BÖTTCHER COORDINATES

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1 BÖTTCHER COORDINATES XAVIER BUFF, ADAM L. EPSTEIN, AND SARAH KOCH Abstract. A well-known theorem of Böttcher asserts that an analytic germ f : (C, 0) (C, 0) which has a superattracting fixed point at 0, more precisely of the form f(z) = az k + o(z k ) for some a C, is analytically conjugate to z az k by an analytic germ φ : (C, 0) (C, 0) which is tangent to the identity at 0. In this article, we generalize this result to analytic maps of several complex variables. Introduction An analytic germ f : (C, 0) (C, 0) with a superattracting fixed point at 0 can be written in the form f(z) = az k + O(z k+1 ), a 0, k 2. In 1904, L. E. Böttcher proved the following theorem. Theorem (Böttcher). If f : (C, 0) (C, 0) has a superattracting point at 0 as above, there exists a germ of analytic map φ : (C, 0) (C, 0), which is tangent to the identity at 0 and conjugates f to the map h : w aw k in some neighborhood of 0, i.e., φ f = h φ. The germ φ is called a Böttcher coordinate for the germ f. Böttcher coordinates have been an essential tool in the study of complex dynamics in one variable; a Böttcher coordinate gives polar coordinates near the associated superattracting fixed point, which are compatible with the dynamics of f. In the case of the superattracting fixed point at infinity for a polynomial, the angular coordinate is called the external angle and its level curves are called external rays. The importance of external angles and external rays was emphasized in [DH]. It is quite natural to ask whether there is an analogue of this theorem in higher dimensions; that is, given an analytic germ F : (C m, 0) (C m, 0) is there an analytic germ Φ : (C m, 0) (C m, 0) which conjugates F to its terms of lowest degree? According to Hubbard and Papadopol in [HP], the map is not in general locally conjugate, even topologically, to its terms of lowest degree; the local geometry near such a point is much too rich for anything like that to be true. Hubbard and Papadopol present the following example to illustrate their point. Example 1. Consider the map F : C 2 C 2 given by (x, y) (x 2 + y 3, y 2 ). Let H : C 2 C 2 be the map (x, y) (x 2, y 2 ). There is no analytic conjugacy between F and H in a neighborhood of the origin because the dynamics of the The research of the first author was supported in part by the grant ANR-08-JCJC-0002 and the IUF. The research of the third author was supported in part by the NSF. 1

2 2 X. BUFF, A.L. EPSTEIN, AND S. KOCH Figure 1. For polynomials of one complex variable, the Böttcher coordinate of extends throughout the entire basin if the filled Julia set is connected. The checkerboard pattern in the pictures above highlights the external rays in each picture. On the right is the filled Julia set of the model map, z z 2, and on the left is the filled Julia set for a quadratic polynomial. maps are incompatible. Indeed, the critical locus of F consists of the the two axes, which is also the critical locus of H. But the map H fixes the two axes, whereas F fixes y = 0 and maps x = 0 to the curve t (t 3, t 2 ). However, there is another explanation which is more relevant to our discussion. The critical value locus of F contains the curve t (t 3, t 2 ) which has a cusp; any smooth conjugacy would have to map this singular curve to a component of the critical value locus of H; however, the critical value locus of H consists of x = 0 and y = 0, which are smooth. So certainly no analytic conjugacy exists. Remark 1. In the example above, the map H is homogeneous. A homogeneous map H : C m+1 C m+1 descends to a map on projective space h : P m P m ; the critical locus and the postcritical locus of H will be cones over the critical locus and postcritical locus of h. Therefore, if F is going to be locally conjugate to a homogeneous map in a neighborhood of 0, each component of the critical locus and postcritical locus of F containing the superattracting fixed point, must be an analytic cone; that is, each of these components must be the image of a cone under an analytic isomorphism. This is a rather strong condition; the homogeneous maps certainly satisfy it, and we will see in section 3 that there are other families of maps which satisfy this criterion. The following people have studied the dynamics of maps with superattracting fixed points in higher dimensions: J.H. Hubbard and P. Papadopol develop a very important theory of Green functions for maps with superattracting fixed points in C m (see [HP]), C. Favre and M. Jonsson classify contracting rigid germs of (C 2, 0) in [FJ]. Both S. Ushiki and T. Ueda have results about a Böttcher theorem in

3 BÖTTCHER COORDINATES 3 higher dimensions: Ushiki presents a local Böttcher theorem in C 2 for maps of a special form in [Us], and Ueda presents both local and global Böttcher theorems in C m for a particular family of maps in [Ue1]. In this article, we begin with an analytic germ F : (C m, 0) (C m, 0), which has an adapted superattracting fixed point at 0. We present necessary and sufficient conditions for such an F to be locally conjugate to its quasihomogeneous part at 0. Our criteria are stated in terms of admissible vector fields; these vector fields detect the shape of the postcritical locus of our map F to see if the analytic cone condition in remark 1 is satisfied. The relevant definitions are given in Section 1.1. Theorem 1 (Local Böttcher Coordinates). Let F : (C m, 0) (C m, 0) be a germ of an analytic map having an adapted superattracting fixed point at 0 C m. Let H : C m C m be the quasihomogeneous part of F at 0 with multidegree (k 1,... k p ). Then the following are equivalent. (1) There is a germ of an analytic map Φ : (C m, 0) (C m, 0) such that Φ(0) = 0, D 0 Φ = id and Φ F = H Φ. (2) There is an admissible p-tuple of germs of vector fields (ξ 1..., ξ p ) such that DF ξ j = k j ξ j F near 0 for all j [1, p]. (3) There is an admissible p-tuple of germs of vector fields (ζ 1,..., ζ p ) such that ζ j is tangent to the germ of the postcritical set of F for all j [1, p]. The proof of theorem 1 is a beautiful confluence of classical results of Euler, Poincaré, Cartan, and Arnol d with the theory of Green functions developed in [HP]. Sections are devoted to presenting these results and adapting them to our setting. In section 2, we present a result about extending a local Böttcher coordinate Φ to a larger a domain. More precisely let Ω be a complex analytic manifold, and F : Ω Ω be an analytic map with a superattracting fixed point at a Ω. The basin of attraction of a for F is the open set of points whose orbits converge to a. The immediate basin is the connected component of the basin containing a; we denote the immediate basin as B a (F ). Theorem 2 (Global Böttcher coordinates). Let Ω be a complex analytic manifold, F : Ω Ω be a proper analytic map with a superattracting fixed point at a Ω and H : T a Ω T a Ω be a quasihomogeneous map. Suppose that there is a local isomorphism Φ : (Ω, a) (T a Ω, 0) with Φ F = H Φ; near a, Φ maps the postcritical set of F : B a (F ) B a (F ) to the postcritical set of H : B 0 (H) B 0 (H). Then, Φ extends to a global isomorphism Φ : B a (F ) B 0 (H) conjugating F to H: B a (F ) F B a (F ) Φ Φ B 0 (H) H B 0 (H). In section 3, we apply our results to a large class of new examples of postcritically finite endomorphisms of P n. And finally, in section 4, we conclude with some remaining questions.

4 4 X. BUFF, A.L. EPSTEIN, AND S. KOCH Acknowledgements. We would like to thank our colleagues for their constant support, and stimulating discussions. In particular, we thank John H. Hubbard, Curtis T. McMullen, John Milnor, and Saeed Zakeri. 1. The Local Result 1.1. Set Up. In section 1.1, we will use the following notation. E is a C-linear space of dimension m 1. is a Hermitian product on E. is the associated norm. E = E 1 E p is the direct sum of p 1 linear spaces. (π j : E E j ) j [1,p] are the projections associated to the direct sum. for v E, we denote v j :=π j (v) E j. For F : (E, 0) (E, 0) an analytic germ, we denote F j :=π j F : E E j. if (H j : E j E j ) j [1,p] are maps, then H 1 H p is the map E v v p H 1 (v 1 ) + + H p (v p ) E. (π j : E E j ) j [1,p] are the projections associated to the direct sum. k 1,..., k p are integers greater than or equal to 2. The word adapted in the definition 1 refers to this data. Recall that a map H : L L on a C-linear space L is homogeneous of degree k 1 if v L, H(v) = φ(v,..., v) }{{} k times for some k-linear map φ : L k L. Equivalently, H is analytic and λ C, v L, H(λv) = λ k H(v). The homogeneous map H is nondegenerate if H 1 {0} = {0}. Definition 1. An analytic germ F : (E, 0) (E, 0) has an adapted superattracting fixed point if for all j [1, p], there is a nondegenerate homogeneous map H j : E j E j of degree k j 2 such that F j (v) = v 0 H j (v j ) + o ( (v j ) kj ). The map H 1 H p is the quasihomogeneous part of F at 0 and (k 1,..., k p ) is its multidegree. Note that the derivative of a map at an adapted superattracting fixed point necessarily vanishes. The particular form of F j implies the following invariance property. Proposition 1. The spaces E j :={v j = 0} and E j are locally totally invariant under F ; that is, there exists a neighborhood V of 0 such that F 1 (E j ) V = E j V and F 1 (E j ) V = E j V. Proof. Let us first prove that Ej is locally totally invariant. First, if v j = 0, then F j (v) = 0. This shows that near 0, Ej F 1 (Ej ). So if E j were not locally totally invariant, we could find an analytic germ γ : (C, 0) (F 1 (E j ), 0) with γ j :=π j γ 0.

5 BÖTTCHER COORDINATES 5 The order of vanishing m of γ j at 0 would be finite. Since H j is nondegenerate with degree k j, the order of vanishing of H j γ j would be k j m. So the order of vanishing of F j γ = H j γ j + o ( γ j kj ) would be kj m < which is a contradiction since by assumption F j γ 0. Next, note that E j = i j E i. Since the spaces Ei are locally totally invariant, it follows that the space E j is locally totally invariant. We want to give necessary and sufficient conditions for the existence of an isomorphism Φ : (E, 0) (E, 0) conjugating F to its quasihomogeneous part H, i.e., such that Φ F = H Φ. Such an isomorphism is called a Böttcher coordinate for F. For v E, there is a canonical isomorphism between the C-linear space E and the tangent space T v E. Together, these isomorphisms induce a canonical bundle isomorphism between T E and E E. If (v, w) E E, we shall denote by (v; w) the corresponding tangent vector in T v E. If F : U E E is an analytic map, we denote D v F : T v E T F (v) E the derivative of F at v U. We denote by DF : T U T X the bundle map (v; w) D v F (v; w). We shall denote by F (v; w) the vector in E corresponding to DF (v; w) T F (v) E. Definition 2. Let ϑ rad and ϑ 1,..., ϑ p be the linear vector fields E T E defined by v E, ϑ rad (v):=(v; v) and ϑ j (v):= ( v; v j ). Note that ϑ rad = ϑ ϑ p. Definition 3. A vector field ξ is asymptotically radial if ξ is defined and analytic near 0 E with ξ(v) = v 0 ϑ rad (v) + o ( v ). A p-tuple of vector fields (ξ 1,..., ξ p ) is admissible if for all j [1, p], ξ j is defined and analytic near 0 in E, ξ j is tangent to E j, ξ j vanishes when v j = 0 and ξ j (v) = v 0 ϑ j (v) + o ( v ), and for all i [1, p] and all j [1, p], the vector fields ξ i and ξ j commute. The tangency condition in Theorem 1 requires some explanation. We say that an analytic vector field ζ on an open set U is tangent to an analytic set A U if ζ(a) belongs to the tangent space T a A for every a in the smooth part of A. We say that a germ of an analytic vector field ζ at 0 is tangent to the germ of the postcritical set of F if there is a neighborhood U of 0 such that F and ζ are defined and analytic on U, F : U F (U) U is proper and the vector field ζ is tangent to the critical value set of F n : U F n (U) for all n 1. When F is not postcritically finite, the postcritical set of F is not analytic and the third condition involves tangency to a priori infinitely many analytic sets.

6 6 X. BUFF, A.L. EPSTEIN, AND S. KOCH 1.2. Quasihomogeneous Maps. Let L be a C-linear space. If H : L L is a homogeneous map of degree k 1, then (1) DH ϑ rad = k ϑ rad H. This is known as Euler s identity. In fact, the converse is true. If H : (L, 0) (L, 0) is a germ of an analytic map, then H is the germ of a homogeneous map of degree k 1 if and only if DH ϑ rad = k ϑ rad H near 0. We shall adapt this result to our setting as follows. Definition 4. A map H : E E is quasihomogeneous of multidegree (k 1,..., k p ) if there are homogeneous maps H j : E j E j of degree k j with H = H 1 H p. The following two lemmas about quasihomogeneous maps will be used later. Lemma 1. Let X E be such that near 0, the set H 1 X coincides with the graph of a function ϕ : (E j, 0) (E j, 0). Then X = H 1 X = E j near 0. Proof. Due to the structure of H = H j Hj : E j Ej E j Ej, and due to the homogeneity of H j, we have that H(e 2πi/kj v j + v j ) = H j (e 2πi/kj v j ) + H j (v j ) = H j (v j ) + H j (v j ) = H(v j + v j ). So v j + v j H 1 X e 2πi/kj v j + v j H 1 X. Since near 0, the set H 1 X coincides with the graph of ϕ, we have v j = ϕ(v j ) e 2πi/kj v j = ϕ(v j ) for v j sufficiently close to 0. Thus ϕ e 2πi/kj ϕ and so ϕ vanishes identically near 0. This shows that H 1 X = E j near 0, which implies that X = E j near 0. Lemma 2. Let H : (E, 0) (E, 0) be a germ of an analytic map. Then, H is the germ of a quasihomogeneous map of multidegree (k 1,..., k p ) if and only if j [1, p], DH ϑ j = k j ϑ j H. Proof. Assume H = H 1 H p with H i : E i E i homogeneous of degree k i 1. Then, H i (v i ) = φ i (v i,..., v i ) with φ i : E ki i E i a symmetric k i -linear map and so, H i (v i; w i ) = k i φ i (v i,..., v i, w i ). Thus, for all v:=v v p E, p H ( (v; v j ) = k i φ i vi,..., v i, π i (v j ) ) = k j φ j (v j,..., v j ) = k j H j (v j ). i=1 This shows that DH ϑ j (v) = k j ϑ j H(v). Conversely, assume DH ϑ j = k j ϑ j H for all j [1, p]. In other words, j [1, p], H (v; v j ) = k j π j H(v). Let (U j E j ) j [1,p] be neighborhoods of 0 such that H is analytic on U:=U U p and if v j U j and λ 1, then λv j U j Let v = v v p be a point in U. Fix j [1, p]. The map χ j : (λ 1,..., λ p ) π j H(λ 1 v λ p v p )

7 BÖTTCHER COORDINATES 7 is defined and analytic in a neighborhood of the closed polydisk D p. In addition, for all i [1, p], χ j λ i (λ 1,..., λ p ) = π j H (λ 1 v λ p v p ; v i ) = 1 λ i π j H (λ 1 v λ p v p ; λ i v i ) = k i λ i π j π i H(λ 1 v λ p v p ). If i j, then χ j / λ i = 0, which shows that χ j only depends on λ j. In addition (2) χ j λ j = k j λ j π j χ j. Let H j : (E j, 0) (E j, 0) be the restriction of π j H to E j. The previous discussion shows that (λ 1,..., λ p ) D p, χ j (λ 1,..., λ p ) = χ j (0,..., 0, λ j, 0,..., 0) = H j (λ j v j ). In particular, taking λ 1 = = λ p = 1, we have that π j H(v) = H j (v j ), and so H = H 1 H p. Finally, the differential equation (2) implies that Consider the map H j (λv j ) λ = k j λ H j(λv j ). ψ : λ λ kj H j (λv j ) which is defined and analytic in a neighborhood of D. Its derivative satisfies ψ (λ) = k j λ kj 1 H j (λv j ) + λ k kj j λ H j(λv j ) = 0. As a consequence, ψ(λ) = ψ(1) for all λ D. So H j (λv j ) = λ kj H j (v j ) and H j is the germ of a homogeneous map of degree k j Linearizability of analytic vector fields. The following result is due to Poincaré. We include a proof for completeness. Lemma 3 (Poincaré). Any asymptotically radial vector field ξ is linearizable: there is a germ of an analytic map Φ : (E, 0) (E, 0) such that Φ(0) = 0, D 0 Φ = id and DΦ ξ = ϑ rad Φ. Proof. Let J be defined in neighborhood of 0 in E by J(v):= Re ξ(v) v Since ξ is asymptotically radial, we have that v 2. J(v) = 1 + O ( v ). v 0 So, there are constants C and r > 0 such that v B(0, r), J(v) 1 C v 1/2. In particular, ξ is outward pointing on the boundary of B(0, r). Let F t (v) = F(t, v) be the flow of the vector field ξ. Since ξ is outward pointing on the boundary of B(0, r), the map F t is defined and analytic on B(0, r) for all t 0.

8 8 X. BUFF, A.L. EPSTEIN, AND S. KOCH For t 0, set G t := log F t. Then, for t 0, G t t = J F t 1/2 and so, G t G 0 + t/2 and F t re t/2. In addition, (G t t) t = J F t 1 C Ft Cre t/2. So, (G t t) (G 0 0) 0 t Cre u/2 du 2Cr and therefore e t F t re 2Cr. For t 0, let Φ t : B(0, r) E be the map defined by Φ t :=e t F t. Note that Φ t (0) = 0, D 0 Φ t = id. Since DF t = ξ F t, we see that DΦ t ξ = e t ξ F t = e t (ϑ rad F t + o ( F t )) = ϑ rad Φ t + o ( Φ t ). The previous estimates show that the family (Φ t ) t 0 is uniformly bounded on B(0, r) by re 2Cr. Thus, it is normal. Any limit value Φ as t linearizes ξ: DΦ ξ = ϑ rad Φ. Corollary 1. Let (ξ 1,..., ξ p ) be an admissible p-tuple of germs of vector fields. Then, the ξ j are simultaneously linearizable, i.e., there is a germ of an analytic map Φ : (E, 0) (E, 0) such that Φ(0) = 0, D 0 Φ = id and DΦ ξ j = ϑ j Φ for all j [1, p]. Proof. The germ of the vector field ξ:=ξ ξ p is asymptotically radial, thus linearizable. Let Φ : (E, 0) (E, 0) be the linearizer. The vector fields Φ ξ j commute with Φ ξ = ϑ rad. It follows that they are linear vector fields and so, Φ ξ j = ϑ j Liftable Vector Fields. Definition 5. Let F : U V be an analytic map. An analytic vector field ξ on V is liftable if there is an analytic vector field ζ on U which satisfies DF ζ = ξ F. We say that ζ lifts ξ. The critical point set of F is the set C F of points x U for which D x F is not invertible. The critical value set of F is V F :=F (C f ). If ξ is a liftable vector field on V and if ζ lifts ξ, then for all x U C F, we have ζ(x) = (D x F ) 1 ξ F (x). Thus, when F has discrete fibers, a liftable vector field ξ on V admits a unique lift ζ on U and we shall use the notation F ξ:=ζ. Lemma 4 (Arnol d). Let F : U V be an analytic map with discrete fibers. Let ξ be a vector field which is analytic on V and tangent to the critical value set V F. Then, ξ is liftable and F ξ is tangent to C F.

9 BÖTTCHER COORDINATES 9 Proof. The vector field F ξ is well defined outside the critical set. According to a lemma of Hartogs, it is enough to show that F ξ extends analytically outside a subset of U of codimension 2 in order to know that it extends globally. Since F has discrete fibers, the critical set C F is either empty or has codimension 1. Thus, outside a codimension 2 subset of U, the critical set C F is smooth. Moreover, it follows from the Constant Rank Theorem that for generic x C F, the kernel of D x F does not intersect the tangent space to C F at x. Thus, near generic points in C F, the map F may be locally expressed as (x 1,..., x m 1, x m ) (y 1,..., y m 1, y m ) = (x 1,..., x m 1, x k m) for some integer k 2. Since the vector field ξ is tangent to the critical value set {y m = 0}, it is of the form and ξ = ξ ξ m 1 + y m ξ m y 1 y m 1 y m F ξ = (ξ 1 F ) ( x k (ξ m 1 F ) + m x 1 x m 1 = (ξ 1 F ) (ξ m 1 F ) + x 1 x m 1 kx k 1 m ( 1 k x mξ m F ) ξ m F x m ) x m which clearly extends analytically through the critical set {x m = 0} and is tangent to the critical set. Lemma 5. Let F : U V be an analytic map with discrete fibers, let ξ be a liftable vector field on V, and let ζ:=f (ξ) be its lift to U. Let φ(t, z) be the flow of ξ, and ψ(t, z) be the flow of ζ. (1) For all z U and for t sufficiently small, we have F (φ(t, z)) = ψ(t, F (z)). (2) If ζ F (z) = 0, then ξ(z) = 0. (3) If ζ is tangent to an analytic set A U, then ξ is tangent to F 1 (A). Proof. If z U C F, then F : (U, z) (V, F (z)) is a local isomorphism. It sends the vector field ξ to the vector field ζ. Thus it conjugates their flows, and the equality in part (1) holds for z U C F and t is sufficiently small. If z C F, it holds by analytic continuation (with respect to z). Parts (2) and (3) follow immediately since when the flow of ζ preserves an analytic set A (which may be reduced to a point if ζ vanishes at this point), then the flow of ξ preserves the analytic set F 1 (A). We shall now study how admissible p-tuple of vector fields behave under pullback. Lemma 6. Assume F : (E, 0) (E, 0) is an analytic germ having an adapted superattracting fixed point. Let (ζ 1,..., ζ p ) be an admissible p-tuple of liftable vector fields. Then, (k 1 F ζ 1,..., k p F ζ p ) is an admissible p-tuple of vector fields. Proof. Fix j [1, p], and set ξ j :=k j F ζ j. Since the vector field ζ j is tangent to E j, the vector field ξ j is tangent to F 1 (E j ). According to proposition 1, F 1 (E j ) coincides with E j in a neighborhood of 0. Thus, the vector field ξ j is tangent to E j.

10 10 X. BUFF, A.L. EPSTEIN, AND S. KOCH Since the vector field ζ j vanishes on Ej :={v j = 0}, the vector field ξ j vanishes on F 1 j (Ej ). According to proposition 1, F 1 (Ej ) coincides with E j in a neighborhood of 0. So ξ j vanishes on Ej. Since ξ j vanishes at 0, we may write ξ j (v) = τ j (v) + o ( v ) with τ j (v):= ( v; A j (v) ) for some linear map A j : E E. It remains to prove that τ j = ϑ j which amounts to showing that A j = π j. Since ξ j vanishes on Ej, the linear map A j vanishes on Ej. Thus it suffices to show that the restriction of A to E j is the identity. The map F restricts to a self-map F j : E j E j. The vectors fields ζ j and ξ j restrict to vector fields on E j (because they are tangent to E j ). It suffices to show that A j (v j ) = v j. We therefore restrict our analysis to the space E j, omitting the index j: F (v) = H(v) + o ( v k) with H : E E a nondegenerate homogeneous map of degree k, ζ(v) = (v; v) + o ( v ), ξ(v) = τ(v) + o ( v ), and DF ξ = k ζ F. On the one hand, DF ξ(v) = ( F (v); F ξ(v) ) = (F (v); H ξ(v) + o ( v k 1 ξ(v) )) = (F (v); H τ(v) + o ( v k)) On the other hand, k ζ F (v) = ( F (v); k F (v) + o ( F (v) )) = It follows that H τ(v) = k H(v), thus DH τ(v) = k (H(v); H(v) ). ( F (v); k H(v) + o ( v k)). According to Euler s identity, we therefore have τ(v) = (v; v) which implies that A(v) = v as required Dynamical Green Functions. We shall use dynamical Green functions introduced by Hubbard and Papadopol [HP]. We will first recall the construction for homogeneous maps, and then explain how this construction may be adapted to our setting. Let H : L L be a nondegenerate homogeneous map of degree k on a C-linear space L of dimension m. Then, the function u H : v 1 k log H(v) log v is defined and bounded on L {0}. It follows that the sequence of plurisubharmonic functions GH:= n 1 n 1 k n log H n = GH 0 u H F i + k i : L R { } i=0 converges uniformly on L to a plurisubharmonic function G H : L R { } which is continuous on L {0} and satisfies G H (v) = log v + O(1) and G H H = k G H.

11 BÖTTCHER COORDINATES 11 We shall adapt this construction to our setting as follows. Lemma 7. Let F : (E, 0) (E, 0) be an analytic germ having an adapted superattracting fixed point. There is a neighborhood U of 0 in E such that for all j [1, p], the sequence of functions Gj n := 1 kj n log π j F n : U R { } converges locally uniformly in U to a plurisubharmonic function G j : U R { } satisfying, G j (v) = v 0 log πj (v) + O(1) and Gj F = k j G j. Proof. Let U E be a sufficiently small neighborhood of 0 so that F is defined and analytic on U, F (U) U and U is contained in the basin of attraction of 0. The functions Gj n := 1 kj n log πj F n : U R { } are then defined and plurisubharmonic on U. By assumption F j (v) = H j (v j ) + o ( v j ) kj. Since H j is nondegenerate, v j kj = O ( H j (v j ) ). Thus πj F (v) = Fj (v) Hj (v j ) = Hj π j (v). v 0 As a consequence, restricting U if necessary, the function log π j F log H j π j is defined and bounded in U Ej. It follows that the function is defined and bounded in U E j. Now, the sequence u j := 1 k j log π j F log π j N 1 Gj N = Gj 0 + n=0 u j F n : U R { } k n j converges uniformly on U to a plurisubharmonic function G j whose difference with Gj 0 = log π j is bounded as required Cartan s Lemma. We shall use the following lemma of Cartan which is a multidimensional version of the Schwarz Lemma. We include a proof for completeness. Lemma 8 (Cartan). Let V be a bounded connected open subset of E containing 0, let Φ : V V be an analytic map such that Φ(0) = 0 and D 0 Φ = id. Then, Φ = id. Proof. The iterates Φ n are defined on V for all n 0. For n 1, let Ψ n : V E be defined as the average Ψ n := 1 n 1 Φ j. n j=0

12 12 X. BUFF, A.L. EPSTEIN, AND S. KOCH Then, Ψ n (0) = 0, D 0 Ψ n = id and Ψ n Φ = Ψ n + Φ n id. n In addition, the sequence (Ψ n ) n 1 is normal. Let Ψ be a limit value. Then, Ψ(0) = 0, D 0 Ψ = id and Ψ Φ = Ψ. In particular, Ψ is invertible at 0 and Φ is equal to the identity near 0, thus in V by analytic continuation. Corollary 2. Let V be a bounded connected open subsets of E containing 0, let Φ : V W E be an isomorphism and Ψ : V W be an analytic map such that Ψ(0) = Φ(0) and D 0 Ψ = D 0 Φ. Then, Ψ = Φ. Proof. Apply Cartan s lemma to Φ 1 Ψ : V V Proof of Theorem 1. (1) (3). For all n 1, H n = H1 n Hp n and the critical value set of H n is V H n = V H n V H n p. For all j [1, p] and all n 1, the critical value set of Hj n : E j E j is homogeneous (i.e., a complex cone with vertex at the origin). Thus, for all j [1, p], the vector field ϑ j is tangent to the critical value set of H n and the vector field ζ j :=Φ ϑ j is tangent to the critical value locus of F n. For all i, j, the vector fields ϑ i and ϑ j commute, thus ζ i and ζ j commute. Since D 0 Φ = id the linear part of ζ j at 0 is ϑ j. Since the vector field ϑ j is tangent to E j and vanishes on Ej, the vector field ζ j is tangent to Φ 1 (E j ) and vanishes on Φ 1 (Ej ). We claim that for all j [1, p], we have that Φ(Ej ) = E j near 0. Indeed, recall that Ej is locally totally invariant by F (see proposition 1). So X:=Φ(Ej ) is locally totally invariant by H. Since D 0 Φ = id, X and thus H 1 X is locally the graph of a function ϕ : Ej E j. According to lemma 1, X = H 1 X = Ej. It follows that Φ 1 (Ej ) = E j near 0. In particular, Φ 1 (E j ) = E j near 0 (because E j is the intersection of all the Ei for i j). Consequently, ζ j is tangent to E j and vanishes on Ej. Thus the p-tuple of vector fields (ζ 1,..., ζ p ) is admissible. (2) (1). According to corollary 1, since (ξ 1,..., ξ p ) is admissible, there exists a germ of an analytic map Φ : (E, 0) (E, 0) such that D 0 Φ = id and DΦ ξ j = ϑ j Φ for all j [1, p]. Then, Φ conjugates F to a map ˇF, defined and analytic near 0 in E, satisfying D ˇF ϑ j = k j ϑ j ˇF. According to lemma 2, ˇF is quasihomogeneous with multidegree (k 1,..., k p ). Since D 0 Φ = id we have that ˇF = H. (3) (2). Let U 0 be a sufficiently small neighborhood of 0 in E so that F is defined and analytic on U 0, F (U 0 ) U 0, and U 0 is contained in the attracting basin of 0, ζ j is defined and analytic on U 0, tangent to the critical value set of F n : U 0 F n (U 0 ) for all n 0 and all j [1, p]. Then, ζ j is liftable by F n and we may define a holomorphic vector field ζ n j on U 0 by ζ n j :=k n j (F n ) ζ j.

13 BÖTTCHER COORDINATES 13 Then, for all n 0, we have that DF ζ n+1 j = k j ζj n F. According to lemma 6, the p-tuple of vector fields (ζ1 n,..., ζp n ) is admissible. We will show that there is a neighborhood V of 0 in E on which the sequence of vector fields (ζj n) n 0 converges uniformly to a vector field ξ j for all j [1, p]. Then, the p-tuple of vector fields (ξ 1,..., ξ p ) is admissible and satisfies DF ξ j = k j ξ j F for all j [1, p]. The proof will be completed. According to lemma 7, the sequence of functions G n F,j:= 1 k n j log π j F n : U 0 R { } converges locally uniformly in U 0 to a function G F,j : U 0 R { } which is plurisubharmonic and satisfies We set G F,j (v) = v 0 log πj (v) + O(1) and GF,j F = k j G Fj. G n F := max j [1,p] Gn F,j and G F := max j [1,p] G F,j. Note that those functions are plurisubharmonic in U 0 and take the value only at 0. In addition, the sequence of functions GF n converges locally uniformly to G F in U 0. In particular, if M > 0 is sufficiently large, the level sets {GF n < M} are compactly contained in U 0. From now on, we assume that M > 0 is sufficiently large so that the set V n := { v U 0 : j [1, p], GF,j(v) n < M } are compactly contained in U 0 for all n 0. Similarly, the sequence of plurisubharmonic functions G n H,j:= 1 k n j log π j H n : E R { } converges locally uniformly in E to a function G Hj : E R { } which is plurisubharmonic and satisfies We set G H,j (v) = v 0 log π j (v) + O(1) and G Hj H = k j G H,j. G n H:= max j [1,p] Gn H,j, G H := max j [1,p] G H,j and W n := { v E : G n H(v) < M }. According to corollary 1, there are germs of analytic maps Φ n : (E, 0) (E, 0) such that D 0 Φ n = id and DΦ n ζj n = ϑ j Φ n for all n 0 and all j [1, p]. According to lemma 2, for all n 0, the map Φ n F Φ 1 n+1 : (E, 0) (E, 0) is quasihomogeneous, thus equal to H. In other words, we have the following commutative diagram: (E, 0) Φn+1 (E, 0) F (E, 0) H Φ n (E, 0). Lemma 9. For all n 0, the linearizing map Φ n is analytic on V n.

14 14 X. BUFF, A.L. EPSTEIN, AND S. KOCH Proof. Let us first show that the linearizing map Φ n is defined on V n. Recall that Φ n is defined as the linearizing map of the asymptotically radial vector field ζ n :=ζ n ζ n p. Note that flowing along ζj 0 during time t < 0 decreases G0 F,j by t. It follows that flowing along (F n ) ζj 0 during time t < 0 decreases G0 F,j F n by t. Thus, flowing along ζj n during time t < 0 decreases GF,j n by t. Finally, since the vector fields (ζj n) j [1,p] commute, flowing along ζ n during time t < 0 decreases GF n by t. In particular, the flow of ζ n is defined on V n for all t < 0, every trajectory remains in V n and converges to 0. If we denote by F n this flow, the linearizing map Φ n may be obtained on V n as Φ n (v) = lim t e t F n (t, x). Since DF ζ n+1 j = k j ζj n F, the analytic germ ˇF defined in a neighborhood of 0 in E by ˇF :=Φ n F Φ 1 n+1 satisfies D ˇF ϑ j = k j ϑ j ˇF. Thus, it is the germ of a quasihomogeneous map of multidegree (k 1,..., k p ). Since, D 0 Φ n = D 0 Φ n+1 = id, we have that ˇF = H. So, we have the following commutative diagram: Lemma 10. The function F n is defined and bounded on U 0 E j. Φ n V n E F n (V n ) Φ0 E. H n u j := log π j Φ 0 log π j Proof. Since D 0 Φ 0 = id, we have the a priori estimate π j Φ 0 (v) = v 0 π j (v) + o ( v ). We claim that X:=Φ 0 (Ej ) coincides with E j near 0. Indeed, Ej is locally totally invariant by F. So H 1 X = Φ 1 (Ej ) is a graph of a function ϕ : (E j, 0) (E j, 0) near 0. According to lemma 1, X = H 1 X = Ej. Consequently, π j Φ 0 vanishes on Ej, and so π j Φ 0 (v) = v 0 π j (v) + o ( π j (v) ). The lemma follows immediately. Observe that Setting GH,j n Φ n = 1 kj n = 1 kj n log π j H n Φ n log π j Φ 0 F n = 1 kj n u j F n + GF,j. n V := { v U 0 : G F (v) < M } and W := { v U 0 : G H (v) < M },

15 BÖTTCHER COORDINATES 15 we deduce that GH n Φ n converges uniformly to GF n on every compact subset of V. Therefore the sequence (Φ n ) is uniformly bounded on every compact subset of V. Similarly, any compact subset of W is contained in the image of Φ n for n large enough, and the sequence (Φ 1 n ) is uniformly bounded on every compact subset of W. Thus, the sequence of maps (Φ n ) is normal on any compact subset of V and the sequence of maps (Φ 1 n ) is normal on any compact subset of W. Extracting a subsequence, we see that there is an isomorphism Φ : V W such that Φ(0) = 0 and D 0 Φ = id. According to Cartan s lemma, any limit value of the sequence (Φ n ) must coincide with Φ. Thus, the whole sequence (Φ n ) converges to Φ locally uniformly in V and the sequence (ζj n = Φ nϑ j ) converges to ξ j = Φ ϑ j. This completes the proof of theorem The Global Result We will now prove Theorem 2. Since F : Ω Ω is proper and since B a (F ) is a connected component of F 1( B a (F ) ), the restriction F : B a (F ) B a (F ) is proper. Let G H,j : E R { } and G H : E R { } be the dynamical Green functions of H introduced above. Let G F,j : B a (F ) R { } be defined by G F,j (x) = 1 kj n G H,j Φ F n (x) where n 0 is chosen sufficiently large so that F n (x) belongs to a neighborhood of a on which Φ is defined. Let G F : B a (F ) R { } be defined by G F := max j [1,p] G F,j. Let M > 0 be sufficiently large so that Φ : (E, 0) (E, 0) has an inverse branch defined on W := { v E : G F (v) < M }. Set V :=Φ 1 (W ), so that Φ : V W is an isomorphism. Increasing M if necessary, we see that V is relatively compact in B a (F ). For j [1, p], set ξ j :=Φ ϑ j which is defined and analytic near 0. Then, ξ j = k j F ξ j near a. Since ϑ j is tangent to the postcritical set of H, and since near a Φ maps the postcritical set of F : B a (F ) B a (F ) to the postcritical set of H : B 0 (H) B 0 (H), the vector field ξ j is tangent to the postcritical set of F : B a (F ) B a (F ). In particular, ξ j is tangent to the critical value set of F n for n large enough. Thus we can extend ξ j to the whole basin of attraction B a (F ) using the formula ξ j = kj n (F n ) ξ j for n large enough. We therefore have a vector field ξ j which is defined and analytic on B a (F ) and satisfies ξ j = k j F ξ j on B a (F ). Let (t, x) F t,j (x) be the flow of the vector field ξ j and let (t, x) F t (x) be the flow of the vector field ξ:=ξ ξ p. Lemma 11. The map F t is defined on B a (F ) for all t 0. For all x B a (F ), we have that F t (x) 0 as t.

16 16 X. BUFF, A.L. EPSTEIN, AND S. KOCH Proof. We first want to prove that F t is defined on B a (F ) for all t 0. The maps F t,j are defined and analytic on V for all t 0. Indeed, Φ : V W conjugates F t,j to the linear map H t,j : v v p v v j 1 + e t v j + v j v p. The equality ξ j = k j F ξ j yields F F t,j = F kjt,j F and so, F n F t,j = F k n j t,j F n. For all x B a (F ), if n is large enough, F k n j t F n (x) remains in a compact subset of B a (F ) for all t 0. Since F n : B a (F ) B a (F ) is proper, F t,j (x) also remains in a compact subset of B a (F ), and so, is defined for all t 0. Since the vector fields ξ j commute, we have that F t = F t,1 F t,p. So, F t is defined on B a (F ) for all t 0. Near 0, we have that G F,j = G H,j Φ and Φ F t,j = H t,j Φ. In addition, G H,j H t,j = G H,j + t. It follows that near 0, we have that G F,j F t,j = G F,j + t. The equality ξ j = k j F ξ j yields F F t,j = F kjt,j F. So, for t 0, the following equality is valid on B a (F ): G F,j F t,j = 1 kj n G F,j F n F t,j = 1 kj n G F,j F k n j t,j F n = 1 kj n G F,j F n + t = G F,j + t. As a consequence, for t 0, the following equality is valid on B a (F ): G F F t = G F F t,1 F t,p = G F + t. We now prove that F 1 {0} = {0}. As we have seen, each trajectory ( F t (x) ) t 0 remains in a compact subset of B a (F ). Since G F F t = G F + t, as t, each trajectory ( F t (x) ) must converge to a point where G t 0 F takes the value, i.e., to a point in the backward orbit of 0. We may therefore partition B a (F ) in the basins of those points (for the flow F t ). The basins are open and since B a (F ) is connected, there is only one such basin: the basin of 0. The linearizer Φ of ξ extends to the whole set B a (F ) by Φ(x) = e t Φ F t (x) where t 0 is chosen sufficiently negative so that F t (x) V. It is injective on B a (F ). Indeed, assume Φ(x 1 ) = Φ(x 2 ) with x 1 and x 2 in B a (F ). Choose t 0 sufficiently negative so that F t (x 1 ) and F t (x 2 ) belong to V. Then, Φ F t (x 1 ) = e t Φ(x 1 ) = e t Φ(x 2 ) = Φ F t (x 2 ) Since Φ : V W is an isomorphism, we have that F t (x 1 ) = F t (x 2 ), and so, x 1 = F t F t (x 1 ) = F t F t (x 2 ) = x 2. The equality Φ F = H Φ holds on B a (F ) by analytic continuation. This shows that Φ(B a (F )) is contained in the basin of attraction B 0 (H). Finally, Φ : B a (F ) B 0 (H) is proper, thus an isomorphism. Indeed, let K B 0 (H) be a compact set and let n 0 be sufficiently large so that K n :=H n (K) W. Since Φ : V W is an isomorphism, Φ 1 (K n ) V is compact. Since F n : B a (F ) B a (F ) is proper, F n( Φ 1 (K n ) ) is compact. This compact set

17 BÖTTCHER COORDINATES 17 contains Φ 1 (K) which is closed since Φ is continuous. It follows that Φ 1 (K) is compact. This completes the proof that Φ : B a (F ) B 0 (H) is an isomorphism. 3. Applications In this section we apply our theorems to a family of postcritically finite endomorphisms of projective space which arose in [K] and were studied in [BEKP]. Let h : P m P m be an endomorphism; that is, a holomorphic map with no indeterminacy points. Let C h be critical locus of h. We define the postcritical locus of h to be P h := n 1 h n (C h ). The endomorphism h is postcritically finite if P h is algebraic. Equivalently (via a Baire category argument), each component of C h is either periodic, or preperiodic to a periodic cycle of components in P h. Postcritically finite endomorphisms of P m were first studied by Fornæss and Sibony in [FS], and by Ueda in [Ue2] Constructing endomorphisms. The following construction is a particular case of a more general construction in [K]. Let I be a finite set of cardinality m 1. Denote by E the C-vector space of functions x : I C whose average is 0. For x E, we use the notation x i :=x(i) and set P x (t):= m + 1 m j I t x j (w x i ) dw. The polynomial P x is the unique monic centered polynomial of degree m + 1 whose critical points are the points (x i ) i I, repeated according to their multiplicities, and for which the barycenter of the critical values ( P x (x i ) ) is 0. i I The function y:=p x x : I C belongs to E and satisfies i I i I y i = P x (x i ). We denote by H : E E the map defined by H(x):=P x x. Proposition 2. For m 2, the map H : E E is a homogeneous map of degree m + 1. For m 3 it induces an endomorphism h : P(E) P(E), where P(E) is the projective space associated to E (isomorphic to P m 2 (C)). Proof. The polynomial P x depends analytically on x E, therefore H is analytic. Since P λx (λt) = m + 1 λt (w λx i ) dw m j I m + 1 = w=λv m j I λx j i I t x j (λv λx i ) d(λv) = λ m+1 P x (t), i I we have H(λx) = λ m+1 H(x), so H is homogeneous of degree m + 1. Assume P x (x i ) = 0 for all i I. This means that P x has only one critical value, namely 0. Therefore P x has only one critical point. Thus x is constant, and the average of x is 0, we have x = 0. This implies that H 1 (0) = {0}, and consequently, H : E E induces an endomorphism h : P(E) P(E).

18 18 X. BUFF, A.L. EPSTEIN, AND S. KOCH The significance of these endomorphisms is that their fixed points correspond to polynomials with fixed critical points. More precisely, we have the following correspondence. Proposition 3. If x is a fixed point of H : E E, then P x is a monic centered polynomial of degree m + 1 with fixed critical points. If P is a monic centered polynomial of degree m + 1 with fixed critical points, then there exists x E such that H(x) = x, and P x = P. Proof. First let x be a fixed point of H. The polynomial P x is monic, centered, and of degree m + 1. The critical points of P x are the x i. Since x is a fixed point of H, x i = P x (x i ) for all i I, and thus P x fixes its critical points. Conversely, let P be a monic centered polynomial of degree m + 1 with fixed critical points. The polynomial P has exactly m critical points (counted with multiplicity). Let x be a bijection from I to the set of critical points of P ; there are m! choices for x. Then x E, and P x = P. Therefore P x (x i ) = P (x i ) = x i, which implies that x is a fixed point of H. A set of particular interest is the noninjectivity locus :={x E : i j with x i = x j }. By proposition 3, the fixed points of H correspond to polynomials with fixed critical points: the fixed points in E correspond to polynomials with simple critical points, whereas the fixed points in correspond to polynomials with at least one multiple critical point. The locus is a union of hyperplanes which are invariant by H; it is a stratified space where each stratum is invariant. More precisely, denote by Part(I) the set of all partitions of I, and set Part (I) = Part(I) {I}. Let I Part (I) be the singleton partition of I: that is I:= { {i} : i I }. Given J Part (I), let L J E be the linear space defined by L J :={x E : x is constant on each element of J }. Note that E = L I and the dimension of L J is J 1. We say that K Part (I) is a contraction of J Part (I) if all elements of K are unions of elements of J. We denote this as K J, and if K = J, we use the notation K J. When K is a contraction of J, the linear space L K is contained in L J, and the codimension of L K in L J is J K. The stratification of is given by = L J. J Part (I) If x L J, then H(x) = P x x is constant on each element of J. Thus H(L J ) L J. In particular, h : P(E) P(E) restricts an endomorphism h : P(L J ) P(L J ). Remark 2. Assume x is a fixed point of H in E ; then by proposition 3, P x has m fixed critical points. The polynomial P x is of degree m + 1, so there is a unique (repelling) fixed point of P x which is not critical. Moreover, since P x is centered and m + 1 3, the fixed points of P x are also centered; this implies that this fixed point is at 0.

19 BÖTTCHER COORDINATES 19 Consequently, the rational map f : w 1/P x (1/w) has degree m+1, a repelling fixed point at, and m + 1 superattracting fixed points. This map is therefore the Newton s method of a polynomial Q of degree m + 1. The critical points of f are the zeroes of Q, and the zeroes of Q. Since f has a critical point of multiplicity m at 0, Q (w) = aw m 1 for some a C and Q vanishes at 0. An elementary computation then shows that P x (z) = m + 1 m z + zm+1. This polynomial is indeed one with simple critical points which are fixed. So H has exactly m! fixed points in E. Figure 2. The polynomial P (z) = 6 5 z +z6 has five critical points, each of which is a superattracting fixed point of P. There is a repelling fixed point at 0. The polynomial P is conjugate to a Newton s method for finding roots of some polynomial Q as discussed in remark The endomorphisms are postcritically finite. Our goal now is to prove that the endomorphisms h : P(L J ) P(L J ) are postcritically finite: we will identify the critical locus of H : L J L J, and show that it is invariant. For this we will use the following observation. Let J be a nonempty proper subset of I. Let E J be the set of functions J C whose average is 0. There is a natural projection π J : E E J given by E x x J Average(x J ) E J. Let H J : E J E J be the homogeneous map constructed as above with the set J instead of the set I. Lemma 12. Assume J Part (I), x L J J, and J J. Then, as v 0 in E, we have the following expansion π J H(x + v) = C J H J π J (v) + O ( v π J (v) J +1) with C J C {0}.

20 20 X. BUFF, A.L. EPSTEIN, AND S. KOCH Proof. Without loss of generality, assume that m J := J 2 since otherwise E J has dimension 0, and the result is vacuous. Set y:=x + v, a y :=Average(y J ) and z:=π J (v) = π J (y) = y J a y. Let Q z be the polynomial defined by Then Thus P y(t) = (m + 1) i I Q z (t):= m J + 1 m J j J t z j (w z i ) dw. i J (t y i ) and Q z(t a y ) = (m J + 1) i J(t y i ). P y(t) = Q z(t a y ) R y (t) with R y (t) = m + 1 m J + 1 i I J (t y i ). Since x L J J, for all i I J we have x i a x, thus R x (a x ) 0. For all i J and j J, as y x, we have: y j y i = z j z i = O ( z ), sup Q z(t a y ) = O ( ) z m J, and t [y i,y j] sup R y (t) R x (a x ) = O ( v ). t [y i,y j] Thus for all i J and j J, P y (y j ) P y (y i ) ( Q z (z j ) Q z (z i ) ) R x (a x ) = = yj y i yj P y(t) Q z(t a y )R x (a x ) dt y i Q z(t a y ) ( R y (t) R x (a x ) ) dt O ( v z m J +1 ). Setting C J :=R x (a x ), we deduce that for all j J, 1 ( 1 P y (y j ) P y (y i ) = C J Q z (z j ) m J + 1 m J + 1 i J The result follows since i J = C J Q z (z j ) + O ( v z m J +1 ). ) Q z (z i ) + O ( v z m J +1 ) π J H = P y y J Average(P y y J ) and H J π J (y) = H J (z) = Q z z. Proposition 4. For J Part (I), the critical set of H : L J L J is J := L K, K Part (I) K J H( J ) = J, and h : P(L J ) P(L J ) is postcritically finite. Proof. It is enough to prove that the critical set of H : L J L J is J as the rest is an immediate consequence. Given J 1, J 2 J, set K:=J 1 J 2 and K = K(J 1, J 2 ):=J {J 1, J 2 } {K}.

21 BÖTTCHER COORDINATES 21 Note that K J and K = J 1, so that L K has codimension 1 in L J. Choose a vector v L J L K so that L J = L K Span(v). According to lemma 12, for all x L K as t 0 we have π K H(x + tv) = O(t K +1 ). It follows that the Jacobian of H : L J L J vanishes with order at least K along L K. Now J = J 1,J 2 J J 1 J 2 L K(J 1,J 2) and L K(J1,J 2) is a critical component of H : L J L J with multiplicity J 1 + J 2. So J is contained in the critical set of H : L J L J as a hypersurface of total degree ( ) J 1 + J 2 = J 1 J = ( J 1 ) m. J 1,J 2 J J J J 1 J 2 Since H : L J L J is a homogeneous map of degree m + 1, its critical locus is a hypersurface of total degree dim(l J ) m = ( J 1 ) m. And therefore the critical set of H : L J L J coincides with J as required. We now analyze the spectrum of D x H : T x E T x E, where x is a fixed point of H : E E. It turns out that for the fixed points in E, we have a complete understanding of this spectrum as outlined in the example below. Example 2. We now demonstrate by way of example that the eigenvalues of D x H at a fixed point x E are precisely λ k :=(m + 1)/k, k [1, m 1], with corresponding eigenspace Span(x k ). 1 According to remark 2, there is a unique polynomial P (z) = m + 1 m z + zm+1 which is monic and centered, with simple fixed critical points, corresponding to a fixed point of H in E. Observe that if c is a critical point of P, then P (c) = (m + 1)/c. Now if P t (z) = P (z) + tq(z) + o(t) with Q(z) C m 1 [z], and if c t = c + tv + o(t) is a critical point of P t, then so that and 0 = P t(c t ) = P (c) + t(p (c)v + Q (c)) + o(t) v = Q (c) P (c) = c m + 1 Q (c), P t (c t ) = P (c) + tq(c) + tp (c)v + o(t) = c + tq(c) + o(t). Therefore v T x E is an eigenvector associated to the eigenvalue λ if and only if i I, Q(x i ) = λv i = λx i m + 1 Q (x i ). This is clearly true if Q(z) = z k, λ = (m + 1)/k, and v i = x k i /λ. 1 Recall that x : I C is a function and x k is the function i x k i C.

22 22 X. BUFF, A.L. EPSTEIN, AND S. KOCH 3.3. Fixed points are super-saddles. Proposition 5. Any fixed point of H : E E is a super-saddle. More precisely, if J Part (I) and if x L J J is a fixed point of H : E E, then: T x E = Ker(D x H) Im(D x H) Im(D x H) = T x L J, and the spectrum of D x H : T x L J T x L J is contained in C D and therefore x is a repelling fixed point of H : L J L J. Proof. According to proposition 4, x is not a critical point of H : L J L J, thus the restriction D x H : T x L J T x L J is invertible. As a consequence the image of D x H contains T x L J and Ker(D x H) T x L J = {0}. The kernel of the projection π:= J J π J : T x E J J T 0 E J is T x L J. According to lemma 12 π J D x H = 0 for all J J. So π D x H = 0, and the image of D x H is contained in Ker(π) = T x L J. This implies that Im(D x H) = T x L J. In addition the codimension of Ker(D x H) is the dimension of T x L J and since Ker(D x H) T x L J = {0}, we conclude that T x E = Ker(D x H) Im(D x H). Since H : L J L J is homogeneous of degree m + 1, there is an obvious eigenvalue m + 1 associated to the eigenspace Span(x). Proposition 6 below asserts that we can endow T x L J with an appropriate norm so that the linear map (D x H) 1 : T x L J T x L J is contracting. As a consequence, the spectrum of (D x H) 1 : T x L J T x L J is contained in D and thus, the spectrum of the linear map D x H : T x L J T x L J is contained in C D. By definition E is a subspace of C I of codimension 1. It is the kernel of the linear form e I : E v 1 v i C. I As a consequence E may be identified with the quotient (C I ) /Span(e I ). Let 1 : I C be the function which is constant and equal to 1. Since C I = E Span(1), the dual space E may also be identified with the orthogonal space ( ) { Span(1) = α (C I ) : α(1) = 0 }. More generally, given J I let e J (C I ) be the linear form defined by e J (v) = 1 v j. J Note that for J Part (I), E = L J J J i I j J Ker(e J ) and therefore L J may be identified with { ( ) Span(1) Span(e J ; J J ) = λ J e J : λ J C and } λ J = 0. J J J J

23 BÖTTCHER COORDINATES 23 If α = J J λ J e J L J then the pairing with v L J is given by α(v) = J J λ J v J with v(j) = {v J }. Given x L J J we will now identify (T x L J ) with a space of quadratic differentials, equip this space with an appropriate norm then identify the transpose of (D x H) 1 : T x L J T x L J, and finally show that this transpose is strictly contracting. So for J J, let q J be the quadratic differential defined on C by q J = dz2. z x J Set { Q x := λ J q J : } λ J = 0. J J J J A quadratic differential q = λ J q J Q x may be paired with a tangent vector v T x L J as follows q, v := J J λ J v J = J J Res xj (q ξ v ) where ξ v is any holomorphic vector field near x(i) which takes the value v J at x J. According to the previous discussion, this pairing gives an identification of (T x L J ) with Q x. Choose R large enough so that Px 1 (D R ) is compactly contained in D R, where D R is the disk centered at 0 of radius R. We equip Q x with the L 1 norm q := q. D R Proposition 6. Assume x L J J. The transpose of the linear mapping (D x H) 1 : T x L J T x L J is identified with the push-forward operator (P x ) : Q x q g q Q x where g ranges over the inverse branches of P x. In addition, q Q x (P x ) q < q. Proof. The space Q x is the set of meromorphic quadratic differentials on P 1 (C) which are holomorphic outside x(i), have at most simple poles along x(i) and at most a double pole at. Set P :=P x. If q Q x then P q is a meromorphic quadratic differential on P 1 (C). Since q has at most a double pole at, P q also has at most a double pole at. The other poles of P q are simple and contained in P ( x(i) ) union the critical value set of P, that is x(i). This shows that P maps Q x to Q x. In addition P q = g q g q = q < q = q. D R P 1 (D R ) D R D R Therefore we only need to prove that for all v T x L J and all q Q x q, v = P q, D x H(v). Fix v T x L J and q Q x. Let U be the complement in D R of pairwise disjoint closed disks centered at the points of x(i), and contained in D R. If ξ is a C

24 24 X. BUFF, A.L. EPSTEIN, AND S. KOCH vector field on C which is holomorphic outside U, vanishes outside D R, and satisfies ξ x = v, then q, v = Res xj (q ξ) = 1 1 q ξ = q ξ = 1 q ξ. 2πi J J U Stokes 2πi U 2πi C With an abuse of notation set Then, x t :=x + tv, P t :=P xt, and y t :=H(x t ) = P t x t. ẋ:= x t t = v and ẏ:= y t = D x H(v). t=0 t t=0 In addition, the critical point set of P t is x t (I) and the critical value set of P t is y t (I). Let (ϕ t : C C) t ( ɛ,ɛ) be an analytic family of C diffeomorphisms such that Note that ϕ 0 = id, ϕ t is the identity outside D R, ϕ t is holomorphic outside U and y t = ϕ t y. ϕ:= ϕ t t is a C vector field on C which is holomorphic outside U, vanishes outside D R, and satisfies ϕ x = D x H(v). Since ϕ t follows the critical value set of P t we can lift the diffeomorphisms ϕ t : C C to diffeomorphisms ψ t : C C so that ψ 0 = id and the following diagram commutes t=0 (C, x(i)) P (C, y(i)) ψ t ϕ t (C, x t (I)) P t (C, y t (I)) Then, ψ:= ψ t t is a C vector field on C which is holomorphic outside U, vanishes outside D R, and satisfies ψ x = v. In addition, the infinitesimal Beltrami differentials ϕ and ψ satisfy the relation t=0 ψ = P ( ϕ). Therefore q, v = q ψ = q P ( ϕ) = P q ϕ = P q, D x H(v) as required. C C C

25 BÖTTCHER COORDINATES 25 Figure 3. On the left is a real slice of the immediate basin B aj (h) in the case I = 4 and J = {J 1, J 2 } with J 1 = 3 and J 2 = 1. The critical set P( ) contains three lines passing through a J. Almost every orbit in B aj (h) converges to a J tangentially along one of these lines; the immediate basin is colored accordingly. Since J 2 = 1, E J2 = {0} and H J1 H J2 = H J1. According to proposition 7 below, there is a Böttcher coordinate Φ : B aj (h) B 0 (H J1 ). On the right is a real slice of the immediate basin B 0 (H J1 ). The isomorphism Φ respects the coloring Superattracting fixed points of h : P(E) P(E). If J Part (I) has cardinality 2, then L J has dimension 1 and its image in P(E) is a point a J. This is a superattracting fixed point for h : P(E) P(E). We will now show that we can apply theorems 1 and 2. Note that H J1 H J2 : E J1 E J2 E J1 E J2 is a quasihomogeneous map of bidegree ( J 1, J 2 ). Proposition 7. Let J :={J 1, J 2 } Part (I) and a J be the image of L J in P(E). Then there is an analytic isomorphism conjugating h to H J1 H J2. Φ : B aj (h) B 0 (H J1 H J2 ) Proof. The analytic map h : P(E) P(E) is proper, and has a superattracting fixed point at a J. We will prove that there is a local isomorphism Φ : ( P(E), a J ) ( EJ1 E J2, 0 ) conjugating h to H J1 H J2. Such a Φ automatically maps the germ of the postcritical set of h at a J, i.e. the germ of P( ) at a J, to the germ of the postcritical set of H J1 H J2 at 0. So near a J, Φ maps the postcritical set of h : B aj (h) B aj (h), that is B aj (h) P( ), to the postcritical set of H J1 H J2. The result then follows from theorem 2. To prove that there is a local conjugacy, we use theorem 1. Choose x L J and let L J E be a subspace of codimension 1 such that E = L J L J. Let σ : (P(E), a J ) (E, x) be the local section with image in x + L J. Then the composition π σ : (P(E), a J ) (E J1 E J2, 0) is a local isomorphism. It conjugates