%--------------------- CUT HERE ------------------------------ % % AUTHORS: GREGERY T. BUZZARD AND FRANC FORSTNERIC % % TITLE: AN INTERPOLATION THEOREM FOR HOLOMORPHIC AUTOMORPHISMS OF C^n % % Last edited: March 1, 1996 % % Plain TeX % % Submitted to: % % Date: % % % This is the Macro file. % % \scrollmode \magnification=\magstep1 \parskip=\smallskipamount \def\demo#1:{\par\medskip\noindent\it{#1}. \rm} \def\ni{\noindent} % noindent \def\np{\hfill\break\indent} % new indented par in horizontal mode \def\npni{\hfill\break\noindent} % new unindented par in horizontal mode \def\ll{\leftline} \def\cl{\centerline} \def\rl{\rightline} % % \beginsection % \outer\def\beginsection#1\par{\bigskip \message{#1}\leftline{\bf\} \nobreak\smallskip\vskip-\parskip\noindent} % % \proclaim % \outer\def\proclaim#1:#2\par{\medbreak\vskip-\parskip \noindent{\bf#1.\enspace}{\sl#2} \ifdim\lastskip<\medskipamount \removelastskip\penalty55\medskip\fi} \def\ba{\medskip\rm} \def\endpr{\hfill $\spadesuit$ \medskip} \def\inner#1#2{\langle{#1},{#2}\rangle} % % % Roman capital characters % % \def\rD{{\rm D}} % roman D in math mode \def\rH{{\rm H}} \def\rT{{\rm T}} % roman T \def\rN{{\rm N}} % roman N in math \def\rH{{\rm H}} % % % Boldface capital characters % % \def\B{{\bf B}} \def\C{{\bf C}} \def\D{{\bf D}} \def\H{{\bf H}} \def\N{{\bf N}} \def\P{{\bf P}} \def\Q{{\bf Q}} \def\R{{\bf R}} \def\T{{\bf T}} \def\Z{{\bf Z}} \def\CP{{\bf CP}} \def\GL{{\bf GL}} \def\U{{\bf U}} \def\SU{{\bf SU}} % \def\O{{\bf O}} \def\SO{{\bf SO}} \def\SL{{\bf SL}} \def\CP{{\bf CP}} \def\Sp{{\bf Sp}} % % % Caligraphic capital characters % % \def\cA{{\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}} \def\cD{{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}} \def\cG{{\cal G}} \def\cH{{\cal H}} \def\cI{{\cal I}} \def\cJ{{\cal J}} \def\cK{{\cal K}} \def\cL{{\cal L}} \def\cM{{\cal M}} \def\cN{{\cal N}} \def\cO{{\cal O}} \def\cP{{\cal P}} \def\cQ{{\cal Q}} \def\cR{{\cal R}} \def\cS{{\cal S}} \def\cT{{\cal T}} \def\cU{{\cal U}} \def\cV{{\cal V}} \def\cW{{\cal W}} \def\cX{{\cal X}} \def\cY{{\cal Y}} \def\cZ{{\cal Z}} % % % Small Greek letters in Math mode % \def\a{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\epsilon} \def\z{\zeta} \def\i{\iota} \def\k{\kappa} \def\l{\lambda} \def\r{\rho} \def\s{\sigma} \def\u{\upsilon} \def\c{\chi} %\def\o{\omega} \def\t{\theta} \def\x{\xi} \def\S{{\Sigma}} \def\L{\Lambda} %\def\O{\Omega} % % % Miscellaneous % % \def\bar{\overline} % conjugate \def\bs{\backslash} % backslash \def\disc{\triangle} % disc \def\cdisc{\bar\triangle} % closed disc \def\di{\partial} % partial derivative \def\dibar{\bar\partial} % di-bar derivative % % Abbreviations % \def\dim{{\rm dim}\,} % dimension \def\holo{holomorphic} % holomorphic \def\auto{automorphism} % automorphism \def\homo{homomorphism} % homomorphism \def\as{analytic subset\ } % analytic subset \def\homeo{homeomorphism} % homeomorphism \def\nbd{neighborhood} % neighborhood \def\psc{pseudoconvex} % pseudoconvex \def\spsc{strongly\ pseudoconvex} % strongly psc \def\ra{real-analytic} % real-analytic \def\psh{plurisubharmonic} % plurisubharmonic \def\spsh{strongly\ plurisubharmonic} \def\tr{totally real} % totally real \def\pc{polynomially convex} % polynomially convex \def\hc{holomorphically convex} % holo convex \def\hf{holomorphic\ function} % holomorphic function \def\ss{\subset\!\subset} % relatively compact subset \def\supp{{\rm supp}\,} % support \def\cne{$\C^n$-equivalent} % C^n equivalent \def\wlog{without\ loss\ of\ generality\ } \def\tcp{This\ concludes\ the\ proof\ of\ } \def\iff{if and only if} \def\hvf{holomorphic vector field} \def\vf{vector field} \def\div{{\rm div}} \def\phe{proper holomorphic embedding} \def\ha{holomorphic automorphism} \def\hra{\hookrightarrow} \def\dist{{\rm dist}} \def\Aut{{\rm Aut}} % roman Aut in math mode \def\Autsp{{\rm Aut}_{\rm sp}} % sypmlectic automorphisms \def\begin{\ll{} \vskip 10mm \nopagenumbers} % beginning of the paper \def\pn{\footline={\hss\tenrm\folio\hss}} % pagenumbers at bottom \def\ii#1{\itemitem{#1}} \begin \cl{\bf AN INTERPOLATION THEOREM FOR} \medskip \cl{\bf HOLOMORPHIC AUTOMORPHISMS OF $\C^{\bf n}$} \bigskip\medskip \cl{\bf Gregery T. Buzzard\footnote{*} {\rm Research at MSRI is supported in part by NSF grant DMS-9022140} and Franc Forstneric\footnote{**} {\rm Supported in part by an NSF grant and by a grant from the Ministry of Science of the Republic of Slovenia}} \bigskip\rm \ll{Department of Mathematics, Indiana University, Bloomington, IN 47405, USA} \ll{Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA} \medskip\bigskip \font\small=cmr9 \midinsert \narrower\narrower \cl{\bf Abstract} \baselineskip=10pt {\small 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.} \endinsert \baselineskip=\normalbaselineskip % % % \beginsection 0. Introduction Let $\C^n$ be complex Euclidean space of dimension $n$ with coordinates $z=(z_1, \ldots, z_n)$. We shall always assume $n>1$. We denote by $\Aut\C^n$ the group of all holomorphic automorphisms of $\C^n$, and by $\Aut_1\C^n$ the group of automorphisms $F\in\Aut\C^n$ with complex Jacobian one: $JF\equiv 1$. 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: \proclaim Definition: A discrete infinite sequence $\{a_j\} \subset\C^n$ (without repetition) is {\it tame} if there exists a \holo\ automorphism $F\in\Aut\C^n$ such that $$ F(a_j)=je_1=(j,0,\ldots,0),\qquad j=1,2,3,\ldots $$ The sequence $\{a_j\}$ is {\it very tame\/} if the above holds for some $F\in\Aut_1\C^n$. 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 $E\subset \C^n$ such that the only nondegenerate \holo\ map $F\colon \C^n\to\C^n$ satisfying $F(\C^n\bs E)\subset \C^n\bs E$ 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. % % % MAIN RESULT % \proclaim 0.1 Theorem: Assume that $n>1$, that $\{a_j\}$ and $\{b_j\}$ are tame sequences in $\C^n$ (without repetitions), and that $P_j\colon\C^n \to \C^n$ is a \holo\ polynomial map of degree at most $m_j \ge 1$ such that $P_j(0) = 0$ and $JP_j(0) \ne 0$. Then there exists $F \in \Aut\C^n$ such that for every $j = 1,2,3,\ldots$ we have $F(a_j)=b_j$ and $$F(z)=b_j+P_j(z-a_j)+O(|z-a_j|^{m_j+1}), \quad z\to a_j. \eqno(0.1)$$ If in addition the polynomials $P_j$ satisfy $$ JP_j(z)=1+O(|z|^{m_j}),\ \ z\to 0,\quad j=1,2,3,\ldots, \eqno(0.2)$$ and the sequences $\{a_j\}$ and $\{b_j\}$ are very tame, we may choose $F \in \Aut_1\C^n$. 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 {\it 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 \pc\ set. \footline={\hss\tenrm\folio\hss} \proclaim 0.2 Theorem: In addition to the hypotheses of theorem 0.1, assume that $\Phi\in\Aut\C^n$ is an automorphism and $K\subset \C^n\bs \{a_j\}_{j=1}^\infty$ is a compact, \pc\ set such that $\Phi(K) \subset \C^n\bs \{b_j\}_{j=1}^\infty$. Then for each $\e>0$ there exists an $F\in\Aut\C^n$ satisfying theorem 0.1 and $$ |F(z)-\Phi(z)|<\e,\qquad z\in K. \eqno(0.3) $$ If the volume preserving assumptions in theorem 0.1 hold and if $\Phi\in\Aut_1\C^n$, we may choose $F\in\Aut_1\C^n$. \medskip 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 \pc\ 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 $\Psi \in \Aut_1 \C^n$ such that at each point $a_j$, $\Psi$ agrees with the translation $z \mapsto c_j + (z-a_j)$ to order $m_j$. In fact, we can find such a $\Psi$ which is a composition of three shears of the form $z \mapsto z + f(\lambda(z)) v$, where $f$ is entire on $\C$, $v \in \C^n\bs\{0\}$, and $\lambda$ is a $\C$-linear form with $\lambda(v) = 0$. 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 $F \in \Aut\C^n$ 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 $2\overline{\B} \cup K_1 \cup F_1(2\overline{\B}) \cup \{c_1\}$. Again the results of section 1 allow us to find an automorphism $\Psi_2$ 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 $F_2 = \Psi_2 \circ F_1$. 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 $(k+1)\overline{\B} \cup K_k \cup F_k((k+1)\overline{\B}) \cup \{c_k\}$, then choosing $\Psi_k$ to be near the identity on $K_{k+1}$, equal to the identity to order $m_j$ at $c_j$ for $j \le k$, 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 $F_1, F_2, \ldots$ with $F_{k+1} = \Psi_{k+1} \circ F_k$. 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, \pc\ set $K\subset \C^n$, 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 $\Psi\in\Aut\C^n$ that approximates $H$ on $K$ and satisfies $\Psi(a_j)=(R+j)e_1$ for each $j$. %---------------------------------------------------------------- % % % \beginsection 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. % and says roughly that given a \pc\ set $K \subset \C^n$ and two % points $p$ and $q$ not in $K$, we can find $F \in \Aut\C^n$ which is near % the identity on $K$ and which moves $p$ to $q$ with prescribed jet at $p$. % This can also be done so that $F$ agrees with the identity to high % order at some given finite set in $K$. \medskip\ni\bf 1.1 Proposition. \sl Let $n>1$. Assume that \item{(a)} $K\subset\C^n$ is a compact, \pc\ set, \item{(b)} $\{a_j\}_{j=1}^k \subset K$ is a finite subset of $K$, \item{(c)} $p$ and $q$ are arbitrary points in $\C^n\bs K$ (not necessarily distinct), \item{(d)} $N$ is a nonnegative integer, and \item{(e)} $P\colon \C^n\to \C^n$ is a holomorphic polynomial map of degree at most $m\ge 1$, satisfying $P(0)=0$ and $JP(0)\ne 0$. \ni Then for each $\e>0$ there exists an automorphism $F\in \Aut\C^n$ satisfying \item{(i)} $F(z)=q+P(z-p)+O(|z-p|^{m+1})$ as $z\to p$, \item{(ii)} $F(z)=z+O(|z-a_j|^N)$ as $z\to a_j$ for each $j=1,2,\ldots,k$, and \item{(iii)} $|F(z)-z|+|F^{-1}(z)-z|<\e$ for each $z\in K$. \ni If in addition the polynomial map $P$ satisfies \item{(e')} $P(0)=0$ and $JP(z)=1+O(|z|^m)$ as $z\to 0$, \ni then there exists a polynomial automorphism $F$ with $JF=1$ satisfying (i)--(iii). \medskip\rm 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. \proclaim 1.2 Proposition: Let $D$ be a connected open set in $\C^n$ which is exhausted by compact sets $K_0\subset K_1\subset K_2\subset\cdots \subset \cup_{j=0}^\infty K_j =D$ such that $K_{j-1} \subset {\rm Int} K_j$ for each $j\in\N$. Choose numbers $\e_j$ ($j=1,2,3,\ldots$) such that $$ 0< \e_j < \dist(K_{j-1},\C^n\bs K_j)\ \ (j\in\N), \qquad \sum_{j=1}^\infty \e_j <\infty. \eqno(1.1) $$ Suppose that for each $j=1,2,3,\ldots$, $\Psi_j$ is a holomorphic automorphism of $\C^n$ satisfying $$ |\Psi_j(z)-z| < \e_j, \quad z\in K_j. \eqno(1.2) $$ Set $\Phi_m=\Psi_m\circ\Psi_{m-1}\circ\cdots \circ \Psi_1$. Then there is an open set $\Omega\subset\C^n$ such that $\lim_{m\to\infty} \Phi_m =\Phi$ exists on $\Omega$ (uniformly on compacts), and $\Phi$ is a biholomorphic map of $\Omega$ onto $D$. In fact, $\Omega = \cup_{m=1}^\infty \Phi_m^{-1}(K_m)$. {}Finally, we include some elementary observations about power series. If $F\colon U\subset\C^n \to\C^n$ is a \holo\ map, $a\in U$, and $m\ge 1$ an integer, we write $$ F(z)=F(a)+ F_{m,a}(z-a) + O(|z-a|^{m+1}),\quad z\to a. $$ 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. \proclaim 1.3 Lemma: If $F$ and $G$ are nondegenerate \holo\ maps on certain open subsets of $\C^n$, with values in $\C^n$, then for each integer $m\ge 1$ we have $$ (G\circ F)_{m,a}(z) = G_{m,F(a)} \circ F_{m,a}(z)+ O(|z|^{m+1}),\quad z\to 0. $$ at each point $a$ where the composition $G\circ F$ is defined. This generalizes to composition of several maps. We also have $$ (F^{-1})_{m,F(a)} \circ F_{m,a}(z)=z+O(|z|^{m+1}), \quad z\to 0. $$ %---------------------------------------------------------------- % % % \beginsection 2. Proof of the interpolation theorem In this section we prove theorem 0.1. The main construction is contained in the following lemma. \proclaim 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 $F \in \Aut\C^n$ such that for all $j$ we have $$ F(z) = c_j + P_j(z-c_j) + O(|z-c_j|^{m_j+1}),\ \ z \to c_j. \eqno(2.1)$$ If in addition each polynomial $P_j$ satisfies (0.2), we may choose $F \in \Aut_1\C^n$. \demo Proof: {}For $j \ge 1$, let $\e_j = 2^{-j}$. We will construct the sequence $\{c_j\}$ inductively and construct $F$ as the limit of a composition of automorphisms, each chosen inductively. Let $K_0 = \emptyset$ and $K_1 = 2\overline{\B}$. Let $c_1 = 3e_1$. By proposition 1.1 with $K = K_1$, there exists $\Psi_1 \in \Aut\C^n$ such that $\Psi_1(z) = c_1 + P_1(z-c_1) + O(|z-c_1|^{m_1+1})$, $z\to c_1$ and $|\Psi_1(z)-z| + |\Psi_1^{-1}(z) - z| < \e_1$ for $z \in K_1$. If also $P_1$ satisfies (0.2), then we may choose $\Psi_1 \in \Aut_1\C^n$. Let $F_0(z) = z$ and $F_1 = \Psi_1$. \medskip We will inductively choose automorphisms $\Psi_j$ and let $F_k = \Psi_k \circ \cdots \circ \Psi_1$. For the induction, suppose we have the following. \ni \item{(1)} Compact, convex sets $K_0 \subset K_1 \subset \cdots \subset K_k$ with $j\B \cup F_{j-1}(j\B) \subset K_j$ and $$\dist(K_{j-1} \cup F_{j-1}(j\B),\C^n\bs K_j) >\e_j$$ for each $1 \le j \le k$. \item{(2)} Points $c_j \in K_{j+1}\bs K_j$ for $1 \le j \le k-1$ and $c_k \in \C^n\bs K_k$ such that each $c_j$ is contained in the $z_1$-axis. \item{(3)} $\Psi_j \in \Aut\C^n$ for $1 \le j \le k$ with $$|\Psi_j(z)-z| + |\Psi_j^{-1}(z) - z| < \e_j$$ for $z \in K_j$ (and $\Psi_j \in \Aut_1\C^n$ if $P_j$ satisfies (0.2)). \item{(4)} $F_k = \Psi_k \circ \Psi_{k-1} \circ \cdots \circ \Psi_1$ satisfying (2.1) for $1 \le j \le k$. \medskip Given this, let $K_{k+1}$ be a compact, convex set in $\C^n$ with $$(k+1)\B \cup K_k \cup F_k((k+1)\B) \cup \{c_k\} \subset K_{k+1}$$ and $$\dist(K_k \cup F_k((k+1)\B),\C^n\bs K_{k+1}) >\e_{k+1}.$$ Since $K_{k+1}$ is compact, we can choose $c_{k+1}$ in the $z_1$-axis so that $c_{k+1}, F_k(c_{k+1}) \in \C^n \bs K_{k+1}$. Let $N = \max\{m_1, \ldots, m_k\}+1$. Let $d_{k+1} = F_k(c_{k+1})$, and let $Q_{k+1} = P_{k+1}\circ (F_k^{-1})_{m_{k+1},d_k}$. Using proposition 1.1, choose $\Psi_{k+1} \in \Aut\C^n$ (with $\Psi_{k+1} \in \Aut_1\C^n$ if $P_{k+1}$ satisfies (0.2) and $F_k \in \Aut_1\C^n$)) so that \ni \item{(i)} $\Psi_{k+1}(z) = c_{k+1} + Q_{k+1}(z-d_{k+1}) + O(|z-d_{k+1}|^{m_{k+1}+1})$ as $z \to d_{k+1}$, \item{(ii)} $\Psi_{k+1}(z) = z+O(|z-c_j|^N)$ as $z \to c_j$, $1\le j \le k$, \item{(iii)} $|\Psi_{k+1}(z) -z| + |\Psi_{k+1}^{-1}(z) - z| <\e_{k+1}$ for each $z \in K_{k+1}$. \medskip\ni Taking $F_{k+1} = \Psi_{k+1} \circ F_k$, we obtain the induction hypotheses at stage $k+1$. By proposition 1.2, the sequence $\{F_j\}$ converges uniformly on compact subsets of $\Omega = \cup_{j=1}^\infty F_j^{-1}(K_j)$ to a biholomorphic map $F$ from $\Omega$ onto $D = \cup_{j=1}^\infty K_j$. Since $j\B \subset K_j$, we see that $D = \C^n$. Moreover, $F_j^{-1}(K_j) = F_{j-1}^{-1} \Psi_j^{-1}(K_j)$, and by Rouch\'e's theorem [1, p.110] and induction hypotheses (1) and (3), we see that $F_{j-1}(j\B) \subset \Psi_j^{-1}(K_j)$. Hence $j\B \subset F_j^{-1}(K_j)$, so also $\Omega =\C^n$. Hence $F \in \Aut\C^n$ satisfies (2.1) for all $j$ since each $F_k$ satisfies (2.1) for $1 \le j \le k$. Finally, if each $P_j$ satisfies (0.2), then each $\Psi_j \in \Aut_1\C^n$, so $F \in \Aut_1\C^n$. \endpr \demo Remark: Given $R>0$, $1>\e>0$, we can replace $K_1$ by $(R+1)\overline{\B}$, $c_1$ by $(R+2)e_1$, and $\e_j$ by $\e (\e_j/2)$ to construct a sequence $\{c_j\}$ and $F$ satisfying the conclusions of the theorem and also so that $|F(z) - z| + |F^{-1}(z) - z| < \e$ on $R\B$. \medskip 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. % % Corollary 2.2 % \proclaim 2.2 Corollary: Let $P_j$ and $m_j$ be as in theorem 0.1. For each $R\ge 0$ and $\e>0$ there exists $F \in \Aut\C^n$ such that $$ \eqalign{ & F(z) = je_1 + P_j(z-je_1) + O(|z-je_1|^{m_j+1}), \quad z \to je_1,\ j>R, \cr & |F(z)-z|+|F^{-1}(z)-z|<\e, \quad |z|0$ with $H(K) \subset R\B$ and let $\e>0$. Then there exists $\Psi \in \Aut\C^n$ such that $|\Psi(z) - H(z)|<\e$ on $K$, $|\Psi^{-1}(z) - H^{-1}(z)| <\e$ on $H(K)$, and $\Psi(a_j) = (R+j)e_1$ for all $j$. If $H \in \Aut_1\C^n$, then we can choose $\Psi \in \Aut_1\C^n$. \demo Proof: Let $r>0$ with $K \subset r\B$. Let $K_0 \subset r\B$ be compact and \pc\ with $K \subset {\rm Int} K_0$, $K_0 \cap \{a_j\} = \emptyset$, and $H(K_0) \subset R\B$. Let $a_{j_1}, \ldots, a_{j_m}$ be the points in $\{a_j\} \cap r\overline{\B}$, and let $\d>0$. Using the fact that the union of $K_0$ with finitely many points is again \pc, we can apply proposition 1.1 $m$ times to find $\Psi_1 \in \Aut_1\C^n$ such that $|\Psi_1(z) - z| + |\Psi_1^{-1}(z) - z| < \d$ for $z \in K_0$ and $\Psi_1(a_{j_k}) = H^{-1}((R+j_k)e_1)$ for $k = 1, \ldots, m$. Let $\pi_2(z) = z_2$. For fixed $v \in \C^n\bs\{0\}$ and $j\in \N$ we consider the 1-variable function $g_j(\z) = \pi_2 H \Psi_1(a_j + \z v)$. Since the kernel of $D_{a_j} (\pi_2 \circ H \circ \Psi_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 $\lambda$ with $\lambda(v)=0$, $\lambda(e_1)=1$, and $\lambda(a_j) \notin \lambda(r\overline{\B})$ if $j \notin \{j_1, \ldots, j_m\}$. Choose $f$ entire on $\C$ such that $|f|<\d/2$ on $\lambda(r\B)$, $f(\lambda(a_{j_k})) = 0$ for each $k=1, \ldots, m$, and $|\pi_2 H \Psi_1(a_j + f(\lambda(a_j)) v)| = R+j$ for each $j \notin \{j_1, \ldots, j_m\}$. Let $\Psi_2(z) = z + f(\lambda(z)) v$. Then $\Psi_2 \in \Aut_1\C^n$, $|\Psi_2(z)-z| + |\Psi_2^{-1}(z)-z| < \d$ on $r\B$, $H \Psi_1 \Psi_2(a_{j_k}) = (R+j_k)e_1$ for $k=1, \ldots, m$, and $|\pi_2 H \Psi_1 \Psi_2(a_j)| = R+j$ for $j \notin \{j_1, \ldots, j_m\}$. Using a composition of two shears as in the proof of corollary 2.2, we can find $\Psi_3 \in \Aut_1\C^n$ such that $|\Psi_3(z)-z| + |\Psi_3^{-1}(z)-z| < \d$ on $R\B$, $\Psi_3((R+j_k)e_1) = (R+j_k)e_1$ for $k=1, \ldots, m$, and $\Psi_3 H \Psi_1 \Psi_2(a_j) = (R+j)e_1$ for each $j \notin \{j_1, \ldots, j_m\}$. Let $\Psi = \Psi_3 \circ H \circ \Psi_1 \circ \Psi_2$. Then $\Psi(a_j) = (R+j)e_1$ for all $j$, and for $\d$ sufficiently small, we have $|\Psi(z) - H(z)| < \e$ on $K$ and $|\Psi^{-1}(z) - H^{-1}(z)| <\e$ on $H(K)$, so the lemma follows. Since each $\Psi_l \in \Aut_1\C^n$, we see that $\Psi \in \Aut_1\C^n$ if $H \in \Aut_1\C^n$. \endpr \demo 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 $\Phi$ by $H_2\circ \Phi \circ 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 \pc\ set $L\subset \C^n\bs \{je_1 \colon j\in \N\}$ such that $K\subset {\rm Int}\, L$. Fix $\eta>0$, and choose an integer $r>0$ such that $L\cup \Phi(L) \subset r\B$. By lemma 3.1 there exist automorphisms $\Psi, \Theta\in\Aut\C^n$ such that $$ \eqalign{ & \Psi(je_1)=\Theta(je_1)= (r+j)e_1, \quad j=1,2,3,\ldots, \cr & |\Psi(z)-\Phi(z)|<\eta, \quad z\in L, \cr % & |\Psi^{-1}(z)-\Phi^{-1}(z)|< \eta,\quad z\in \Phi(L), \cr & |\Theta(z)-z| + |\Theta^{-1}(z)-z| <\eta, \quad z\in \Phi(L).\cr} $$ By corollary 2.2, there exists $G \in \Aut\C^n$ such that $G((r+j)e_1)=(r+j)e_1$ for $j \in \N$, % % $|G(z)-z|+|G^{-1}(z)-z| <\eta$ for % $|G(z)-z|<\eta$ for $|z|\le r$, and such that for each $j\in \N$ the jet $Q_j=G_{m_j,(r+j)e_1}$ satisfies $$ P_j(z)= (\Theta^{-1})_{m_j,(r+j)e_1} \circ Q_j \circ \Psi_{m_j,je_1}(z) + O(|z|^{m_j+1}),\quad z\to 0. $$ Let $F=\Theta^{-1}\circ G\circ \Psi$. Then $F(je_1)=je_1$ for each $j\ge 1$, $F$ satisfies (0.3) provided that $\eta>0$ is chosen sufficiently small (depending on $\e$ and $\dist(K,\C^n\bs L)$), and lemma 1.3 shows that $F$ satisfies (0.1), with $a_j=b_j=je_1$. 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