Complete Conjugacy Invariants of Nonlinearizable Holomorphic Dynamics

Perez-Marco proved the existence of non-trivial totally invariant connected compacts called hedgehogs near the fixed point of a nonlinearizable germ of holomorphic diffeomorphism. We show that if two nonlinearisable holomorphic germs with a common indifferent fixed point have a common hedgehog then they must commute. This allows us to establish a correspondence between hedgehogs and nonlinearizable maximal abelian subgroups of Diff$(\bf C,0)$. We also show that two nonlinearizable germs are conjugate if and only if their rotation numbers are equal and a hedgehog of one can be mapped conformally onto a hedgehog of the other. Thus the conjugacy class of a nonlinearizable germ is completely determined by its rotation number and the conformal class of its hedgehogs.


Introduction.
We consider the dynamics of a holomorphic germ f (z) = e 2πiα z + O(z 2 ), α ∈ R − Q near the indifferent irrational fixed point 0. The germ f is said to be linearizable if there is a holomorphic change of variables z = h(w) = w + O(w 2 ) such that where R α (w) = e 2πiα w is the rigid rotation. The maximal linearization domain of f is called the Siegel disk of f .
The problem of linearization, or determining when f is linearizable, is intimately linked to the arithmetic of the rotation number α, and has a long and interesting history.The work of H. Cremer ([Cr1], [Cr2]) in the 1920's showed the existence of nonlinearizable germs for rotation numbers very well approximable by rationals, while that of C.L. Siegel ([Si]) in 1942 and A.D.Brjuno in the 1960's showed linearization was always possible for for germs with rotation numbers poorly approximated by rationals. The matter was settled definitively by J.C.Yoccoz ( [Yo]) in 1987, when he showed that Brjuno's arithmetic condition was the optimal one for linearizability. The reader is referred to R.Perez-Marco's Bourbaki Seminar [PM4] for a complete account of the story.
When f is linearisable, the closures of the linearization domains h({|w| < r}) for r small are completely invariant connected compacts for f . It is not obvious however whether nonlinearizable germs have completely invariant non-trivial connected compacts, but Perez-Marco showed ( [PM1]) that in fact for any germ f there are always completely invariant, non-trivial connected compacts near the fixed point. These invariant compacts K are called Siegel compacta. If K is not contained in the closure of a linearisation domain it is called a hedgehog. The hedgehog is called linearizable or non-linearizable depending on whether it contains a linearization domain or not.
Perez-Marco has studied the topology (in [PM2]) and the dynamics (in [PM3]) of hedgehogs. The results in [PM2] show that the topology of hedgehogs is complex. Nonlinearizable hedgehogs have empty interior. They are not locally connected at any point except possibly at the fixed point. They always contain points inaccessible from their complement.
The results in [PM3] show that the dynamics on hedgehogs has many features in common with linearizable dynamics. For example, there are no periodic points on the hedgehog, and every point is recurrent (as is the case in the linearizable situation). Any hedgehog contains a continuous nested one-parameter family of sub-hedgehogs (recalling the filtration of a Siegel disk by invariant sub-disks). If two germs commute then they preserve the same hedgehogs (commuting linearizable germs have common linearization domains).
These results suggest that hedgehogs should be thought of as "degenerate linearization domains" in some sense, though we are unable at present to give this heuristic notion a precise mathematical formulation. Nevertheless it is one of the main motivations behind Perez-Marco's results and the results of this article. We prove the converse of the last result mentioned above. If two germs preserve a common hedgehog then they must commute (recall that linearizable germs with a common linearization domain must commute). So the group of germs preserving the hedgehog is commutative, and equal to the centralizer in Diff(C, 0) of any element of the group. As Perez-Marco shows in [PM3], there is a unique continuous nested one-parameter family of hedgehogs associated to a nonlinearisable germ, indeed to its centralizer since commuting germs have the same hedgehogs. On the other hand since germs preserving the same hedgehogs commute, to every family of hedgehogs we can associate an abelian subgroup of germs. We show that this gives a one-to-one correspondence between hedgehogs and the nonlinearizable maximal abelian subgroups of Diff(C, 0) (nonlinearizable meaning not conjugate to the subgroup (R λ (z) = λz) λ∈C * ).
The correspondence is natural with respect to change of variables: two such subgroups are conjugate by an element φ of Diff(C, 0) if and only if φ maps one family of hedgehogs to the other. Two nonlinearisable germs are conjugate if and only they have the same rotation number and a hedgehog of one can be 'conformally mapped' to the other (ie there is a conformal mapping between neighbourhoods of the hedgehogs taking one hedgehog to the other). Thus the geometry of the hedgehog (in some sense its conformal geometry) completely determines the nonlinearizable dynamics. It would be interesting (and morally satisfying) to find an intrinsic notion of conformal structure for singular spaces like hedgehogs characterizing their conformal equivalence. It is interesting to compare this with the description of conjugacy classes of germs tangent to the identity given by Martinet-Ramis in [Ma-Ra]. They describe a complete set of invariants as a formal invariant λ (a complex number) together with a singular quotient object, the "chapelet de spheres", and define an appropriate notion of conformal equivalence of these objects, so that the conjugacy class of a germ is determined by the conformal class of the associated chapelet.

Preliminaries.
Here we collect the basic definitions and results about hedgehogs due to Perez-Marco which we will require. We let f (z) = e 2πiα z + O(z 2 ), α ∈R, be a germ of holomorphic diffeomorphism with an indifferent fixed point at 0.   We now restrict ourselves to the case of germs with irrational rotation number (so any Siegel compact is either a linearization domain, or a linearizable hedgehog, or a nonlinearizable hedgehog). If two admissible domains are nested, U ⊂ V , then the non-escaping points of U are of course non-escaping points of V , and it follows from the above Theorem that the associated Siegel compacta are nested, K(U ) ⊂ K(V ). In fact any Siegel compact is filtered by a nested family of sub-Siegel compacta: Theorem 2.4 ( [PM3]).Let K be a Siegel compact. Given an admissible neighbourhood U such that K = K(U ) and a continuous monotone increasing one-parameter family of admissible neighbourhoods is a continuous (for the Hausdorff topology on compact sets), monotone increasing family of sub-Siegel compacta of K, such that K t → {0} as t → 0 and K t → K 1 = K as t → 1. Moreover any Siegel compact contained in U (in particular any sub-Siegel compact of K) belongs to the family (K t ) 0<t≤1 .
Thus the sub-Siegel compacta of a given Siegel compact form a continuous, monotone, one-parameter family (which is a trivial fact for linearization domains, but quite a remarkable one for hedgehogs, given their complex topological structures). The parametrization of the family is not unique; different choices of the admissible domain U and the filtration (U t ) 0<t≤1 lead to different parametrizations. In the case of a linearizable germ and a linearization domain, the sub-linearization domains admit a natural parametrization given by the conformal radius, which is conformally invariant. An interesting problem is to find such a conformally invariant parametrization for sub-hedgehogs of a hedgehog; is there an appropriate notion of conformal radius for hedgehogs?
Any linearizable hedgehog of a germ f must contain the Siegel disk of f ; however, it can have no other interior points: The interior of a linearizable hedgehog is equal to the Siegel disk. The interior of a nonlinearizable hedgehog is empty.
The family of all Siegel compacta of f , is a linearly ordered family with respect to inclusion: given Siegel compacta K 1 , K 2 , they are both sub-Siegel compacta of their union K 1 ∪ K 2 (which is full, since by the maximum principle and Theorem 2.5 the complement cannot have any bounded components) and hence by Theorem 2.4 one must be contained in the other. For a nonlinearizable germ f the union ∪ K∈K(f ) K of all its hedgehogs is, morally speaking, the analogue of the Siegel disk; note that the closure of this union is not a hedgehog, since every hedgehog is strictly contained in some admissible domain and hence in a larger Siegel compact.
While the family of all Siegel compacta K(f ) of a germ f is a natural object to consider, it is not an invariant of the local dynamics; for any germ there are of course local changes of variables which do not extend univalently to neighbourhoods of all the Siegel compacta of the germ. However, since any change of variables is univalent on a neighbourhood of every sufficiently small Siegel compact, the situation is easily remedied by considering the germ of the family of Siegel compacta.
Definition 2.6 (Germ of Siegel compact, Germ of Hedgehog). Consider all families K of compacta in the plane such that 0 ∈ K for all K ∈ K, and such that for any ǫ > 0 there is a K ∈ K with diam(K) < ǫ. We define the following equivalence relation on such familes: We call an equivalence class [K] a germ of compact.
There is a natural action of germs of homeomorphisms of C fixing 0 on germs of compacta, where φ([K]) is defined as follows: it is possible to pick a representative K ′ ∼ K such that φ is defined in a neighbourhood of all K ∈ K ′ . Set φ( We can restate the following result from [PM3] in terms of germs of hedgehogs: Theorem 2.8 ( [PM3]).If two nonlinearisable germs f, g commute then any sufficiently small hedgehog of f is also a hedgehog of g, so they define the same germ of hedgehog 3. Results.
The key result from which the others follow easily is the following: Theorem 3.1. Let K be a hedgehog for a nonlinearizable germ f . If a germ T (z) = z + O(z 2 ) tangent to the identity is univalent on a neighbourhood of K and K is either forward or backward invariant for T , then T is equal to the identity.
For notational convenience, given quantities a, b which are either positive sequences or functions of z near 0, we write a b if a ≤ Cb for some constant C > 0 for all n sufficiently large (or all z sufficiently small as the case may be). We write a ∼ b if a b and b a.
is a nondegenerate parabolic germ. Let U 1 ⊂ U 2 , U 1 = U 2 be admissible domains for f and T such K ∩ ∂U 1 = φ, K = K(U 1 ). Let F ⊂ U 2 be a Siegel compact of T such that F ∩ ∂U 2 = φ. As Perez-Marco shows in [PM1], F is a invariant Fatou flower of T , meaning that int(F ) = P 1 ∪ . . . P d where P 1 , . . . , P d (the 'petals' of the 'flower') are pairwise disjoint Jordan domains invariant under T whose boundaries intersect only at the origin, and T |P i is a parabolic automorphism of P i having 0 as the unique fixed point on the boundary. Pick a point z 0 ∈ int(F ) − U 1 and a small ball B 0 around z 0 contained in int(F ) − U 1 . Since B 0 ⊂ P i for some i, T n (B 0 ) → {0} as n → ±∞.
We may assume wlog (considering T −1 if necessary) that T (K) ⊂ K. For n ≥ 0 let z n = T −n (z 0 ), B n = T −n (B 0 ). Then points in B n escape from U 1 under T n , while by hypothesis points of the hedgehog K remain in K ⊂ U 1 , so we will arrive at the desired contradiction if we can show that B n ∩ K = φ for some n. We need the following estimate on the asymptotic size of the B n 's: Proof: Taking a covering of a neighbourhood of the origin by attracting and repelling Fatou petals, we see that the domains B n converge to 0 through a repelling petal P ; there is a Fatou coordinate w = χ(z) defined on P which maps P to the right half-plane {Re w > 0} and conjugates T −1 to the translation w → w + 1. Moreover, χ has an asymptotic expansion of the form χ(z) = c log(z) + n≥−d c n z n with c −d = 0 (see for example [Ec]), so |χ ′ (z)| ∼ |z| −(d+1) and |(χ −1 ) ′ (w)| ∼ |w| −1/d−1 . Fix an n 0 such that B n 0 ⊂ P . Note that for n > n 0 , all the domains χ(B n ) are translates of χ(B n 0 ) and have the same constant diameter, so for large n, for all w ∈ χ(B n ), |w| ∼ |χ(z n )|, and hence |(χ −1 ) ′ (w)| ∼ |(χ −1 ) ′ (χ(z n ))| = |χ ′ (z n )| −1 | ∼ |z n | d+1 . It follows that for all z ∈ B n , for n large, putting w = χ(z), |χ ′ (z)| −1 = |(χ −1 ) ′ (w)| ∼ |z n | d+1 . So taking z ′ n ∈ ∂B n such that |z n − z ′ n | = d(z n , ∂B n ), for n > n 0 we have, for some z ′′ n ∈ B n , and the Lemma follows. ⋄ The points z n converge slowly to 0, in the sense that |z n+1 |/|z n | = |T −1 (z n )|/|z n | = 1 + O(z d n ) → 1 as n → ∞. To prove the Theorem it suffices to prove the following Proposition: Proposition 3.3 Let K be a hedgehog of a nonlinearizable germ f . Let (z n ) be a sequence converging to 0 such that for n large enough |z n+1 | ≥ ǫ|z n | for some ǫ > 0 , and (B n ) a sequence of domains such that z n ∈ B n and d(z, ∂B n ) |z n | d+1 for some d ≥ 1. Then for some subsequence (B n k ), K ∩ B n k = φ for all large k.
Proof: Note that for any change of variables φ(z) = z + O(z 2 ), the sequences (φ(z n )) and (φ(B n )) satisfy the hypotheses of the Proposition, thus we may assume wlog, that f is of the form f (z) = e 2πiα z +O(z N ) for some large N which we will choose appropriately in the course of the proof (any irrationally indifferent germ can always be analytically conjugated to germs tangent to the rotation R α upto arbitrarily high orders). We need the following two estimates on how long we can iterate such a germ close to the origin, and how close the orbits stay to that of the rotation: Proof: There are constants C 1 > 0 and ǫ 0 > 0 such that for |z| ≤ ǫ 0 we have where φ(t) = t + C 1 t N . To estimate |f k (z)| for small z, we estimate φ k (t) for t = |z| close to 0. It is convenient to conjugate the mapping t → φ(t) to a mapping s →φ(s), in terms of the variables s = 1/t N−1 andφ(s) = 1/φ(t) N−1 close to +∞. A calculation gives for s ≥ s 0 sufficiently large, for some constants s 0 , C 2 . It follows that for s ≥ 2s 0 and k ≤ s/(2C 2 ), we haveφ k (s) ≥ s − kC 2 In terms of the variables φ(t), t, this means that for t ≤ t 0 sufficiently small and k ≤ 1/( For k ≤ 1/(2C 2 t N−1 ), it is easy to see from the above that φ k (t) ≤ 2t. So for |z| ≤min(t 0 , ǫ 0 /2), we have φ k (|z|) ≤ ǫ 0 for k ≤ C|z| −N+1 where C = 1/(2C 2 ). Since φ is monotone increasing, it follows from (1) by induction that Proof of Proposition 3.3: For all n sufficiently large, the circle {|z| = |z n |} intersects the hedgehog K since it is a non-trivial connected set containing the origin; let w n be a point of K on this circle. Let (p k /q k ) k≥0 be the continued fraction convergents of α. For all k, it follows from the continued fraction algorithm that any point on the circle {|z| = |z n |} is at distance at most 2q −1 k 2π|z n | from the first q k points R m (w n ), m = 0, . . . , q k of the orbit of w n under the rotation R α . If for each k we take z n k to be be the first element of the sequence (z n ) such that |z n k | < q , and 2q −1 k 2π|z n k | ∼ |z n k | d+1 |z n k | = |z n k | d+2 . So for all k there is some 0 ≤ m k ≤ q k such |z n k − R m k (w n k )| |z n k | d+2 . By Lemma 3.5, the orbit of w n k under f stays close to the orbit under R α for at least M (w n k ) iterates, and M (w n k ) ∼ |w n k | −(N−1) = |z n k | −(N−1) ∼ q Since d(z n k , ∂B n k ) |z n k | d+1 , it follows that f m k (w k ) ∈ B n k for all large k and K ∩B n k = φ. ⋄ Proof of Theorem 3.1. By Proposition 3.3 we can pick a point w in K ∩ B n for some n > 0; then T n (w) ∈ B 0 ⊂ U 1 c whereas by hypothesis T n (w) ∈ K ⊂ U 1 , a contradiction.

⋄
Theorem 3.6. A nonlinearizable germ f cannot commute with a nondegenerate parabolic germ g.
Proof: While this is easy to see using formal power series arguments, we can give a dynamical proof using the previous result. Indeed if f and g commute, then for a small hedgehog K of f on a neighbourhood of which g, g −1 are univalent, f (g(K)) = g(f (K)) = g(K), so g(K) is also a hedgehog for f . Since the hedgehogs of f are linearly ordered with respect to inclusion, K is either forward or backward invariant under g, contradicting Theorem 3.1, since g q is tangent to the identity for some q but g q = id. ⋄ Proof: which implies that φ(K 1 ) is a hedgehog of f 2 . This proves the "only if" part.
For the "if" part, given φ(K 1 ) = K 2 , letf 2 be the conjugatef 2 := φ • f 1 • φ −1 . Theñ f 2 (K 2 ) = K 2 , so the germ g =f 2 • f −1 2 preserves K 2 . Since the rotation numbers of f 1 and f 2 are equal, g is tangent to the identity and hence by Theorem 3.1 is equal to the identity. Sof 2 = f 2 , i.e. φ conjugates f 1 to f 2 . ⋄ We have the converse of Theorem 2.8: Theorem 3.8. If two nonlinearizable germs f and g have a common hedgehog K then they commute. In particular if f and g have the same germ of hedgehog (so all sufficiently small hedgehogs of one are hedgehogs of the other) then they commute.   ) commutes with f and hence maps sufficiently small hedgehogs K of f to hedgehogs of f since f (g(K)) = g(f (K)) = g(K), thus g([K]) = [K] and g ∈Aut([K]).
It is straightforward to check that the two maps are mutual inverses. The second assertion of the Theorem follows easily from Theorem 3.7. ⋄ Remark. It is well known that the unique maximal abelian subgroup containing an irrationally indifferent germ f is its centralizer Cent(f ), thus Aut([K]) =Cent(f ) for any nonlinearizable f whose germ of hedgehog is [K].
5. Acknowledgements. I thank Ricardo Perez-Marco for suggesting the method of proof of Theorem 3.8 (via consideration of the parabolic commutator germ and Theorem 3.1), and for numerous helpful discussions about hedgehogs. I am also grateful to Arnaud Cheritat for suggesting the proof of Proposition 3.3 via coordinate changes and elementary estimates; the original proof was based on Yoccoz's renormalization for germs, which gives much sharper but non-elementary estimates. I thank Xavier Buff as well for many helpful discussions.