Quasi-conformal deformations of nonlinearizable germs

Let $f(z) = e^{2\pi i \alpha}z + O(z^2), \alpha \in \mathbb{R}$ be a germ of holomorphic diffeomorphism in $\mathbb{C}$. For $\alpha$ rational and $f$ of infinite order, the space of conformal conjugacy classes of germs topologically conjugate to $f$ is parametrized by the Ecalle-Voronin invariants (and in particular is infinite-dimensional). When $\alpha$ is irrational and $f$ is nonlinearizable it is not known whether $f$ admits quasi-conformal deformations. We show that if $f$ has a sequence of repelling periodic orbits converging to the fixed point then $f$ embeds into an infinite-dimensional family of quasi-conformally conjugate germs no two of which are conformally conjugate.


Introduction
Let f (z) = e 2πiα z + O(z 2 ), α ∈ R/Z be a germ of holomorphic diffeomorphism fixing the origin in C. We consider the question of when f admits quasi-conformal deformations, i.e. when do there exist germs g which are quasi-conformally but not conformally conjugate to f ? If f is linearizable (i.e. analytically conjugate to the rigid rotation R α (z) = e 2πiα z) then any germ topologically conjugate to f is linearizable. In the nondegenerate parabolic case (i.e. α = p/q ∈ Q, f q = id), the quasi-conformal conjugacy class of f contains an infinite dimensional family of conformal conjugacy classes parametrized by the Ecalle-Voronin invariants ( [Eca75], [Vor81]). In the irrationally indifferent nonlinearizable case (α irrational, f not linearizable), it seems to be unknown whether quasi-conformal deformations are possible. We show the following: λ n ∈ C, |λ n | > 1} such that all conjugates of g Λ = h Λ • f • h −1 Λ are holomorphic, and g Λ1 , g Λ2 are conformally conjugate if and only if all but finitely many terms of Λ 1 and Λ 2 agree. For fixed z, h Λ (z) depends holomorphically on each λ n .
The proof proceeds as follows: given Λ = (λ n ) the germ g Λ is obtained by quasiconformally deforming f near its periodic orbits. Each periodic orbit is attracting for f −1 , and it is possible to construct an f -invariant Beltrami differential on each basin of attraction such that the quasi-conformal map h Λ rectifying the Beltrami differential conjugates f to a holomorphic germ g Λ having multipliers λ n at the periodic orbits. The multipliers at periodic orbits being invariant under conformal conjugacies, the conclusion of the theorem follows.
Examples of germs satisfying the hypothesis of the theorem are germs of rational maps of degree d with α satisfying the Cremer condition of degree d (see for example Milnor [Mil99], Ch. 8) lim sup log log q n+1 q n > log d (where (p n /q n ) are the continued fraction convergents of α) which ensures that the fixed point is accumulated by periodic orbits (only finitely many of which can be repelling for a rational map).
Perez-Marco has shown ([PM97]) for any nonlinearizable germ f the existence of a unique monotone one-parameter family (K t ) t>0 of full, totally invariant continua called hedgehogs containing the fixed point. In [Bis09] it is proved that any conformal mapping in a neighbourhood of a hedgehog K of a germ f 1 mapping K to a hedgehog of a germ f 2 necessarily conjugates f 1 to f 2 . As a corollary of Theorem 1.1 we have Corollary 1.2. There exists a holomorphic motion φ : D * ×Ĉ →Ĉ ofĈ over D * and a hedgehog K such that all the sets φ(t, K) are hedgehogs, all of which are quasi-conformal images of K, but for s = t, φ(s, K) cannot be conformally mapped to φ(t, K) .
Acknowledgements. The author thanks Ricardo Perez-Marco for many helpful discussions and comments. This article was written in part during a visit to the Chennai Mathematical Institute. The author is very grateful to the organizers for their warmth and hospitality.

Deformations.
We fix a germ f (z) = e 2πiα z + O(z 2 ), α ∈ R − Q and a neighbourhood U of the origin such that f and f −1 are univalent on a neighbourhood of U . By a periodic orbit or cycle of f of order q ≥ 1 we mean a finite set The periodic orbit is called attracting, indifferent and repelling according as |λ| < 1, |λ| = 1 and |λ| > 1 respectively. A periodic orbit for f of multiplier λ is a periodic orbit for f −1 of multiplier λ −1 . The basin of attraction of an attracting periodic cycle is defined by We observe that f can have only finitely many cycles of a given order q in U (by the uniqueness principle). Since f is asymptotic to an irrational rotation, i.e. f (z)/z → e 2πiα as z → 0, it follows that if f has small cycles then the orders of the cycles must go to infinity.
We let µ be the Beltrami differential ofK • L • φ λ restricted to a small neighbourhood D of z 1 ∈ O. The mapK •φ λ conjugates f q to the translation ξ → ξ +τ ′ . So at points z, z ′ = f q (z) in D, µ satisfies the invariance condition for z ∈ D, j = 1, . . . , q−1 is an f -invariant Beltrami differential. Similarly the above equation allows us to extend µ to an f -invariant Beltrami differential µ(O, λ, λ ′ ) on Lemma 2.1. Any quasi-conformal homeomorphism h with Beltrami coefficient Proof: For i = 1, . . . , q we let ψ i be the branch of φ −1 sending 0 to z i . By construction the map k defined by k = ψ i • L −1 •K • L • φ(z) for z in a neighbourhood of z i has Beltrami coefficient equal to µ and conjugates f on a neighbourhood of O to a holomorphic map f 1 = k • f • k −1 with periodic orbit k(O) and multiplier λ ′ .
Since h and k have the same Beltrami coefficient the map h • k −1 is holomorphic, and conjugates f 1 to g, so the multipliers of f 1 and g are equal. The Beltrami differential ofK is constant equal to − log(λ ′ /λ) 2 log |λ|+log(λ ′ /λ) which depends holomorphically on λ ′ , so the Beltrami differential µ(O, λ, λ ′ ) ofK • L • φ λ depends holomorphically on λ ′ . ⋄ 2.2. Deforming germs with small cycles. We use the Beltrami differentials µ(O, λ, λ ′ ) to deform a germ with small cycles as follows: Proof of Theorem 1.1: Let f be a germ with a sequence of repelling small cycles of orders (q n ) (which we may assume to be strictly increasing). The basins of attraction A(O, f −1 ) of distinct repelling cycles of f are disjoint, so for any Λ = (λ n ) ∈ M we can define an f -invariant Beltrami differential µ Λ on U by putting µ Λ (z) = µ(O, λ, λ n )(z) when z belongs to A(O, f −1 ) for a cycle O of order q n and multiplier λ, and µ Λ (z) = 0 otherwise. Let h Λ be the unique quasi-conformal homeomorphism with Beltrami coefficient µ Λ fixing 0, 1, ∞ given by the Measurable Riemann Mapping Theorem. Then Λ is a holomorphic germ fixing the origin with small cycles. By Naishul's theorem [Nai83] the multiplier at an indifferent fixed point is a topological conjugacy invariant so g ′ Λ (0) = e 2πiα . By Lemma 2.1, the multiplier of g Λ at all its repelling cycles of order q n is equal to λ n .
If φ is a holomorphic germ conjugating two such germs g Λ1 , g Λ2 , φ must take all repelling cycles of g Λ1 of order q n (in its domain) to repelling cycles of g Λ2 of order q n and preserve multipliers, so the sequences of multipliers Λ 1 , Λ 2 must agree for all but finitely many terms. It follows from Lemma 2.1 that µ Λ depends holomorphically on each λ n , hence by the Measurable Riemann Mapping Theorem so does h Λ (z) for fixed z. ⋄ Proof of Corollary 1.2: Let f be as above and K ⊂ U be a hedgehog of f . For t ∈ D * we let Λ t be the constant sequence (λ n = 1/t). Then µ Λt depends holomorphically on t, and φ : (t, z) → h Λt (z) gives the required holomorphic motion. ⋄