%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Document in latex using package 'amstex' %% Run latex 3 times to get all references \documentclass[12pt]{amsart} \usepackage{amsmath} \def\CC{{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em C}} \def\PP{{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em P}} \def\ZZ{{\rm\kern.26em\vrule width.02em height0.5ex depth0ex\kern.04em\vrule width.02em height1.47ex depth-1ex\kern-.34em Z}} \def\BB{{\rm\kern.24em\vrule width.02em height1.4ex depth-.05ex\kern-.26em B}} \def\noi{\noindent} \newcommand {\BOX} {\rule{2mm}{2mm} \bigskip} \newcommand{\ra}{\rightarrow} \newcommand {\ov} {\overline} \newcommand {\eps} {\epsilon} \newcommand {\ol} {\overline} \newtheorem{example}{Example}[section] \newtheorem{lem}{Lemma}[section] \newtheorem{theo}[lem]{Theorem} \newtheorem{claim}[lem]{Claim} \newtheorem{coro}[lem]{Corollary} \newtheorem{prop}[lem]{Proposition} \newtheorem{definition}[lem]{Definition} \newtheorem{remark}[lem]{Remark} \newtheorem{question}[lem]{Question} \newtheorem{questions}[lem]{Questions} \newtheorem{remarks}[lem]{Remarks} \numberwithin{equation}{section} \begin{document} \title{Compositional roots of H{\'e}non maps} \author{Gregery T. Buzzard and John Erik Forn\ae ss} \date{} %\tableofcontents \begin{abstract} Let $H$ denote a composition of complex H\'enon maps in $\CC^2$. In this paper we show that the only possible compositional roots of $H$ are also compositions of H\'enon maps, and that $H$ can have compositional roots of only finitely many distinct orders. \end{abstract} \renewcommand{\thefootnote}{} \footnote{Research at MSRI is supported in part by NSF grant DMS-9022140} \maketitle \let\cal\mathcal %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} Following \cite{bs1}, we say that a generalized H{\'e}non map is a map of the form % $$ H(z,w) = (w, p(w) - az), $$ % where $p$ is a monic polynomial of degree $d \geq 2$ and $a \in \CC-\{0\}$, and we let ${\cal G}$ denote the space of finite compositions of such maps. From \cite{fm}, we know that any polynomial diffeomorphism of $\CC^2$ is conjugate either to one of the maps in ${\cal G}$ or to an elementary map which preserves each line of the form $w = \text{const}$. In \cite{bf}, we classified, up to conjugacy, all polynomial diffeomorphisms which arise as the time-1 map of a holomorphic vector field. In particular, each of these maps is an elementary map and has compositional roots of all orders. Moreover, in some cases, these roots can be nonpolynomial. See \cite{forst} for information about such cases. \bigskip In this paper we treat the question of the existence of compositional roots for the remaining cases. In particular, we show that any root of a map in ${\cal G}$ must be a polynomial map and that any map in ${\cal G}$ can have roots of only a finite number of distinct orders. For the remaining elementary maps which are not the time-1 map of a flow, we show that such maps have roots of arbitrarily high order and nonpolynomial roots, but that any root of such a map is conjugate to a polynomial elementary map. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Dynamical behavior and Green's functions} Fix $H \in {\cal G}$. Let $K^+ (= K^+(H))$ and $K^- (= K^-(H))$ denote the set of points $p$ in $\CC^2$ such that the orbit of $p$ under $H$ is bounded with respect to forward or backward iteration, respectively. Also, for $R>0$, define sets % \begin{align*} V^- & := \{(z,w): |w|>R \text{ and } |w|>|z|\}, \\ V^+ & := \{(z,w): |z|>R \text{ and } |w|<|z|\},\\ V & := \{(z,w): |z| \leq R \text{ and } |w| \leq R\}. \end{align*} % A simple argument shows that $K^\pm \subseteq V^\pm \cup V$ for $R$ sufficiently large. Finally, we let $d$ denote the degree of $H$ as a polynomial map, and define functions % $$ G^\pm(z,w) := \lim_{n \ra \infty} \frac{1}{d^n} \log^+\|H^{\pm n}(z,w)\|. $$ \bigskip It is immediate that $G^+$ is continuous and plurisubharmonic (psh) on $\CC^2$, identically $0$ on $K^+$, and strictly positive and pluriharmonic on $\CC^2 - K^+$. Analogous statements are true with $G^-$ and $K^-$ in place of $G^+$ and $K^+$. See, for example, \cite{bs1} for a more systematic discussion. Note also that $G^\pm \circ H = d^{\pm 1} G^\pm$. Moreover, corollary~2.6 in \cite{bs1} implies the following. \begin{lem} \label{lemma:bounds} There exist $R>0$, $C>0$ such that if $(z,w) \in \ol{V^- \cup V}$, then % $$ \log^+|w| - C \leq G^+(z,w) \leq \log^+|w| + C,$$ % and if $(z,w) \in \ol{V^+ \cup V}$, then % $$ \log^+|z| - C \leq G^-(z,w) \leq \log^+|z| + C. $$ % \end{lem} Since $G^+(z_0,w)$ is subharmonic in $w$ for each fixed $z_0$, we see that it is bounded from above in $\{|w|<|z_0|\}$ by its maximum on the boundary of this disk, which is contained in $\ol{V^- \cup V}$. Applying a similar argument to $G^-$ gives the following. \begin{lem} \label{lemma:max} There exists $C>0$ such that for $(z,w) \in \CC^2$, % $$ G^\pm (z,w) \leq \max\{\log^+|z|, \log^+|w|\} + C. $$ % \end{lem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Compositional roots and Green's functions} Fix $H \in {\cal G}$ and let $d$ be the degree of $H$. In this section we show that if $F^n = H$ in the sense of composition, then $G^\pm \circ F = d^{\pm 1/n} G^\pm$. First a simple lemma. \begin{lem} Suppose $F$ is an automorphism of $\CC^2$ and $F^n = H$. Then $K^+$ and $K^-$ are the same for $F$ as for $H$, and $F$ is a diffeomorphism of $K^+$ and of $K^-$. \end{lem} {\bf Proof:} Take $p \in K^+(H)$ and let ${\cal O}^+_H(p)$ denote the forward orbit of $p$ under $H$. Then $\cup_{j=0}^{n-1} F^j(\ol{{\cal O}^+_H(p)})$ is compact, and ${\cal O}^+_F(p)$ is contained in this set, hence is bounded. Thus $K^+(H) \subseteq K^+(F)$. If $p \not \in K^+(H)$, then the forward orbit is not bounded for $H = F^n$, hence is not bounded for $F$. Thus $K^+(H) = K^+(F)$, and a similar argument applies to $K^-$. The fact that $F$ is a diffeomorphism of $K^\pm = K^\pm(F)$ is clear from the definition of these sets. $\BOX$ \bigskip \begin{lem} \label{lemma:mult} Let $F$ be as in the previous lemma. Then $G^\pm \circ F = d^{\pm 1/n} G^\pm$. \end{lem} {\bf Proof:} Since $F$ is holomorphic on $\CC^2$ and preserves $K^+$, we see that $G^+ \circ F$ is $0$ on $K^+$, plurisubharmonic and continuous on $\CC^2$, and strictly positive and pluriharmonic on $\CC^2 - K^+$. Fix $z_0$ and define $g_{z_0}(w) := G^+ \circ F(z_0,w)$. Note that $K^+ \cap (\{z_0\} \times \CC)$ is a compact set and that $g$ is harmonic on the complement of this set. Hence, outside a large disk, $g_{z_0}$ has a harmonic conjugate in a neighborhood of any point. Using analytic continuation in the exterior of this disk, we obtain a harmonic conjugate with periods. Hence for some $r>0$, some constant $c_{z_0}$, and a real harmonic function $h_{z_0}$, we get a function % $$ \phi_{z_0}(w) = g_{z_0}(w) - c_{z_0} \log|w| + i h_{z_0}(w)$$ % which is holomorphic for $|w| > r$. Since $g_{z_0} \geq 0$, we have $|\exp(-\phi_{z_0}(w))| \leq |w|^{c_{z_0}}$. Hence $\exp(-\phi_{z_0}(w))$ has at most a pole at infinity, so we can write % $$ \exp(-\phi_{z_0}(w)) = w^N \exp(f(w))$$ % for some integer $N$ and some $f$ holomorphic in $|w|>r$ with a removable singularity at infinity. Taking absolute value and $\log$, we get $g_{z_0}(w) - c_{z_0} \log|w| = -N \log|w| - \text{Re}(f(w))$. Hence $g_{z_0}(w) = b_{z_0} \log|w| +O(1)$ in $\{|w|>2r\}$, for some $b_{z_0}$. Since $g_{z_0} \geq 0$, we have $g_{z_0}(w) = b_{z_0} \log^+|w| + O(1)$ in $\CC$. \bigskip We claim that $b_z$ is independent of $z$. Note that $2 \pi b_{z_0}$ is the period for the harmonic conjugate of $g_{z_0}$ in $|w|>r$, and that $g_z(w)$ is pluriharmonic in $(z,w)$ near $(z_0,w_0)$ for any $|w_0|>r$. Fix $|w_0|>r$. We can construct the harmonic conjugate for $g$ in the bidisk $\Delta(z_0;r_0) \times \Delta(w_0;r_0)$ for some $r_0$ small. For $w \in \Delta(w_0;r_0)$, we can use analytic continuation as above to extend $g_z$ around a circle in $\{z\} \times \{|w|>r\}$. Doing this for each such $w$ gives a new harmonic conjugate for $g$ in the bidisk, which must differ from the original by a constant. Thus $b_z = b_{z_0}$ for $z$ near $z_0$. \bigskip Hence $g_{z_0}(w) = b \log^+|w| + O(1)$, and from lemma~\ref{lemma:bounds}, we see $G^+(z_0,w) = \log^+|w| + O(1)$. Thus $g_{z_0}(w) - b G^+(z_0,w)$ is continuous for $w \in \CC$ and harmonic for $w$ such that $(z_0, w) \notin K^+$, has a removable singularity at $\infty$, and is $0$ for $w$ such that $(z_0, w) \in K^+$, which is a nonempty set. Hence $g_{z_0}(w) \equiv b G^+(z_0,w)$ for all $w \in \CC$. Similarly, $G^+ \circ F(z,w) - b G^+(z,w) \equiv 0$ for all $(z,w) \in \Delta(z_0;r_0) \times \CC$. Since this difference is pluriharmonic in $\CC^2-K^+$, which is connected, it must be $0$ throughout $\CC^2 - K^+$, hence throughout $\CC^2$ since both terms are $0$ on $K^+$. Finally, $G^+ \circ F^n(z,w) = G^+ \circ H(z,w) = d G^+(z,w)$, while induction with the above result shows that $G^+ \circ F^n(z,w) = b^n G^+(z,w)$ Hence $b^n = d$, and $b>0$ since $G^+\geq 0$. This gives the lemma for $G^+$, and the same proof applies to $G^-$. $\BOX$ \bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Polynomial roots} In the proof of the following theorem, we use the terminology and results of \cite{fm}. In particular, we use the fact that the group of polynomial automorphisms of $\CC^2$ is the amalgamated product of the group ${\cal A}$ of affine linear automorphisms and the group ${\cal E}$ of elementary automorphisms which preserve the set of lines of the form $w = const$. A {\em reduced word} is an automorphism of the form $g_1 \cdots g_k$, $k \geq 1$, where each $g_k$ is in ${\cal A}$ or ${\cal E}$ but not in the intersection of these two groups and no two adjacent $g_j$'s are in the same group. We say that $k$ is the {\em length} of this reduced word. Also, we need to know that the identity cannot be written as a reduced word. \begin{theo} Suppose $H \in {\cal G}$ is a composition of generalized H{\'e}non maps and $F$ is an automorphism of $\CC^2$ with $F^n = H$. Then $F \in {\cal G}$. \end{theo} {\bf Proof:} Let $(z,w) \in \CC^2$, and let $F=(F_1,F_2)$. If $F(z,w) \in \ol{V^- \cup V}$, then from lemmas~\ref{lemma:bounds}, \ref{lemma:mult}, and \ref{lemma:max}, we see % \begin{align*} \log^+|F_2| - C_1 & \leq G^+(F(z,w)) \\ & = d^{1/n} G^+(z,w) \\ & \leq d^{1/n}( \max \{\log^+|z|, \log^+|w|\} + C_2). \end{align*} % Exponentiating and using $|F_1| \leq |F_2|+R$, we obtain % $$ |F(z,w)| \leq C \max\{ (|z|+1)^{d^{(1/n)}}, (|w|+1)^{d^{(1/n)}} \}. $$ % Similarly, if $F(z,w) \in V^+$, then % \begin{align*} \log^+|F_1| - C_1 & \leq G^-(F(z,w)) \\ & = 1/d^{(1/n)} G^-(z,w) \\ & \leq 1/d^{(1/n)} (\max \{\log^+|z|, \log^+|w|\} + C_2). \end{align*} % Exponentiating and using $|F_2| \leq |F_1|$, we obtain % $$ |F(z,w)| \leq C \max\{ (|z|+1)^{1/d^{(1/n)}}, (|w|+1)^{1/d^{(1/n)}} \}. $$ % Hence $F$ has polynomial growth throughout $\CC^2$, hence must be a polynomial. \bigskip We show next that $F \in {\cal G}$. Let $\tau(z,w) := (w,z)$. By \cite{fm}, we can write $H = \tau e_1 \cdots \tau e_m$, for some elementary maps $e_j$. Since $F^n=H$, $F$ must be a reduced word with length at least 2. There are four possibilities for the form of $F$. The first is % $$ F = a_1 e_1' \cdots a_l e_l' $$ % for some affine, non-elementary maps $a_j$ and some elementary, non-affine maps $e_j'$. By \cite{fm} or \cite{ar}, each $a_j$ can be written $a_j = a_j^1 \tau a_j^2$, where $a_j^1$ and $a_j^2$ are affine and elementary, and $a_1^1$ has the form % $$ a_1^1(z,w) = (bz +c w, w). $$ % Now, since $a_j^k$ is elementary, $F^n$ has the form $a_1^1 \circ (p,q)$, where $p$ and $q$ are polynomials and $(p,q) = \tau e_1'' \cdots \tau e_{nl}''$. By \cite{fm}, we have $\deg(q) > \deg(p)$, and likewise the degree of the second coordinate function of $H=F^n$ is larger than the degree of the first coodinate function. This implies that $c=0$, so $a_1^1(z,w) = (bz, w)$. Replacing $a_2^1$ by $\sigma a_2^1$, where $\sigma(z,w) = (z,bw)$, we obtain % $$ F=\tau E_1 \cdots \tau E_l.$$ % Hence $F \in {\cal G}$. The second case is % $$ F= e_1' a_2 \cdots a_l e_l'.$$ % In this case, we can replace each $a_j$ by $a_j^1 \tau a_j^2$ as before, and hence relabeling, we may assume $F = e_1' \tau \cdots \tau e_l'$. But then $H= F^n = e_1' \tau \cdots \tau e_{nl}'$, which implies that the degree of the first coordinate of $H$ is larger than the degree of the second coordinate, which is impossible. Hence $F$ cannot have this form. The third case is % $$ F = e_1' a_2 \cdots e_{l-1}' a_l. $$ % As before, we may relabel to assume that $F = e_1' \tau \cdots e_{l-1}' \tau a_l^2$. But then $H = F^n = e_1' \tau \cdots e_k' \tau a_l^2$, but also $H = \tau e_1 \cdots \tau e_m$. Hence % $$ I = (F^n)^{-1} H = (a_l^2)^{-1} (\tau (e_k')^{-1} \cdots (e_1')^{-1}) (\tau e_1 \cdots \tau e_m).$$ % But then $I$ has been written as a reduced word, which is impossible from \cite{fm}. Thus $F$ cannot have this form. In the final case, we have % $$ F = a_1 e_1' \cdots e_{l-1}' a_l. $$ % Again we may relabel and collect terms and assume $H=F^n = a_1^1 \tau e_1' \cdots e_{k-1}' \tau a_l^2. $ Since $a_l^2$ is linear, we can use an argument like that in the first case to relabel and replace $a_1^1$ by the identity. Since $a_l^2$ is elementary, we can write $a_l^2(z,w) = (az +bw+c, dw +e)$ with $a \neq 0$. Applying $\tau e_1' \cdots e_{k-1}' \tau$ to this, we see that the homogeneous polynomial of highest degree in $F^n$ depends on $z$. But a simple inductive argument shows that the corresponding polynomial for $H$ is independent of $z$. Hence $F$ cannot have this form. Thus $F \in {\cal G}$. $\BOX$ \bigskip \begin{remark} In general, a map $H$ can have distinct roots of a given order. For example, the map $H$ given by squaring $F(z,w)=(w, z+w^2)$ has three square roots. This is true because $(F \circ s)^2 =H$ for $s(z,w) = (\omega^2 z, \omega w)$, where $\omega^3=1$. In fact, one can check that these are the only possible square roots of $H$. \end{remark} \section{Roots of elementary maps} In \cite{bf}, we showed that no H{\'e}non map can be the time-1 map of the flow of a holomorphic vector field and gave a precise classification of those maps which can be the time-1 map of such a flow. In \cite{forst} and \cite{afv}, it was shown that any flow of a holomorphic vector field whose time-1 map is an elementary map is in fact conjugate to a flow which is polynomial for all time. In this section, we consider the set of elementary maps which are not the time-1 map of any holomorphic flow and show that such maps have roots of arbitrarily high order but that any root is conjugate to a polynomial map. The elementary maps which cannot be the time-1 map of a flow have the form % $$F(z,w) = (\beta^\mu(z+w^\mu q(w^r)), \beta w), $$ % where $\beta$ is a primitive $r$th root of unity, $q(w) = w^k + q_{k-1} w^k + \cdots q_1 w + q_0$, and $k \geq 1$. A simple check shows that if we replace $w^\mu q(w^r)$ by $(w^\mu q(w^r))/(lr+1)$ for any $l \in \ZZ^+$, then the resulting map is an $(lr+1)$st root of $F$. In general, maps of this form can have nonpolynomial roots. For instance, let $F(z,w) = (-(z+w(w^4+1)), -w)$ and let $k$ be any entire function of one variable. Define $\phi(z,w) = (i (z+w(w^4+1)/2 +w^3 k(w^4)), iw)$. A simple check shows that $\phi^2 = F$, and $\phi$ is nonpolynomial whenever $k$ is transcendental. \bigskip We claim that any root of $F$ is conjugate to a polynomial automorphism. Suppose that $\phi$ is an automorphism of $\CC^2$ with $\phi^n = F$. Then $\phi F^r \phi^{-1} = F^r$, so an argument like that in \cite[theorem 6.10]{fm} shows that $\phi(z,w) = (e^{g(w)} z +h(w), aw + b)$ for some entire $g$, $h$, and some $a, b \in \CC$, $a \neq 0$. Since $\phi^n = F$, we see that $a^n = \beta$ and $b(a^n-1)/(a-1) = 0$, so that $b=0$. Using this form for $\phi$ and the fact that $\phi F^r = F^r \phi$, it follows that $e^{g(w)}$ is a nonzero rational function, hence is a constant, $c \neq 0$. Moreover, since $\phi F = F \phi$, we see that $c = a^\mu$. Thus $\phi(z,w) = (a^\mu z + h(w), a w)$. Now, since $\phi^{rn} = F^r$, it follows that $\sum_{j=0}^{rn-1} (a^\mu)^{-j} h(a^j w) = r w^\mu q(w^r)$. Write $h = h_1 + h_2$, where $h_2 = w^{\mu + kr} \tilde{h}_2$ for some entire $\tilde{h}_2$. Then the sum just given is valid with $h_2$ in place of $h$ and $0$ in place of $r w^\mu q(w^r)$. Hence by \cite{forst}, there exists $f$ entire such that $f(a w) - a^\mu f(w) = h_2(w)$. A simple check shows that if $\psi(z,w) = (z + f(w), w)$, then $\psi^{-1} \phi \psi$ is a polynomial, and in fact, $\psi^{-1} \phi^n \psi = F$. Thus any root of $F$ is conjugate to a polynomial automorphism. \bigskip Given any elementary map $F$ and a root $\phi^n=F$, one can ask if $\phi$ is conjugate to a polynomial automorphism. There are a few cases such as the above where this result is relatively straightforward, but in general, this seems to be a hard question. For some results along these lines in the case $F = I$, see \cite{ar}. \begin{thebibliography}{9999} \bibitem[AF]{forst} Ahern, P., and Forstneric, F., One parameter automorphism groups on $\CC^2$, {\em Complex Variables}, to appear. \bibitem[AFV]{afv} Ahern, P, Forstneric, F., and Varolin, D., Flows on $\CC^2$ with polynomial time one map, preprint, 1995. \bibitem[AR]{ar} Ahern, P., and Rudin, W., Periodic automorphisms of $\CC^2$, preprint, 1994. \bibitem[BS]{bs1} Bedford, E., and Smillie, J., Polynomial diffeomorphisms of $\CC^2$: currents, equilibrium measure and hyperbolicity, {\em Inv. Math.}, 103 (1991), 69-99. \bibitem[BF]{bf} Buzzard, G., and Forn\ae ss, J.E., Complete holomorphic vector fields and time-1 maps, preprint. \bibitem[FM]{fm} Friedland, S., and Milnor, J., Dynamical properties of plane polynomial automorphisms, {\em Ergod. Th. and Dynam. Sys.}, 9 (1989), 67-99. \end{thebibliography} \bigskip \small \noindent Gregery T. Buzzard \\ Department of Mathematics\\ The University of Michigan\\ Ann Arbor, MI 48109, USA\\ and\\ MSRI \\ 1000 Centennial Drive\\ Berkeley, CA 94720, USA\\ \noindent John Erik Forn\ae ss\\ Department of Mathematics\\ The University of Michigan\\ Ann Arbor, MI 48109, USA\\ \end{document}