Non-linearizability of cubic-perturbed analytic germs 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

September 17, 1999

1991 Mathematics Subject Classification. Primary 58F23, 30C62, 30D05, 39B12

Abstract

In this paper, we consider the non-linearizability of analytic germs with irrationally indifferent fixed points. Assume that an analytic germ f has an irrationally indifferent fixed point at the origin and its multiplier satisfies the Tortrat condition, which is a generalization of the Cremer condition, of degree three. Then a cubic perturbation of f is non-linearizable at the origin if this perturbation is large enough.

1 Introduction

For an analytic germ f at the origin with a fixed point f(0) = 0, we call λ := f′(0) the multiplier of f at the origin. If λ = exp(2πiα) for some α ∈ ℝ \ ℚ, the origin is said to be an irrationally indifferent fixed point of f. The linearization problem of f at the origin is whether there exists a holomorphic local change of coordinate z = h(w) with h(0) = 0 and h′(0) ≠ 0 which conjugates f to the linear map w ↦ λw. If such h exists, f is called (analytically) linearizable at the origin.

In this paper, we consider the non-linearizability of analytic germs at irrationally indifferent fixed points.

∗Partially supported by JSPS Research Fellowships for Young Scientists

From now on, we always assume λ := exp(2πiα) for α ∈ (ℝ\ℚ)∩(0, +1). For a precise analysis of α, we consider the continued fraction expansion of α:

α = \frac{1}{a_{1}} + \frac{1}{a_{2}} + ⋯

where the a_{j} is uniquely determined positive integers and also consider the n-th convergent of it:

\frac{p_{n}}{q_{n}} = \frac{1}{a_{1}} + \frac{1}{a_{2}} + ⋯ + \frac{1}{a_{n}}

where (p_{n}, q_{n}) = 1 and p_{n}, q_{n} > 0. We set for d ≥ 2,

R_{λ,d} := {f; rational map of degree d, f(0) = 0 and f′(0) = λ} and (1)
P_{λ,d} := {f ∈ R_{λ,d}; f is a polynomial}.  (2)

Brjuno showed in [2] that if α satisfies the Brjuno condition

∑_{n} \frac{log q_{n+1}}{q_{n}} < +∞,  (B)

then any analytic germ f at the origin with f(0) = 0 and f′(0) = λ is linearizable at the origin. We call α satisfying B a Brjuno number. Conversely, Yoccoz showed in [9] that the quadratic polynomial Q(z) = exp(2πiα)z + z^{2} is non-linearizable at the origin if α is not a Brjuno number. Therefore Q is linearizable at the origin if and only if α is Brjuno number. However it remains open whether there exists a non-linear rational map linearizable at an irrationally indifferent fixed point with a non-Brjuno multiplier.

On the other hand, there is a well-known sufficient condition to obtain the non-linearizability of any element of R_{λ,d}. Under the Cremer condition

sup_{n} \frac{log q_{n+1}}{d^{q_{n}}} = +∞,  (Cr_{d})

any rational map f ∈ R_{λ,d} is non-linearizable at the origin (see [3]). Furthermore, Tortrat showed in [8] strictly weaker condition than Cr_{d}

lim sup_{n→+∞} \frac{log q_{n+1}}{d^{q_{n}}} > 0  (T_{d})

implies that any polynomial f ∈ P_{λ,d} is non-linearizable at the origin. However these conditions depend on d ≥ 2. In fact, Cr_{d+1} (resp. T_{d+1}) is strictly stronger than Cr_{d} (resp. T_{d}). Can we obtain a weaker non-linearizability condition independent of d?

The main theorem in this paper is the following.

Main Theorem. Let λ = exp(2πiα) for α ∈ ℝ \ ℚ and f be an analytic germ with f(0) = 0 and f′(0) = λ. There exists an absolute constant c_{1} > 0 such that if α satisfies T_{3}, then a cubic-perturbation f(z) + bz^{3} of f is nonlinearizable at the origin for any complex number b with |b| > c_{1}/(r_{f}^{2}). Here r_{f} > 0 is the radius of univalency of f at the origin, i.e.

r_{f} := sup{r > 0; f is injective on |z| < r}.

In Section 2, we shall prove Main Theorem.

2 Proof of Main Theorem

Let S be the set of holomorphic functions univalent on 𝔻 with f(0) = 0 and |f′(0)| = 1 and let S_{λ} be the set of elements f of S which satisfies f′(0) = λ. We set 𝔻_{r} := {z; |z| < r} for r > 0.

For f ∈ S, we define a cubic-perturbation of f:

f_{a,b}(z) := a^{−1}f(az) + bz^{3}

where 0 < |a| ≤ 1 and b ∈ ℂ. Furthermore we set W := 𝔻_{15/2}, W̃ := 𝔻_{1/3} ∩ f_{a,b}^{−1}(W) and a constant c_{0} := 225. Recall that a triplet (Ũ, U, f) is called a cubic-like map if Ũ and U are simply connected proper subdomains of ℂ, Ũ is relatively compact in U and f : Ũ → U is a proper holomorphic map of degree three.

The following is a special case of Lemma 2.1 in [6].

Lemma 2.1. If |b| ≥ c_{0}, the triplet (W̃, W, f_{a,b}) is a cubic-like map.

We take a smooth function η : ℝ → [0, 1] which is identically 1 on (−∞, 1/3] and 0 on [15/2, +∞) and define

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

for f ∈ S_{λ}, 0 < |a| < 2/15 and |b| ≥ c_{0}. Then f̃_{a,b} : ℂ → ℂ is in C^{∞}, coincides with f_{a,b} on \overline{𝔻_{1/3}} and with λz + bz^{3} on ℂ \ 𝔻_{15/2}. Moreover it converges to λz +bz^{3} in C^{∞}-topology on ℂ if a tends to 0, and this convergence is uniform in f ∈ S_{λ} and |b| ≥ c_{0}.

From now on we assume |b| = c_{0}. We can conclude the following.

Lemma 2.2. There exist an a_{0} ∈ (0, 2/15] and a continuous function k : [0, a_{0}] → [0, 1) with k(0) = 0 such that for f ∈ S_{λ}, |b| = c_{0} and 0 < |a| < a_{0}, the map f̃_{a,b} is a branched covering map of ℂ of degree three and it satisfies

|\frac{¯∂f̃_{a,b}(z)}{∂f̃_{a,b}(z)}| ≤ k(|a|) (1/3 ≤ |z| ≤ 15/2).

We identify a Beltrami coefficient on an open set U with a function μ ∈ L^{∞}(U) such that ∥μ∥_{∞} < 1. 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}, |b| = c_{0} and 0 < |a| < a_{0}, there exists a unique Beltrami coefficient μ = μ_{f,a,b} on ℂ which is invariant under the pullback by f̃_{a,b} and agrees with \frac{¯∂f̃_{a,b}}{∂f̃_{a,b}} on 1/3 ≤ |z| ≤ 15/2 and is 0 on (ℂ − W) ∪ ⋂_{n≥0} f_{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,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 2.3 (cf. [4]). There exists an A ∈ ℂ such that ϕ ∘ f̃_{a,b} ∘ ϕ^{−1}(z) = λz + Az^{2} + bz^{3}.

Proof. Since μ(ϕ∘f_{a,b}) = μ(ϕ), the map ϕ∘f̃_{a,b} ∘ϕ^{−1} : ℂ → ℂ is holomorphic and fixes the origin. Thus it is a branched holomorphic covering map of ℂ of degree three fixing the origin so we can write ϕ ∘ f̃_{a,b} ∘ ϕ^{−1}(z) = λ′z + Az^{2} + b′z^{3}. The multiplier of a fixed point of a holomorphic germ is topologically invariant if its modulus is equal to 1, so we have λ′ = λ (cf. [5], see also [7]). On a neighborhood of the point at infinity, ϕ_{f,a,b}(z) = z + (lower terms) and f̃_{a,b}(z) = λz + bz^{3}. On the other hand, ϕ(f̃_{a,b}(z)) = λϕ(z) + A(ϕ(z))^{2} + b′(ϕ(z))^{3}. Thus ϕ(λz + bz^{3}) − (λz + bz^{3}) = (b′ − b)z^{3} + (lower terms) when |z| is sufficiently large. Since this quantity remains bounded as |z| → +∞ by (iii), it is necessary that b′ − b = 0.

We set c_{1} := c_{0}/(a_{0}^{2}). The following is equivalent to Main Theorem.

Proposition. Let λ = exp(2πiα) (α ∈ ℝ \ ℚ) and f ∈ S_{λ}. If α ∈T_{3}, then f_{1,b} is non-linearizable at the origin for any |b| > c_{1}.

Proof. Noting that f̃_{a,b} ≡ f_{a,b} on 𝔻_{1/3}, we can see f_{a,b} is non-linearizable at the origin if and only if the cubic polynomial ϕ ∘ f̃_{a,b} ∘ ϕ^{−1} ∈ P_{λ,d} is so. If α ∈T_{3}, it is non-linearizable by the Tortrat Theorem. On the other hand, af_{a,b}(z/a) = f_{1,b/(a^{2})} and for any |b_{0}| > c_{1}, there exist a ∈ {0 < |a| < a_{0}} and b ∈ {|b| = c_{0}} such that b/(a^{2}) = b_{0}.

Consequently the proof of Main Theorem is completed.

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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