(|z|))(λz + Abz^{2} + b^{2}z^{3}).

Then f̃_{a,A,b} : ℂ → ℂ is in C^{∞}.

Lemma 3.1. If a → 0, then f̃_{a,A,b}(z) converges to λz + Abz^{2} + b^{2}z^{3} in C^{∞}- topology on ℂ, and this convergence is uniform in f ∈ S_{λ}, A ∈ 𝔻_{M} and b ∈ A(M).

Proof. On ℂ − 𝔻_{R_{A,b}}, f̃_{a,A,b}(z) ≡ λz + Abz^{2} + b^{2}z^{3}. If we write f(z) = λz + ∑_{n=2}^{∞} a_{n}z^{n} then |a_{n}| ≤ n. Noting that R_{A,b} < R(M), we see that on 𝔻_{R_{A,b}},

|f_{a,A,b} − (λz + Abz^{2} + b^{2}z^{3})| ≤ ∑_{n=2}^{∞} |a_{n}||a|^{n−1}|z|^{n}

≤ |z| ∑_{n=2}^{∞} n|az|^{n−1} ≤ R(M) ∑_{n=2}^{∞} n|aR(M)|^{n−1}

< R(M)\frac{|a|R(M)(2 − |a|R(M))}{(1 − |a|R(M))^{2}}

so lim_{a→0} f_{a,A,b}(z) = λz + Abz^{2} + b^{2}z^{3} uniformly on 𝔻_{R_{A,b}} and in f ∈ S_{λ}, A ∈ 𝔻_{M} and b ∈ A(M). Since f̃_{a,A,b}(z)−(λz+Abz^{2}+b^{2}z^{3}) = η(z){f_{a,A,b}(z)− (λz + Abz^{2} + b^{2}z^{3})}, it follows that lim_{a→0} f̃_{a,A,b}(z) = λz + Abz^{2} + b^{2}z^{3} uniformly on ℂ and in f ∈ S_{λ}, A ∈ 𝔻_{M} and b ∈ A(M). Similarly, we can see the convergence of the derivatives of f̃_{a,A,b}.

Lemma 3.2. Both of two finite critical points of f̃_{a,A,b}(z) belong to {z; |z| < 1/3} if |a| is sufficiently small.

Proof. From Lemma 3.1, it is sufficient to prove that two critical points of z ↦ λz + Abz^{2} + b^{2}z^{3} is included in {z; |z| < 1/3}. Note that (λz + Abz^{2} + b^{2}z^{3})′ = λ + 2Abz + 3b^{2}z^{2}. Since |b| > N(M) i.e. |b|^{2} > B_{A,b}, we can see on |z| = 1/3,

|3b^{2}z^{2}| − |λ + 2Abz| ≥ \frac{B_{A,b}}{3} − 1 − \frac{2}{3}|A||b| > 0.

By the theorem of Rouché, both of critical points of λz + Abz^{2} + b^{2}z^{3} are in 𝔻_{1/3}.

We can conclude the following by the previous lemmas.

Lemma 3.3. There exist an a_{0} ∈ (0, \frac{1}{R(M)}] and a continuous function k : [0, a_{0}] → [0, 1) such that k(0) = 0 and for any f ∈ S_{λ}, A ∈ 𝔻_{M}, b ∈ A(M) and a ∈ \overline{𝔻}_{a_{0}} − \{0\}, the map f̃_{a,A,b} is a branched covering map of ℂ of degree 3 and it satisfies

|\frac{¯∂f̃_{a,A,b}(z)}{∂f̃_{a,A,b}(z)}| ≤ k(|a|) (1/3 ≤ |z| ≤ R_{A,b}).

Moreover, the Beltrami coeffitient \frac{¯∂f̃_{a,A,b}(z)}{∂f̃_{a,A,b}(z)} holomorphically depends on f ∈ S_{λ}, A ∈ 𝔻_{M}, b ∈ A(M) and a ∈ 𝔻_{a_{0}} − {0}.

For an open set U ⊂ ℂ, we identify a Beltrami coefficient on U with a function μ ∈ L^{∞}(U) such that ∥μ∥_{∞} < 1. And for a C^{1}-function f : U → V and a Beltrami coefficient μ on V, we define the pullback f^{∗}μ of μ on U by

(f^{∗}μ)(z) = \frac{\overline{∂f(z)}μ(f(z)) + ¯∂f(z)}{\overline{¯∂f(z)}μ(f(z)) + ∂f(z)}.

For f ∈ S_{λ}, A ∈ 𝔻_{M}, b ∈ A(M) and a ∈ 𝔻_{a_{0}} − {0}, there exists a unique Beltrami coefficient μ = μ_{f,a,A,b} on ℂ which is invariant under the pullback by f̃_{a,A,b} and agrees with \frac{¯∂f̃_{a,A,b}}{∂f̃_{a,A,b}} on 1/3 ≤ |z| ≤ R_{A,b} and is 0

on (ℂ − W) ∪ ⋂_{n≥0} f_{a,A,b}^{−n}(W̃). Since supp μ ⊂ W and ∥μ∥_{∞} ≤ k(a) < 1, by the Ahlfors-Bers theorem [1], there exists a unique quasiconformal homeomorphism ϕ = ϕ_{f,a,A,b} of ℂ onto itself which satisfies the following

(i) for a.e.z ∈ ℂ, ¯∂ϕ(z) = μ(z)∂ϕ(z),

(ii) ϕ(0) = 0 and

(iii) ϕ(z) − z is bounded on ℂ.

Lemma 3.4 (cf. [3]). There exists an A′ ∈ ℂ such that ϕ ∘ f̃_{a,A,b} ∘ ϕ^{−1}(z) = λz + A′z^{2} + b^{2}z^{3}, where A′ ∈ ℂ holomorphically depends on f ∈ S_{λ}, A ∈ 𝔻_{M}, b ∈ A(M) and a ∈ 𝔻_{a_{0}} − {0}.

Proof. Since μ(ϕ ∘ f_{a,A,b}) = μ(ϕ), it follows that ϕ ∘ f̃_{a,A,b} ∘ ϕ^{−1} : ℂ → ℂ is holomorphic and fixes the origin. So it is a branched covering map of ℂ of degree 3 fixing the origin. Thus we can write

ϕ ∘ f̃_{a,A,b} ∘ ϕ^{−1}(z) = λ′z + A′z^{2} + b′z^{3} (λ′, A′, b′ ∈ ℂ).

By the theorem of Naishul [6], the multiplier of the fixed point of a holomorphic map is topologically invariant when its modulus is 1. So we have λ′ = λ. Next, we show b′ = b^{2}. According to (iii), we have

ϕ_{f,a,A,b}(z) = z + c + (lower terms)  (2)

on a neighborhood of the point at infinity. When |z| is sufficiently large, f̃_{a,A,b}(z) = λz + Abz^{2} + b^{2}z^{3} by definition, and we note that ϕ(f̃_{a,A,b}(z)) = λϕ(z) + A′(ϕ(z))^{2} + b′(ϕ(z))^{3}. Therefore it follows that

ϕ(λz + Abz^{2} + b^{2}z^{3}) − (λz + Abz^{2} + b^{2}z^{3}) = (b′ − b^{2})z^{3} + {(A′ − Ab) + 3b′c}z^{2} + (lower terms).

Since this quantity remains bounded as |z| → +∞, it is necessary that b′ − b^{2} = 0 and A′ − Ab + 3b′c = 0. Thus it follows that b′ = b^{2} and A′ = Ab − 3b^{2}c.

Lemma 3.5. c = c(f, a, A, b) in (2) holomorphically depends on f ∈ S_{λ}, A ∈ 𝔻_{M}, b ∈ A(M) and a ∈ 𝔻_{a_{0}} − {0}. And c → 0 uniformly in f ∈ S_{λ}, A ∈ 𝔻_{M} and b ∈ A(M) as a → 0.

Proof. The former part follows from the Ahlfors-Bers theorem. Noting that ∥μ∥_{∞} ≤ k(|a|) and lim_{a→0} k(|a|) = 0 uniformly in f ∈ S_{λ}, A ∈ 𝔻_{M} and b ∈ A(M), we can see the latter part holds.

4 Proof of Main Theorem

Let α ∉ B and λ = e^{2πiα}. For A_{0} ∈ ℂ, we take M = M_{0} := 2|A_{0}| + 1. By Lemma 3.5, for a fixed 0 < ϵ < 1/3, there exists an a_{1} ∈ (0, a_{0}] which is independent of f ∈ S_{λ}, A ∈ 𝔻_{M_{0}} and b ∈ A(M_{0}) such that

3|b||c(f, a, A, b)| < ϵ (0 < |a| < a_{1}).

Then |A_{0}| < M_{0} − 2ϵ. We define a holomorphic map F_{f,a,b} on 𝔻_{M_{0}}:

A ↦ A − 3bc(f, a, A, b).

By the theorem of Rouché, there exists A_{1} = A_{1}(f, a, b) ∈ 𝔻_{M_{0}−ϵ} such that F_{f,a,b}(A_{1}) = A_{0}. From the implicit function theorem, A_{1} = A_{1}(f, a, b) holomorphically depends on f ∈ S_{λ}, b ∈ A(M_{0}) and a ∈ 𝔻_{a_{1}} − {0}. Thus we have the following.

Theorem 4.1. Fix λ = e^{2πiα} (α ∉ B). For A_{0} ∈ ℂ, we set M_{0} := 2|A_{0}| + 1. Then there exist an a_{1} ∈ (0, a_{0}], a holomorphic function A_{1} = A_{1}(f, a, b) with |A_{1}| < M_{0} of f ∈ S_{λ}, a ∈ 𝔻_{a_{1}} − {0}, b ∈ A(M_{0}), and also exists a quasiconformal homeomorphism ϕ = ϕ_{f,a,A_{1},b} of ℂ onto itself such that

ϕ ∘ f̃_{a,A_{1},b} ∘ ϕ^{−1}(z) = λz + A_{0}bz^{2} + b^{2}z^{3} = \frac{1}{b}P_{λ,A_{0}}(bz).

If P_{λ,A_{0}} is linearizable at the origin, then f_{a,A_{1},b}(z) = a^{−1}f(az)+A_{1}bz^{2} +b^{2}z^{3} is linearizable at the origin.

Proof. The former part has been proved in the above and the latter part follows from Lemma 3.4. Assume P_{λ,A_{0}} is linearizable at the origin and let D be the Siegel disk of \frac{1}{b}P_{λ,A_{0}}(bz). We have f_{a,A_{1},b}(ϕ^{−1}(D)) = ϕ^{−1}(D), so the fixed point z = 0 of f_{a,A_{1},b} is contained in the Fatou set of f_{a,A_{1},b}.

Proof of Main Theorem. Suppose P_{λ,A_{0}} is linearizable at the origin. If d = 3, Main Theorem is trivial. We consider, for d > 3, the family

U_{d} := {P (z) = λz + a_{2}z^{2} + ⋯ + a_{d}z^{d}; ∑_{n=2}^{d} n|a_{n}| ≤ 1} ⊂ S_{λ}.

We fix b ∈ A(M_{0}) and a ∈ 𝔻_{a_{1}} −{0}. If f(z) = λz +a_{2}z^{2} +a_{3}z^{3} +⋯+a_{d}z^{d} ∈ U_{d} and a_{d} ≠ 0, then we can see

f_{a,A_{1},b}(z) = λz + (a_{2}a + A_{1}b)z^{2} + (a_{3}a^{2} + b^{2})z^{3} + ∑_{n=4}^{d} a_{n}a^{n−1}z^{n}

is linearizable at the origin. The map

(a_{2}, a_{3}, …, a_{d}) ↦ (a_{2}a + A_{1}(a, b, a_{2}, a_{3}, …, a_{d})b, a_{3}a^{2} + b^{2}, a_{4}a^{3}, …, a_{d}a^{d−1})

is holomorphic on {(a_{2}, a_{3}, …, a_{d}); ∑_{n=2}^{d} n|a_{n}| < 1 and a_{d} ≠ 0} which is an open subset of ℂ^{d−1}. Note that the Jacobian of the map (a_{3}, …, a_{d}) ↦ (a_{3}a^{2} +b^{2}, a_{4}a^{3}, …, a_{d}a^{d−1}) is not zero. Therefore {f_{a,A_{1},b}; f ∈ U_{d}} is at least a d − 2 dimensional holomorphic subfamily of 𝒫_{λ,d} and all of the elements are linearizable at the origin. Consequently we have completed the proof of Main Theorem in Section 1.

In Proof of Main Theorem, we do not use Pérez-Marco’s result. Combining Pérez-Marco’s result and the argument in Proof of Main Theorem, we have the following.

Theorem 4.2. Fix λ = e^{2πiα} (α ∉ B). The cubic polynomial P_{λ,A_{0}}(z) = λz + A_{0}z^{2} + z^{3} is non-linearizable at the origin if the pair (α, A_{0}) satisfies the fulldimension property : For some d > 3, the holomorphic map

(a, b, a_{2}, …, a_{d}) ↦

(a_{2}a + A_{1}(a, b, a_{2}, a_{3}, …, a_{d})b, a_{3}a^{2} + b^{2}, a_{4}a^{3}, …, a_{d}a^{d−1}) (3)

has the maximal rank d − 1.

Proof. Suppose that P_{λ,A_{0}} is linearizable at the origin. Then the image of the map (3) is a d − 1 dimensional holomorphic subfamily of 𝒫_{λ,d} consisting of elements linearizable at the origin. It contradicts Pérez-Marco’s result.

Corollary 4.1. If any pair (α, A_{0}) (α ∉ B and A_{0} ∈ ℂ) has the fulldimension property, it holds that a cubic polynomial with an irrationally indifferent fixed point with multiplier e^{2πiβ} is linearizable there if and only if β is a Brjuno number.

ACKNOWLEDGEMENT. The author would like to express his gratitude to Prof. Masahiko Taniguchi and Prof. Toshiyuki Sugawa for many valuable discussions and advices.

References

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