AN INTERPOLATION THEOREM FOR

HOLOMORPHIC AUTOMORPHISMS OF C^{n}

Gregery T. Buzzard* and Franc Forstneric**

Department of Mathematics, Indiana University, Bloomington, IN 47405, USA Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA

Abstract

We construct automorphisms of C^{n} which map certain discrete sequences one onto another with prescribed finite jet at each point, thus solving a general Mittag-Leffler interpolation problem for automorphisms. Under certain circumstances, this can be done while also approximating a given automorphism on a compact set.

&0. Introduction

Let C^{n} be complex Euclidean space of dimension n with coordinates z = (z_{1}, …, z_{n}). We shall always assume n > 1. We denote by AutC^{n} the group of all holomorphic automorphisms of C^{n}, and by Aut_{1}C^{n} the group of automorphisms F ∈ AutC^{n} with complex Jacobian one: JF ≡ 1. For some recent developments regarding these groups see the survey [2]. By B we denote the open unit ball in C^{n}.

In this paper we construct automorphisms of C^{n} which map certain discrete sequences one onto another with prescribed finite jet at each point, thus solving a general Mittag-Leffler interpolation problem for automorphisms.

In approaching such an interpolation problem one must consider the limitations of automorphisms in mapping one sequence to another. Rosay and Rudin [5] showed that, in general, one cannot map an infinite discrete set in C^{n} onto another discrete set by an automorphism of C^{n}. In fact, the infinite discrete sets in C^{n} form uncountably many different equivalence classes under this relation [5, sect. 5]. They introduced the following notion:

Definition. A discrete infinite sequence {a_{j}} ⊂ C^{n} (without repetition) is tame if there exists a holomorphic automorphism F ∈ AutC^{n} such that

F (a_{j}) = je_{1} = (j, 0, …, 0), j = 1, 2, 3, …

* Research at MSRI is supported in part by NSF grant DMS-9022140 ** Supported in part by an NSF grant and by a grant from the Ministry of Science of the Republic of Slovenia The sequence {a_{j}} is very tame if the above holds for some F ∈ Aut_{1}C^{n}.

Rosay and Rudin [5] showed that points in a tame sequence are permutable by automorphisms of C^{n}. Hence one can speak about the tameness of an infinite discrete set. On the other hand, they showed that there exist discrete sets E ⊂ C^{n} such that the only nondegenerate holomorphic map F: C^{n} → C^{n} satisfying F (C^{n}\E) ⊂ C^{n}\E is the identity map [5, theorem 5.1].

In view of this, it is reasonable to consider tame sequences in order to get positive results on jet interpolation. The main result of this paper is the following.

0.1 Theorem. Assume that n > 1, that {a_{j}} and {b_{j}} are tame sequences in C^{n} (without repetitions), and that P_{j}: C^{n} → C^{n} is a holomorphic polynomial map of degree at most m_{j} ≥ 1 such that P_{j}(0) = 0 and JP_{j}(0) ≠ 0. Then there exists F ∈ AutC^{n} such that for every j = 1, 2, 3, … we have F (a_{j}) = b_{j} and

F (z) = b_{j} + P_{j}(z − a_{j}) + O(|z − a_{j}|^{m_{j}+1}), z → a_{j}.  (0.1)

If in addition the polynomials P_{j} satisfy

JP_{j}(z) = 1 + O(|z|^{m_{j}}), z → 0, j = 1, 2, 3, …,  (0.2)

and the sequences {a_{j}} and {b_{j}} are very tame, we may choose F ∈ Aut_{1}C^{n}.

A version of this theorem in which the degrees of the polynomials P_{j} are uniformly bounded can be found in [3, theorem 3.1]. In that result, with a_{j} = b_{j} = je_{1} for all j, the automorphism F is a finite composition of shears and generalized shears. See the survey [2] for further information on shears and generalized shears.

Under certain circumstances we can obtain the above result while at the same time approximating a given automorphism on a polynomially convex set.

0.2 Theorem. In addition to the hypotheses of theorem 0.1, assume that Φ ∈ AutC^{n} is an automorphism and K ⊂ C^{n}\{a_{j}}_{j=1}^{∞} is a compact, polynomially convex set such that Φ(K) ⊂ C^{n}\{b_{j}}_{j=1}^{∞}. Then for each ϵ > 0 there exists an F ∈ AutC^{n} satisfying theorem 0.1 and

|F (z) − Φ(z)| < ϵ,  z ∈ K.  (0.3)

If the volume preserving assumptions in theorem 0.1 hold and if Φ ∈ Aut_{1}C^{n}, we may choose F ∈ Aut_{1}C^{n}.

The paper is organized as follows. In section 1 we recall a result concerning holomorphic automorphisms of C^{n} which have prescribed jets of finite order at a finite set of points, and which are close to the identity on a given polynomially convex set, as well as a result about convergence of compositions of automorphisms. Both of these results are proved in [3].

In section 2 we provide the proof of theorem 0.1. The outline of the proof is as follows. Since {a_{j}} and {b_{j}} are tame, we may first assume that they both lie in the z_{1}-axis. Next, if {c_{j}} is a tame set in the z_{1}-axis, we can find Ψ ∈ Aut_{1}C^{n} such that at each point a_{j}, Ψ agrees with the translation z ↦ c_{j} + (z − a_{j}) to order m_{j}. In fact, we can find such a Ψ which is a composition of three shears of the form z ↦ z + f(λ(z))v, where f is entire on C, v ∈ C^{n}\{0}, and λ is a C-linear form with λ(v) = 0.

Hence it suffices to solve the following simpler problem: Given polynomials P_{j} as above, construct a tame sequence {c_{j}} contained in the z_{1}-axis and F ∈ AutC^{n} such that F fixes each c_{j} and has prescribed jet P_{j} at c_{j} (with appropriate modifications in the volume-preserving case).

To construct F and {c_{j}}, we use an inductive procedure. We take c_{1} = 3e_{1} and use the results of section 1 to find an automorphism F_{1} with prescribed jet P_{1} at c_{1} and which is near the identity on K_{1} = 2\overline{B}, where B is the unit ball in C^{n}. We then choose a point c_{2} on the z_{1}-axis so that c_{2} and F_{1}(c_{2}) lie outside the convex hull K_{2} of 2\overline{B}∪K_{1} ∪F_{1}(2\overline{B})∪\{c_{1}\}. Again the results of section 1 allow us to find an automorphism Ψ_{2} which is near the identity on K_{2} and which agrees with the identity to order m_{1} at c_{1}, which maps F_{1}(c_{2}) to c_{2}, and which has an appropriate jet at F_{1}(c_{2}). Then F_{2} = Ψ_{2} ∘ F_{1}. We then repeat this procedure, choosing c_{k+1} so that c_{k+1} and F_{k}(c_{k+1}) lie outside the convex hull K_{k+1} of (k + 1)\overline{B} ∪ K_{k} ∪ F_{k}((k + 1)\overline{B}) ∪ \{c_{k}\}, then choosing Ψ_{k} to be near the identity on K_{k+1}, equal to the identity to order m_{j} at c_{j} for j ≤ k, mapping F (c_{k+1}) to c_{k+1} and having appropriate jet at F_{k}(c_{k+1}).

In this way we define a sequence of automorphisms F_{1}, F_{2}, … with F_{k+1} = Ψ_{k+1} ∘ F_{k}. Then the second result of section 1 implies that this sequence converges to an automorphism of C^{n} with the desired properties.

In section 3 we prove theorem 0.2. We use the result of theorem 0.1, but must take extra care to insure that we can in addition approximate the given map. The main addition here is lemma 3.1: given a compact, polynomially convex set K ⊂ C^{n}, an automorphism H of C^{n}, and a discrete sequence {a_{j}} contained in the z_{1}-axis outside K, there is an R > 0 and an automorphism Ψ ∈ AutC^{n} that approximates H on K and satisfies Ψ(a_{j}) = (R + j)e_{1} for each j.

&1. Interpolation on a finite set and convergence of automorphisms

The results of this section are taken directly from [3]. We do not reproduce the proofs here. The first result concerns jet interpolation by automorphisms on a finite set.

1.1 Proposition. Let n > 1. Assume that

(a) K ⊂ C^{n} is a compact, polynomially convex set,

(b) {a_{j}}_{j=1}^{k} ⊂ K is a finite subset of K,

(c) p and q are arbitrary points in C^{n}\K (not necessarily distinct),

(d) N is a nonnegative integer, and

(e) P: C^{n} → C^{n} is a holomorphic polynomial map of degree at most m ≥ 1, satisfying P (0) = 0 and JP (0) ≠ 0.

Then for each ϵ > 0 there exists an automorphism F ∈ AutC^{n} satisfying

(i) F (z) = q + P (z − p) + O(|z − p|^{m+1}) as z → p, (ii) F (z) = z + O(|z − a_{j}|^{N}) as z → a_{j} for each j = 1, 2, …, k, and (iii) |F (z) − z| + |F^{−1}(z) − z| < ϵ for each z ∈ K. If in addition the polynomial map P satisfies (e’) P (0) = 0 and JP (z) = 1 + O(|z|^{m}) as z → 0, then there exists a polynomial automorphism F with JF = 1 satisfying (i)–(iii).

The next result concerns convergence of a sequence of automorphisms. It gives sufficient conditions for such a sequence to converge to a biholomorphic map in some domain and describes the domain of convergence and the image of the limit map.

1.2 Proposition. Let D be a connected open set in C^{n} which is exhausted by compact sets K_{0} ⊂ K_{1} ⊂ K_{2} ⊂ ⋯ ⊂ ∪_{j=0}^{∞}K_{j} = D such that K_{j−1} ⊂ IntK_{j} for each j ∈ N. Choose numbers ϵ_{j} (j = 1, 2, 3, …) such that

0 < ϵ_{j} < dist(K_{j−1}, C^{n}\K_{j}) (j ∈ N),  ∑_{j=1}^{∞} ϵ_{j} < ∞.  (1.1)

Suppose that for each j = 1, 2, 3, …, Ψ_{j} is a holomorphic automorphism of C^{n} satisfying

|Ψ_{j}(z) − z| < ϵ_{j}, z ∈ K_{j}.  (1.2)

Set Φ_{m} = Ψ_{m}∘Ψ_{m−1}∘⋯∘Ψ_{1}. Then there is an open set Ω ⊂ C^{n} such that lim_{m→∞} Φ_{m} = Φ exists on Ω (uniformly on compacts), and Φ is a biholomorphic map of Ω onto D. In fact, Ω = ∪_{m=1}^{∞}Φ_{m}^{−1}(K_{m}).

Finally, we include some elementary observations about power series. If F: U ⊂ C^{n} → C^{n} is a holomorphic map, a ∈ U, and m ≥ 1 an integer, we write

F (z) = F (a) + F_{m,a}(z − a) + O(|z − a|^{m+1}), z → a.

Thus F_{m,a} is just the Taylor polynomial of F of order m at a without the constant term. The following lemma is evident by composing the power series.

1.3 Lemma. If F and G are nondegenerate holomorphic maps on certain open subsets of C^{n}, with values in C^{n}, then for each integer m ≥ 1 we have

(G ∘ F)_{m,a}(z) = G_{m,F (a)} ∘ F_{m,a}(z) + O(|z|^{m+1}), z → 0.

at each point a where the composition G ∘ F is defined.

This generalizes to composition of several maps. We also have

(F^{−1})_{m,F (a)} ∘ F_{m,a}(z) = z + O(|z|^{m+1}), z → 0.

&2. Proof of the interpolation theorem In this section we prove theorem 0.1. The main construction is contained in the following lemma.

2.1 Lemma. Given P_{j} and m_{j} as in theorem 0.1, there exists a discrete sequence {c_{j}} contained in the z_{1}-axis and F ∈ AutC^{n} such that for all j we have

F (z) = c_{j} + P_{j}(z − c_{j}) + O(|z − c_{j}|^{m_{j}+1}), z → c_{j}.  (2.1)

If in addition each polynomial P_{j} satisfies (0.2), we may choose F ∈ Aut_{1}C^{n}.

Proof. For j ≥ 1, let ϵ_{j} = 2^{−j}. We will construct the sequence {c_{j}} inductively and construct F as the limit of a composition of automorphisms, each chosen inductively.

Let K_{0} = ∅ and K_{1} = 2\overline{B}. Let c_{1} = 3e_{1}. By proposition 1.1 with K = K_{1}, there exists Ψ_{1} ∈ AutC^{n} such that Ψ_{1}(z) = c_{1} + P_{1}(z − c_{1}) + O(|z − c_{1}|^{m_{1}+1}), z → c_{1} and |Ψ_{1}(z) − z| + |Ψ_{1}^{−1}(z) − z| < ϵ_{1} for z ∈ K_{1}. If also P_{1} satisfies (0.2), then we may choose Ψ_{1} ∈ Aut_{1}C^{n}. Let F_{0}(z) = z and F_{1} = Ψ_{1}.

We will inductively choose automorphisms Ψ_{j} and let F_{k} = Ψ_{k} ∘ ⋯ ∘ Ψ_{1}. For the induction, suppose we have the following.

(1) Compact, convex sets K_{0} ⊂ K_{1} ⊂ ⋯ ⊂ K_{k} with jB ∪ F_{j−1}(jB) ⊂ K_{j} and

dist(K_{j−1} ∪ F_{j−1}(jB), C^{n}\K_{j}) > ϵ_{j}

for each 1 ≤ j ≤ k.

(2) Points c_{j} ∈ K_{j+1}\K_{j} for 1 ≤ j ≤ k −1 and c_{k} ∈ C^{n}\K_{k} such that each c_{j} is contained in the z_{1}-axis.

(3) Ψ_{j} ∈ AutC^{n} for 1 ≤ j ≤ k with

|Ψ_{j}(z) − z| + |Ψ_{j}^{−1}(z) − z| < ϵ_{j}

for z ∈ K_{j} (and Ψ_{j} ∈ Aut_{1}C^{n} if P_{j} satisfies (0.2)).

(4) F_{k} = Ψ_{k} ∘ Ψ_{k−1} ∘ ⋯ ∘ Ψ_{1} satisfying (2.1) for 1 ≤ j ≤ k.

Given this, let K_{k+1} be a compact, convex set in C^{n} with

(k + 1)B ∪ K_{k} ∪ F_{k}((k + 1)B) ∪ {c_{k}} ⊂ K_{k+1}

and

dist(K_{k} ∪ F_{k}((k + 1)B), C^{n}\K_{k+1}) > ϵ_{k+1}.

Since K_{k+1} is compact, we can choose c_{k+1} in the z_{1}-axis so that c_{k+1}, F_{k}(c_{k+1}) ∈ C^{n}\K_{k+1}. Let N = max{m_{1}, …, m_{k}} + 1. Let d_{k+1} = F_{k}(c_{k+1}), and let Q_{k+1} = P_{k+1} ∘ (F_{k}^{−1})_{m_{k+1},d_{k}}. Using proposition 1.1, choose Ψ_{k+1} ∈ AutC^{n} (with Ψ_{k+1} ∈ Aut_{1}C^{n} if P_{k+1} satisfies (0.2) and F_{k} ∈ Aut_{1}C^{n})) so that

(i) Ψ_{k+1}(z) = c_{k+1} + Q_{k+1}(z − d_{k+1}) + O(|z − d_{k+1}|^{m_{k+1}+1}) as z → d_{k+1}, (ii) Ψ_{k+1}(z) = z + O(|z − c_{j}|^{N}) as z → c_{j}, 1 ≤ j ≤ k, (iii) |Ψ_{k+1}(z) − z| + |Ψ_{k+1}^{−1}(z) − z| < ϵ_{k+1} for each z ∈ K_{k+1}.

Taking F_{k+1} = Ψ_{k+1} ∘ F_{k}, we obtain the induction hypotheses at stage k + 1.

By proposition 1.2, the sequence {F_{j}} converges uniformly on compact subsets of Ω = ∪_{j=1}^{∞}F_{j}^{−1}(K_{j}) to a biholomorphic map F from Ω onto D = ∪_{j=1}^{∞}K_{j}. Since jB ⊂ K_{j}, we see that D = C^{n}. Moreover, F_{j}^{−1}(K_{j}) = F_{j−1}^{−1}Ψ_{j}^{−1}(K_{j}), and by Rouché’s theorem [1, p.110] and induction hypotheses (1) and (3), we see that F_{j−1}(jB) ⊂ Ψ_{j}^{−1}(K_{j}). Hence jB ⊂ F_{j}^{−1}(K_{j}), so also Ω = C^{n}.

Hence F ∈ AutC^{n} satisfies (2.1) for all j since each F_{k} satisfies (2.1) for 1 ≤ j ≤ k. Finally, if each P_{j} satisfies (0.2), then each Ψ_{j} ∈ Aut_{1}C^{n}, so F ∈ Aut_{1}C^{n}. ♠

Remark. Given R > 0, 1 > ϵ > 0, we can replace K_{1} by (R + 1)\overline{B}, c_{1} by (R + 2)e_{1}, and ϵ_{j} by ϵ(ϵ_{j}/2) to construct a sequence {c_{j}} and F satisfying the conclusions of the theorem and also so that |F (z) − z| + |F^{−1}(z) − z| < ϵ on RB.

We next use a classical 1-variable interpolation result together with theorem 0.1 to find an automorphism fixing each je_{1} and having prescribed jet there. Our technique of using shears to map a given discrete set in the z_{1}-axis to another was also used in the proof of proposition 3.1 in [5]. However, for the current application, we need the map to be tangent to a translation to high order at each point in the discrete set.

2.2 Corollary. Let P_{j} and m_{j} be as in theorem 0.1. For each R ≥ 0 and ϵ > 0 there exists F ∈ AutC^{n} such that

\begin{matrix}F (z) = je_{1} + P_{j}(z − je_{1}) + O(|z − je_{1}|^{m_{j}+1}), z → je_{1}, j > R, \\ |F (z) − z| + |F^{−1}(z) − z| < ϵ, |z| < R. \end{matrix}

If each P_{j} satisfies (0.2), then we can choose F ∈ Aut_{1}C^{n}.

Proof. We shall first prove the corollary without the last condition on F (i.e., taking R = 0). By lemma 2.1, there exists G ∈ AutC^{n} and a sequence {c_{j}} contained in the z_{1}-axis so that G(z) = c_{j} + P_{j}(z − c_{j}) + O(|z − c_{j}|^{m_{j}+1}) as z → c_{j}. Also, G ∈ Aut_{1}C^{n} if each P_{j} satisfies (0.2).

To obtain the corollary we need only find Ψ ∈ Aut_{1}C^{n} which maps each je_{1} to c_{j} with Ψ_{m_{j},je_{1}}(z) = z − je_{1}, since then lemma 1.3 implies that Ψ^{−1} ∘ G ∘ Ψ has the desired properties.

To do this, let ξ_{j} ∈ C such that c_{j} = ξ_{j}e_{1}. By a standard 1-variable interpolation result [4, cor. 1.5.4], there exists an entire function f_{1} of one variable with f_{1}(ζ) = j+O(|ζ−j|^{m_{j}+1}) as ζ → j for all j ≥ 1. Let Ψ_{1}(z) = z+f_{1}(z_{1})e_{2}, where e_{2} = (0, 1, 0, …, 0). Then Ψ_{1} ∈ Aut_{1}C^{n} and Ψ_{1}(z) = j(e_{1} + e_{2}) + (z − je_{1}) + O(|z − je_{1}|^{m_{j}+1}) as z → je_{1}. Thus Ψ_{1} maps je_{1} to j(e_{1} + e_{2}) and agrees with a translation to order m_{j} + 1 at je_{1}.

Likewise, choosing f_{2} entire with f_{2}(ζ) = ξ_{j} − j + O(|ζ − j|^{m_{j}+1}) as ζ → j, and taking Ψ_{2}(z) = z + f_{2}(z_{2})e_{1}, we see that Ψ_{2} maps j(e_{1} + e_{2}) to ξ_{j}e_{1} + je_{2} and agrees with a translation to order m_{j} + 1 at j(e_{1} + e_{2}). Similarly, we can find Ψ_{3}(z) = z_{1} + f_{3}(z_{1})e_{2} so that Ψ_{3} maps ξ_{j}e_{1} + je_{2} to ξ_{j}e_{1} = c_{j} and agrees with a translation to order m_{j} + 1 at ξ_{j}e_{1} + je_{2}.

Let Ψ = Ψ_{3} ∘ Ψ_{2} ∘ Ψ_{1}. Then Ψ(z) = c_{j} + (z − je_{1}) + O(|z − je_{1}|^{m_{j}+1}) as z → je_{1}, and by lemma 1.3, we see that Ψ^{−1}(z) = je_{1} + (z − c_{j}) + O(|z − c_{j}|^{m_{j}+1}) as z → c_{j}. Also, Ψ ∈ Aut_{1}C^{n} since each Ψ_{l} has Jacobian one.

Hence taking F = Ψ^{−1} ∘ G ∘ Ψ, and applying lemma 1.3, we obtain the desired automorphism.

Our proof shows that the sequence {je_{1}} could clearly be replaced by any discrete sequence {d_{j}} without repetition in the z_{1}-axis. Moreover, to get the last condition on F, suppose that \{d_{j}\} and \{c_{j}\} lie outside the ball R\overline{B} (in the z_{1} axis). A simple argument using Runge’s theorem and the Weierstrass theorem shows that we can choose each f_{l} as above and so that |f_{l}| is small on a neighborhood of the closed disk of radius R in C. Hence by the remark after the proof of lemma 2.1, we see that we may choose F satisfying the conclusions of corollary 2.2 with d_{j} in place of je_{1} and so that |F (z)−z|+|F^{−1}(z)−z| < ϵ on RB. ♠

Proof of theorem 0.1. Since {a_{j}} and {b_{j}} are tame, there exist H_{1}, H_{2} ∈ AutC^{n} such that H_{1}(a_{j}) = H_{2}(b_{j}) = je_{1}, and if the sequences are very tame, then H_{1}, H_{2} ∈ Aut_{1}C^{n}.

Hence it suffices to construct G ∈ AutC^{n} (or G ∈ Aut_{1}C^{n}) such that G(je_{1}) = je_{1} and (H_{2}^{−1}GH_{1})_{m_{j},a_{j}} = P_{j}. By lemma 1.3, this latter condition can be satisfied by making

G_{m_{j},je_{1}}(z) = (H_{2})_{m_{j},b_{j}} ∘ P_{j} ∘ (H_{1}^{−1})_{m_{j},je_{1}}(z) + O(|z|^{m_{j}+1}), z → 0

for each j. By corollary 2.2, we can find G ∈ AutC^{n} satisfying this condition and G(je_{1}) = je_{1}. Moreover, if each P_{j} satisfies (0.2) and H_{1}, H_{2} ∈ Aut_{1}C^{n}, then we can choose G ∈ Aut_{1}C^{n}.

Finally, taking F = H_{2}^{−1} ∘ G ∘ H_{1}, we obtain the desired automorphism. ♠

&3. Interpolation with approximation

We begin with the following lemma.

3.1 Lemma. Let H ∈ AutC^{n}, let K be a compact, polynomially convex set in C^{n}, and let {a_{j}}_{j=1}^{∞} be a discrete sequence disjoint from K and contained in the z_{1}-axis. Let R > 0 with H(K) ⊂ RB and let ϵ > 0. Then there exists Ψ ∈ AutC^{n} such that |Ψ(z)−H(z)| < ϵ on K, |Ψ^{−1}(z) − H^{−1}(z)| < ϵ on H(K), and Ψ(a_{j}) = (R + j)e_{1} for all j. If H ∈ Aut_{1}C^{n}, then we can choose Ψ ∈ Aut_{1}C^{n}.

Proof. Let r > 0 with K ⊂ rB. Let K_{0} ⊂ rB be compact and polynomially convex with K ⊂ IntK_{0}, K_{0} ∩ {a_{j}} = ∅, and H(K_{0}) ⊂ RB. Let a_{j_{1}}, …, a_{j_{m}} be the points in \{a_{j}\} ∩ r\overline{B}, and let δ > 0. Using the fact that the union of K_{0} with finitely many points is again polynomially convex, we can apply proposition 1.1 m times to find Ψ_{1} ∈ Aut_{1}C^{n} such that |Ψ_{1}(z) − z| + |Ψ_{1}^{−1}(z) − z| < δ for z ∈ K_{0} and Ψ_{1}(a_{j_{k}}) = H^{−1}((R + j_{k})e_{1}) for k = 1, …, m.

Let π_{2}(z) = z_{2}. For fixed v ∈ C^{n}\{0} and j ∈ N we consider the 1-variable function g_{j}(ζ) = π_{2}HΨ_{1}(a_{j} + ζv). Since the kernel of D_{a_{j}}(π_{2} ∘ H ∘ Ψ_{1}) is an (n − 1)-dimensional subspace for each j, we may choose v arbitrarily near e_{2} such that g_{j} is nonconstant for each j and hence so that the image of g_{j} omits at most one point in C. In particular, we may choose such a v so that there exists a C-linear form λ with λ(v) = 0, λ(e_{1}) = 1, and λ(a_{j}) ∉ λ(r\overline{B}) if j ∉ \{j_{1}, …, j_{m}\}.

Choose f entire on C such that |f| < δ/2 on λ(rB), f(λ(a_{j_{k}})) = 0 for each k = 1, …, m, and |π_{2}HΨ_{1}(a_{j} + f(λ(a_{j}))v)| = R + j for each j ∉ {j_{1}, …, j_{m}}. Let Ψ_{2}(z) = z + f(λ(z))v. Then Ψ_{2} ∈ Aut_{1}C^{n}, |Ψ_{2}(z) − z| + |Ψ_{2}^{−1}(z) − z| < δ on rB, HΨ_{1}Ψ_{2}(a_{j_{k}}) = (R + j_{k})e_{1} for k = 1, …, m, and |π_{2}HΨ_{1}Ψ_{2}(a_{j})| = R + j for j ∉ {j_{1}, …, j_{m}}.

Using a composition of two shears as in the proof of corollary 2.2, we can find Ψ_{3} ∈ Aut_{1}C^{n} such that |Ψ_{3}(z) − z| + |Ψ_{3}^{−1}(z) − z| < δ on RB, Ψ_{3}((R + j_{k})e_{1}) = (R + j_{k})e_{1} for k = 1, …, m, and Ψ_{3}HΨ_{1}Ψ_{2}(a_{j}) = (R + j)e_{1} for each j ∉ {j_{1}, …, j_{m}}.

Let Ψ = Ψ_{3} ∘ H ∘ Ψ_{1} ∘ Ψ_{2}. Then Ψ(a_{j}) = (R + j)e_{1} for all j, and for δ sufficiently small, we have |Ψ(z)−H(z)| < ϵ on K and |Ψ^{−1}(z)−H^{−1}(z)| < ϵ on H(K), so the lemma follows. Since each Ψ_{l} ∈ Aut_{1}C^{n}, we see that Ψ ∈ Aut_{1}C^{n} if H ∈ Aut_{1}C^{n}. ♠

Proof of theorem 0.2. Since the sequences {a_{j}} and {b_{j}} are tame, there exist automorphisms H_{1}, H_{2} of C^{n} such that H_{1}(a_{j}) = je_{1} and H_{2}(b_{j}) = je_{1} for each j. Replacing Φ by H_{2} ∘ Φ ∘ H_{1}^{−1}, K by H_{1}(K), and adjusting the jets P_{j} as in the proof of theorem 0.1, we reduce the problem to the case when a_{j} = b_{j} = je_{1} for all j.

Choose a larger polynomially convex set L ⊂ C^{n}\{je_{1}: j ∈ N} such that K ⊂ Int L. Fix η > 0, and choose an integer r > 0 such that L ∪ Φ(L) ⊂ rB. By lemma 3.1 there exist automorphisms Ψ, Θ ∈ AutC^{n} such that

Ψ(je_{1}) = Θ(je_{1}) = (r + j)e_{1}, j = 1, 2, 3, …,
|Ψ(z) − Φ(z)| < η, z ∈ L,
|Θ(z) − z| + |Θ^{−1}(z) − z| < η, z ∈ Φ(L).

By corollary 2.2, there exists G ∈ AutC^{n} such that G((r + j)e_{1}) = (r + j)e_{1} for j ∈ N, |G(z) − z| < η for |z| ≤ r, and such that for each j ∈ N the jet Q_{j} = G_{m_{j},(r+j)e_{1}} satisfies

P_{j}(z) = (Θ^{−1})_{m_{j},(r+j)e_{1}} ∘ Q_{j} ∘ Ψ_{m_{j},je_{1}}(z) + O(|z|^{m_{j}+1}), z → 0.

Let F = Θ^{−1} ∘ G ∘ Ψ. Then F (je_{1}) = je_{1} for each j ≥ 1, F satisfies (0.3) provided that η > 0 is chosen sufficiently small (depending on ϵ and dist(K, C^{n}\L)), and lemma 1.3 shows that F satisfies (0.1), with a_{j} = b_{j} = je_{1}.

Finally, if the sequences a_{j} and b_{j} are very tame, the P_{j}’s satisfy (0.2), and Φ ∈ Aut_{1}C^{n}, then the automorphisms Ψ, Θ, and G can be chosen with Jacobian one. ♠

References.

1. Chirka, E.: Complex Analytic Sets. Kluwer, Dordrecht 1989 2. Forstneric, F.: Holomorphic automorphism groups of C^{n}: A survey. (The Proceedings

‘Complex Analysis and Geometry’, Ed. V. Ancona, E. Ballico, A. Silva; pp. 173–200) Lecture Notes in Pure and Applied Mathematics 173, Marcel Dekker, New York 1996

3. Forstneric, F.: Interpolation by holomorphic automorphisms and embeddings in C^{n}. Preprint, 1996

4. Hörmander, L.: An Introduction to Complex Analysis in Several Variables, 3rd ed. Amsterdam: North Holland 1990

5. Rosay, J.-P., Rudin, W.: Holomorphic maps from C^{n} to C^{n}. Trans. Amer. Math. Soc. 310, 47–86 (1988)