COMPOSITIONAL ROOTS OF HÉNON MAPS

GREGERY T. BUZZARD AND JOHN ERIK FORNÆSS

Abstract. Let H denote a composition of complex Hénon maps in C^{2}. In this paper we show that the only possible compositional roots of H are also compositions of Hénon maps, and that H can have compositional roots of only finitely many distinct orders.

1. Introduction

Following [BS], we say that a generalized Hénon map is a map of the form

H(z, w) = (w, p(w) − az),

where p is a monic polynomial of degree d ≥ 2 and a ∈ C − {0}, and we let G denote the space of finite compositions of such maps. From [FM], we know that any polynomial diffeomorphism of C^{2} is conjugate either to one of the maps in G or to an elementary map which preserves each line of the form w = const. In [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 [AF] for information about such cases.

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 G must be a polynomial map and that any map in 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.

2. Dynamical behavior and Green’s functions

Fix H ∈ G. Let K^{+}(= K^{+}(H)) and K^{−}(= K^{−}(H)) denote the set of points p in C^{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

Research at MSRI is supported in part by NSF grant DMS-9022140

V^{−} := {(z, w) : |w| > R and |w| > |z|},
V^{+} := {(z, w) : |z| > R and |w| < |z|},
V := {(z, w) : |z| ≤ R and |w| ≤ R}.

A simple argument shows that K^{±} ⊆ V^{±} ∪ V for R sufficiently large. Finally, we let d denote the degree of H as a polynomial map, and define functions

G^{±}(z, w) := lim_{n→∞} \frac{1}{d^{n}} log^{+} ∥H^{±n}(z, w)∥.

It is immediate that G^{+} is continuous and plurisubharmonic (psh) on C^{2}, identically 0 on K^{+}, and strictly positive and pluriharmonic on C^{2} − K^{+}. Analogous statements are true with G^{−} and K^{−} in place of G^{+} and K^{+}. See, for example, [BS] for a more systematic discussion.

Note also that G^{±} ∘ H = d^{±1}G^{±}. Moreover, corollary 2.6 in [BS] implies the following.

Lemma 2.1. There exist R > 0, C > 0 such that if (z, w) ∈ \overline{V^{−} ∪ V}, then

log^{+} |w| − C ≤ G^{+}(z, w) ≤ log^{+} |w| + C,

and if (z, w) ∈ \overline{V^{+} ∪ V}, then

log^{+} |z| − C ≤ G^{−}(z, w) ≤ log^{+} |z| + C.

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 \overline{V^{−} ∪ V}. Applying a similar argument to G^{−} gives the following.

Lemma 2.2. There exists C > 0 such that for (z, w) ∈ C^{2},

G^{±}(z, w) ≤ max{log^{+} |z|, log^{+} |w|} + C.

3. Compositional roots and Green’s functions

Fix H ∈ 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^{±} ∘ F = d^{±1/n}G^{±}. First a simple lemma.

Lemma 3.1. Suppose F is an automorphism of C^{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^{−}.

COMPOSITIONAL ROOTS OF HÉNON MAPS 3

Proof: Take p ∈ K^{+}(H) and let O_{H}^{+}(p) denote the forward orbit of p under H. Then ∪_{j=0}^{n−1}F^{j}(\overline{O_{H}^{+}(p)}) is compact, and O_{F}^{+}(p) is contained in this set, hence is bounded. Thus K^{+}(H) ⊆ K^{+}(F).

If p ∉ 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^{±} = K^{±}(F) is clear from the definition of these sets.

Lemma 3.2. Let F be as in the previous lemma. Then G^{±} ∘ F = d^{±1/n}G^{±}.

Proof: Since F is holomorphic on C^{2} and preserves K^{+}, we see that G^{+}∘F is 0 on K^{+}, plurisubharmonic and continuous on C^{2}, and strictly positive and pluriharmonic on C^{2} − K^{+}.

Fix z_{0} and define g_{z_{0}}(w) := G^{+} ∘F (z_{0}, w). Note that K^{+} ∩({z_{0}}×C) 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

ϕ_{z_{0}}(w) = g_{z_{0}}(w) − c_{z_{0}} log |w| + ih_{z_{0}}(w)

which is holomorphic for |w| > r.

Since g_{z_{0}} ≥ 0, we have | exp(−ϕ_{z_{0}}(w))| ≤ |w|^{c_{z_{0}}}. Hence exp(−ϕ_{z_{0}}(w)) has at most a pole at infinity, so we can write

exp(−ϕ_{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|− 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}} ≥ 0, we have g_{z_{0}}(w) = b_{z_{0}} log^{+} |w| + O(1) in C.

We claim that b_{z} is independent of z. Note that 2πb_{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 Δ(z_{0}; r_{0}) × Δ(w_{0}; r_{0}) for some r_{0} small. For w ∈ Δ(w_{0}; r_{0}), we can use analytic continuation as above to extend g_{z} around a circle in {z} × {|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}.

Hence g_{z_{0}}(w) = b log^{+} |w| + O(1), and from lemma 2.1, we see G^{+}(z_{0}, w) = log^{+} |w| + O(1). Thus g_{z_{0}}(w) − bG^{+}(z_{0}, w) is continuous for w ∈ C and harmonic for w such that (z_{0}, w) ∉ K^{+}, has a removable singularity at ∞, and is 0 for w such that (z_{0}, w) ∈ K^{+}, which is a nonempty set. Hence g_{z_{0}}(w) ≡ bG^{+}(z_{0}, w) for all w ∈ C.

Similarly, G^{+} ∘F (z, w)−bG^{+}(z, w) ≡ 0 for all (z, w) ∈ Δ(z_{0}; r_{0})×C. Since this difference is pluriharmonic in C^{2}−K^{+}, which is connected, it must be 0 throughout C^{2} − K^{+}, hence throughout C^{2} since both terms are 0 on K^{+}.

Finally, G^{+} ∘ F^{n}(z, w) = G^{+} ∘ H(z, w) = dG^{+}(z, w), while induction with the above result shows that G^{+} ∘ F^{n}(z, w) = b^{n}G^{+}(z, w) Hence b^{n} = d, and b > 0 since G^{+} ≥ 0. This gives the lemma for G^{+}, and the same proof applies to G^{−}.

4. Polynomial roots

In the proof of the following theorem, we use the terminology and results of [FM]. In particular, we use the fact that the group of polynomial automorphisms of C^{2} is the amalgamated product of the group A of affine linear automorphisms and the group E of elementary automorphisms which preserve the set of lines of the form w = const. A reduced word is an automorphism of the form g_{1} ⋯ g_{k}, k ≥ 1, where each g_{k} is in A or 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 length of this reduced word. Also, we need to know that the identity cannot be written as a reduced word.

Theorem 4.1. Suppose H ∈ G is a composition of generalized Hénon maps and F is an automorphism of C^{2} with F^{n} = H. Then F ∈ G.

Proof: Let (z, w) ∈ C^{2}, and let F = (F_{1}, F_{2}). If F (z, w) ∈ \overline{V^{−} ∪ V}, then from lemmas 2.1, 3.2, and 2.2, we see

log^{+} |F_{2}| − C_{1} ≤ G^{+}(F (z, w))
= d^{1/n}G^{+}(z, w)
≤ d^{1/n}(max{log^{+} |z|, log^{+} |w|} + C_{2}).

Exponentiating and using |F_{1}| ≤ |F_{2}| + R, we obtain

|F (z, w)| ≤ C max{(|z| + 1)^{d^{(1/n)}}, (|w| + 1)^{d^{(1/n)}}}.

Similarly, if F (z, w) ∈ V^{+}, then

log^{+} |F_{1}| − C_{1} ≤ G^{−}(F (z, w))
= 1/d^{(1/n)}G^{−}(z, w)
≤ 1/d^{(1/n)}(max{log^{+} |z|, log^{+} |w|} + C_{2}).

Exponentiating and using |F_{2}| ≤ |F_{1}|, we obtain

|F (z, w)| ≤ C max{(|z| + 1)^{1/d^{(1/n)}}, (|w| + 1)^{1/d^{(1/n)}}}.

Hence F has polynomial growth throughout C^{2}, hence must be a polynomial.

We show next that F ∈ G. Let τ(z, w) := (w, z). By [FM], we can write H = τe_{1} ⋯ τe_{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} ⋯ a_{l}e′_{l}

for some affine, non-elementary maps a_{j} and some elementary, nonaffine maps e′_{j}. By [FM] or [AR], each a_{j} can be written a_{j} = a_{j}^{1}τa_{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 + cw, w).

Now, since a_{j}^{k} is elementary, F^{n} has the form a_{1}^{1} ∘ (p, q), where p and q are polynomials and (p, q) = τe′′_{1} ⋯ τe′′_{nl}. By [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 σa_{2}^{1}, where σ(z, w) = (z, bw), we obtain

F = τE_{1} ⋯ τE_{l}.

Hence F ∈ G.

The second case is

F = e′_{1}a_{2} ⋯ a_{l}e′_{l}.

In this case, we can replace each a_{j} by a_{j}^{1}τa_{j}^{2} as before, and hence relabeling, we may assume F = e′_{1}τ ⋯ τe′_{l}. But then H = F^{n} = e′_{1}τ ⋯ τe′_{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} ⋯ e′_{l−1}a_{l}.

As before, we may relabel to assume that F = e′_{1}τ ⋯ e′_{l−1}τa_{l}^{2}. But then H = F^{n} = e′_{1}τ ⋯ e′_{k}τa_{l}^{2}, but also H = τe_{1} ⋯ τe_{m}. Hence

I = (F^{n})^{−1}H = (a_{l}^{2})^{−1}(τ(e′_{k})^{−1} ⋯ (e′_{1})^{−1})(τe_{1} ⋯ τe_{m}).

But then I has been written as a reduced word, which is impossible from [FM]. Thus F cannot have this form.

In the final case, we have

F = a_{1}e′_{1} ⋯ e′_{l−1}a_{l}.

Again we may relabel and collect terms and assume H = F^{n} = a_{1}^{1}τe′_{1} ⋯ e′_{k−1}τa_{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 ≠ 0. Applying τe′_{1} ⋯ e′_{k−1}τ 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 ∈ G.

Remark 4.2. 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 ∘ s)^{2} = H for s(z, w) = (ω^{2}z, ωw), where ω^{3} = 1. In fact, one can check that these are the only possible square roots of H.

5. Roots of elementary maps

In [BF], we showed that no Hénon 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 [AF] and [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) = (β^{μ}(z + w^{μ}q(w^{r})), βw),

where β is a primitive rth root of unity, q(w) = w^{k} +q_{k−1}w^{k} +⋯ q_{1}w + q_{0}, and k ≥ 1. A simple check shows that if we replace w^{μ}q(w^{r}) by

COMPOSITIONAL ROOTS OF HÉNON MAPS 7

(w^{μ}q(w^{r}))/(lr+1) for any l ∈ Z^{+}, 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 ϕ(z, w) = (i(z + w(w^{4} + 1)/2 + w^{3}k(w^{4})), iw). A simple check shows that ϕ^{2} = F, and ϕ is nonpolynomial whenever k is transcendental.

We claim that any root of F is conjugate to a polynomial automorphism. Suppose that ϕ is an automorphism of C^{2} with ϕ^{n} = F. Then ϕF^{r}ϕ^{−1} = F^{r}, so an argument like that in [FM, theorem 6.10] shows that ϕ(z, w) = (e^{g(w)}z + h(w), aw + b) for some entire g, h, and some a, b ∈ C, a ≠ 0. Since ϕ^{n} = F, we see that a^{n} = β and b(a^{n} − 1)/(a − 1) = 0, so that b = 0.

Using this form for ϕ and the fact that ϕF^{r} = F^{r}ϕ, it follows that e^{g(w)} is a nonzero rational function, hence is a constant, c ≠ 0. Moreover, since ϕF = F ϕ, we see that c = a^{μ}. Thus ϕ(z, w) = (a^{μ}z + h(w), aw).

Now, since ϕ^{rn} = F^{r}, it follows that ∑_{j=0}^{rn−1}(a^{μ})^{−j}h(a^{j}w) = rw^{μ}q(w^{r}). Write h = h_{1} + h_{2}, where h_{2} = w^{μ+kr}h̃_{2} for some entire h̃_{2}. Then the sum just given is valid with h_{2} in place of h and 0 in place of rw^{μ}q(w^{r}). Hence by [AF], there exists f entire such that f(aw)−a^{μ}f(w) = h_{2}(w).

A simple check shows that if ψ(z, w) = (z + f(w), w), then ψ^{−1}ϕψ is a polynomial, and in fact, ψ^{−1}ϕ^{n}ψ = F. Thus any root of F is conjugate to a polynomial automorphism.

Given any elementary map F and a root ϕ^{n} = F, one can ask if ϕ 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 [AR].

References

[AF] Ahern, P., and Forstneric, F., One parameter automorphism groups on C^{2}, Complex Variables, to appear.

[AFV] Ahern, P, Forstneric, F., and Varolin, D., Flows on C^{2} with polynomial time one map, preprint, 1995.

[AR] Ahern, P., and Rudin, W., Periodic automorphisms of C^{2}, preprint, 1994.

[BS] Bedford, E., and Smillie, J., Polynomial diffeomorphisms of C^{2}: currents, equilibrium measure and hyperbolicity, Inv. Math., 103 (1991), 69-99.

[BF] Buzzard, G., and Fornæss, J.E., Complete holomorphic vector fields and time-1 maps, preprint.

[FM] Friedland, S., and Milnor, J., Dynamical properties of plane polynomial automorphisms, Ergod. Th. and Dynam. Sys., 9 (1989), 67-99.

Gregery T. Buzzard Department of Mathematics The University of Michigan Ann Arbor, MI 48109, USA and MSRI 1000 Centennial Drive Berkeley, CA 94720, USA

John Erik Fornæss Department of Mathematics The University of Michigan Ann Arbor, MI 48109, USA