Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 22, 1997, 113–121

POLYMORPHISMS AND LINEARIZATION OF NONLINEAR POLYNOMIALS

Timo Erkama

University of Joensuu, Department of Mathematics P.O. Box 111, FIN-80101 Joensuu, Finland; erkama@joyl.joensuu.fi

Abstract. We prove that if a nonconstant holomorphic function f satisfies a non-discrete family of functional equations of the form f(az + b) = P (f(z)) where a ≠ 0 and b are constants and P is a polynomial of degree k ≥ 2, then f must be an exponential function.

Let D be a domain of the complex plane C, and let Aff(D) be the group of all holomorphic polynomials of degree one mapping D onto itself. Every element ϕ ∈ Aff(D) is an affine conformal automorphism of D, and Aff(D) is a closed subgroup of the Lie group Aff(C).

Let f: D → C be a holomorphic function, and let ϕ ∈ Aff(D). We say that ϕ is a polymorphism of f if there exists a polynomial P such that f ∘ ϕ = P ∘ f. If f is nonconstant, then P is uniquely determined by ϕ; we say that P is the polynomial associated with the polymorphism ϕ, and the degree deg P of P is the f-degree of ϕ.

Let Π(f) be the set of polymorphisms of f. Then Π(f) is a topological semigroup: if f ∘ ϕ_{j} = P_{j} ∘ f for j = 1, 2, then f ∘ (ϕ_{1} ∘ ϕ_{2}) = (P_{1} ∘ P_{2}) ∘ f. The identity element of Π(f) is the trivial polymorphism of f.

Many elementary functions have nontrivial polymorphisms. For example, nonconstant homogeneous polynomials are associated with polymorphisms of e^{z}, and Chebyshev polynomials are associated with polymorphisms of cos z. The power function f(z) = z^{κ} has a non-discrete group of polymorphisms of f-degree one for each κ ≠ 0. In general, if f admits nontrivial polymorphisms of f-degree one, then f is a polymorphic function in the sense of Pommerenke [7].

Each polymorphism of a nonconstant holomorphic function f gives rise to a linearization of the associated polynomial. The linearizing map f linearizes simultaneously all polynomials associated with elements of Π(f). For entire functions the problem of simultaneous linearization has been studied already by Fatou [4] and Julia [6].

We say that two holomorphic functions f: D → C and g: D → C are conjugate if there exist σ ∈ Aff(D) and τ ∈ Aff(C) such that f ∘ σ = τ ∘ g. If f and g are conjugate, then Π(f) and Π(g) are isomorphic.

1991 Mathematics Subject Classification: Primary 30D05; Secondary 58F23.

Let Π_{k}(f) be the subset of Π(f) of polymorphisms of f-degree k. In this article we show that Π_{k}(f) is always discrete for k ≥ 2 unless f is either constant or conjugate to the exponential function. This result answers affirmatively a question raised in [3] where a similar theorem was proved in the case k = 2.

Theorem 1. Let f: D → C be a nonconstant holomorphic function, and suppose that Π_{k}(f) is not discrete for some k ≥ 2. Then D is either the whole plane C or a half plane and f is conjugate to the exponential function e^{z}.

The idea of proof of Theorem 1 differs slightly from the approach used in [3] where we obtained f as a solution of a differential equation. Now we show, by using analytic continuation, that under the hypotheses of Theorem 1 f must be the restriction of an entire function.

For entire functions Theorem 1 can be proved under weaker assumptions. It suffices to assume that f is locally one-to-one and that there exist k ≥ 2 and ϕ ∈ Π_{k}(f) such that ϕ has a fixed point in C. By using Picard’s theorem as in [3] we can then show that the polynomial P associated with ϕ is conjugate to a homogeneous polynomial and that f satisfies the same differential equation as the exponential function e^{z}.

Before proving Theorem 1 we introduce some additional terminology. We say that an element ϕ ∈ Aff(C) is a pseudopolymorphism of f: D → C if there is a subdomain D′ ⊂ D and a polynomial P such that ϕ(D′) ⊂ D and f(ϕ(z)) = P (f(z)) for each z ∈ D′. If I is an open interval of R and α ∈ R, we define a half-strip

S(α, I) = {x + iy | x > α and y ∈ I}

and a corresponding strip

S(I) = {x + iy | y ∈ I}.

The following lemma contains the key argument on analytic continuation needed in the proof of Theorem 1.

Lemma 1. Let g be holomorphic and one-to-one in S(α, I), and suppose that there exists β ∈ R such that the translation z ↦ z + β is a pseudopolymorphism of g.

(a) If β < 0, then g has a holomorphic extension to S(I).

(b) If β > 0, then I can be divided into finitely many subintervals I_{1}, …, I_{m} such that for each j ∈ {1, …, m} the restriction of g to S(α, I_{j}) has a holomorphic extension to S(I_{j}).

Proof. By hypothesis there exists a polynomial P such that

(1)  g(z + β) = P (g(z))

for each z in a subdomain of S(α, I).

(a) If β < 0, we can define an extension of g to S(α + β, I) so that

(2)  g(z) = P (g(z − β))

for each z ∈ S(α + β, I). This follows from the fact that the right hand side of (2) is defined and holomorphic in S(α + β, I) and agrees with g in a subdomain of S(α, I). Since z ↦ z + β is a pseudopolymorphism of this extension of g, we may proceed by induction and obtain an extension of g to S(α + nβ, I) for each n > 0.

(b) The forward orbit of a point w ∈ C under P consists of all points of the form P^{∘n}(w) where P^{∘n} is the nth iterate of P. Let Y be the set of all y ∈ I such that g(x + iy) is contained in the forward orbit of a critical point of P for some x > α. Suppose that g(x_{1} + iy_{1}) and g(x_{2} + iy_{2}) are contained in the same orbit so that e.g. g(x_{1} + iy_{1}) = P^{∘n}g(x_{2} + iy_{2}) for some n > 0. Then by iteration of (1) it follows that g(x_{1} + iy_{1}) = g(x_{2} + nβ + iy_{2}), and hence y_{1} = y_{2} because g is one-to-one. Thus different points of Y correspond to different critical points of P, and we conclude that Y is finite.

Let J be any open subinterval of I contained in the complement of Y . It remains to prove that the restriction of g to S(α, J) has a holomorphic extension to S(J).

Let D be the set of all w ∈ C such that the forward orbit of w under P contains points of g(S(α, J)). There is an obvious equivalence relation in D: points w_{1} , w_{2} ∈ D are equivalent if their forward orbits intersect each other. The quotient space under this relation is a Riemann surface X, and the canonical projection ψ: D → X is a covering map; this follows from the fact that P is one-to-one in g(S(α, J)) and orbits of all critical points of P lie outside of D.

A similar equivalence relation can be defined in S(α, J) and in S(J): two points are equivalent if their difference is a multiple of β. These relations in S(α, J) and in S(J) have the same quotient space E, and g maps equivalent points of S(α, J) to equivalent points of D. Hence g induces a holomorphic quotient map g_{∗}: E → X, and the restriction of g to S(α, J) is actually a lifting of the composite map S(α, J) →^{π_{α}} E →^{g_{∗}} X where π_{α} is the natural projection. This projection extends to the canonical projection π: S(J) → E. Since S(J) is simply connected and ψ: D → X is a covering map, the map g_{∗}∘π has a lifting g̃: S(J) → D which agrees with g in S(α, J) and so is the desired holomorphic extension of g. This completes the proof of Lemma 1.

Lemma 2. If k ≥ 2 and f is nonconstant, then each element of Π_{k}(f) is of infinite order. Moreover, the intersection of Π_{k}(f) with every abelian subgroup of Aff (D) is discrete.

Proof. If ϕ ∈ Π_{k}(f), there is a polynomial P of degree k such that f ∘ ϕ = P ∘ f. Then f ∘ ϕ^{∘n} = P^{∘n} ∘ f for each positive integer n, and we conclude that ϕ^{∘n} ∈ Π_{kn}(f), because deg P^{∘n} = kn. Thus the polymorphisms ϕ^{∘n} are all distinct, because they have different f-degrees.

To prove the second assertion of the lemma let Γ be an abelian subgroup of Aff(D). Let 𝒫 be the set of all polynomials associated with an element of Γ ∩ Π_{k}(f). If P_{j} ∈ 𝒫 for j = 1, 2, there exist ϕ_{j} ∈ Γ ∩ Π_{k}(f) such that f ∘ ϕ_{j} = P_{j} ∘ f. Since f ∘ (ϕ_{i} ∘ ϕ_{j}) = (P_{i} ∘ P_{j}) ∘ f for i, j = 1, 2 and since Γ is abelian, it follows that P_{1} ∘ P_{2} = P_{2} ∘ P_{1} . Therefore all elements of 𝒫 commute with each other. Thus 𝒫 is finite, because the number of polynomials of degree k commuting with a given polynomial of degree k is finite [5].

It might happen that a polynomial P ∈ 𝒫 is associated with several polymorphisms in the set Γ∩Π_{k}(f). However, the set of such polymorphisms is always a discrete subset of Aff(D) [3, Lemma 4]. Since 𝒫 is finite, we conclude that Γ ∩ Π_{k}(f) is discrete.

In the proof of Theorem 1 we shall also need the following result which deals with contracting polymorphisms of a function defined in the upper half plane H = {z ∈ C | Im z > 0}. We say that a polymorphism ϕ ∈ Π(f) is contracting if |ϕ′| < 1.

Theorem 2. Suppose that f: H → C is holomorphic and that Π(f) contains two contracting polymorphisms which do not commute. Then f has a holomorphic extension to the boundary of H only if f is a polynomial of degree ≤ 1.

Proof. Let ϕ_{1}, ϕ_{2} ∈ Π(f) be contracting polymorphisms which do not commute. Then there exist polynomials P_{1}, P_{2} such that

(3)  f ∘ ϕ_{j} = P_{j} ∘ f  (j = 1, 2).

Suppose that f has a holomorphic extension to the boundary of H, and let ξ_{j} be the attracting fixed point of ϕ_{j}. Note that ξ_{1} ≠ ξ_{2} because ϕ_{1} and ϕ_{2} do not commute. Since ϕ_{1} and ϕ_{2} are contracting, both ξ_{1} and ξ_{2} lie on the real axis; we may assume that ξ_{1} < ξ_{2} .

We prove first that the holomorphic extension of f to any domain containing ξ_{1} and ξ_{2} is either constant or one-to-one. If f is not one-to-one, there exist z_{1} and z_{2} such that z_{1} ≠ z_{2} and f(z_{1}) = f(z_{2}). Then iteration of (3) shows that f(ϕ_{2}^{∘n}(z_{1})) = f(ϕ_{2}^{∘n}(z_{2})) for each n ≥ 1, and it follows that f is not locally one-to-one at ξ_{2} = lim_{n→∞} ϕ_{2}^{∘n}(z_{1}) = lim_{n→∞} ϕ_{2}^{∘n}(z_{2}). Thus f′(ξ_{2}) = 0, and by differentiation of f ∘ ϕ_{1}^{∘n} = P_{1}^{∘n} ∘ f we conclude that f′(ϕ_{1}^{∘n}(ξ_{2})) = 0 for each n ≥ 1. But then the zeros of f′ accumulate at ξ_{1} = lim_{n→∞} ϕ_{1}^{∘n}(ξ_{2}) and we see that f must be constant.

Next assume that the extension of f is one-to-one. It suffices to prove that f is the restriction of an entire function, because every holomorphic injection of C is an element of Aff(C).

The function g defined by

(4)  g(w) = f(e^{−w} + ξ_{2})

satisfies the hypotheses of Lemma 1 (b) for β = log 1/ϕ′_{2} . In fact, since f has a holomorphic extension to a neighborhood V of ξ_{2} , g is defined and holomorphic in S(α, (0, π)) if α is sufficiently large. Moreover, because (3) holds in V by analytic continuation and because ϕ_{2}(z) = (z − ξ_{2})ϕ′_{2} + ξ_{2} for each z ∈ C, we have

(5) g(w + β) = f(e^{−(w+β)} + ξ_{2}) = f(ϕ_{2}(e^{−w} + ξ_{2})) = P_{2}(f(e^{−w} + ξ_{2})) = P_{2}g(w)

for each w ∈ S(α, (0, π)).

Let Y be the set of all y ∈ (0, π) such that the restriction of g to S(α, (0, y)) has a holomorphic extension to S((0, y)). By Lemma 1 (b) Y is not empty, and it is evident that Y is closed in (0, π). We show next that Y is also open.

The domain of the equations (3), (4) and (5) can be enlarged by analytic continuation. In fact, g can be defined by (4) not only in S(α, (0, π)) but also in S((−π, 0)) and on the real axis. By analytic continuation (5) holds also in this larger domain. For j = 1 equation (3) is equivalent to

(6)  g(log \frac{1}{ϕ_{1}(z) − ξ_{2}} ) = P_{1}g(log \frac{1}{z − ξ_{2}} )

where z ∈ H. If y ∈ Y , the right side of (6) is defined and holomorphic in the domain Δ_{y} = {z ∈ C \ {ξ_{2}} | −y < arg(z − ξ_{2}) < π}, and therefore (6) defines a holomorphic extension of g to the domain

(7)  \{log \frac{1}{ϕ_{1}(z) − ξ_{2}} | z ∈ Δ_{y}\}.

Since ξ_{1} < ξ_{2} and ϕ_{1} is contracting, Δ_{y} contains all points of the form ϕ_{1}^{−1}(z) where arg(z − ξ_{2}) = −y. Consequently the domain (7) contains all points w with Im w = y. Hence Y is open.

We conclude that Y = (0, π), because Y is nonempty, open and closed in (0, π). Thus g is holomorphic in S((−π, π)), and it follows that f has a holomorphic extension to the whole complex plane C. Note that branch points cannot occur, because by assumption f has a single-valued holomorphic extension to the real axis. The proof of Theorem 2 is now complete.

Proof of Theorem 1. Let k ≥ 2 be such that Π_{k}(f) is not discrete. Then there exists a sequence of distinct polymorphisms ϕ_{j} ∈ Π_{k}(f) such that lim_{j→∞} ϕ_{j} = ψ ∈ Π_{k}(f). Moreover, since Π_{k}(f) is contained in Aff(D), the group Aff(D) is not discrete. Domains with this property have been classified in [2]. The classification shows that if D is not the whole plane C or a half plane, then either Aff(D) is abelian or D is the image of a strip under an element of Aff(C).

If D is affinely equivalent to a strip, then each element of Π_{k}(f) is either a translation or an involution of order two. However, in view of Lemma 2 no involutions can occur, so that each element of Π_{k}(f) belongs to the abelian group of translations of D. By Lemma 2 this is impossible, because Π_{k}(f) is not discrete. It also follows from Lemma 2 that Aff(D) cannot be abelian if Π_{k}(f) is not discrete. We conclude that D is either the whole plane C or a half plane.

Next we show that f′(z) ≠ 0 for each z ∈ D. For each j choose a polynomial P_{j} such that deg P_{j} = k and

(8)  f ∘ ϕ_{j} = P_{j} ∘ f,

and let X = {z ∈ D | f′(z) = 0}. Differentiation of (8) shows that ϕ_{j}(X) ⊂ X for each j. Since {ϕ_{j}} is not discrete in Aff(D), Lemma 3 of [3] implies that there exists at most one point z_{0} ∈ D such that the set of points ϕ_{j}(z_{0}) is discrete in D. On the other hand X is discrete in D because f is nonconstant. It follows that X contains at most one point. However, such a point would be a fixed point of each ϕ_{j}, because ϕ_{j}(X) ⊂ X for each j. This contradicts Lemma 2, because the stabilizer of a point of D is an abelian subgroup of Aff(D). We conclude that X must be empty.

To examine the dependence of P_{j} on ϕ_{j} we write

P_{j}(w) = a_{j0} + a_{j1}w + ⋯ + a_{jk}w^{k}

and try to express the coefficients a_{j0}, …, a_{jk} of P_{j} in terms of ϕ_{j}. To this end we differentiate (8) k times with respect to the independent variable z ∈ D. Since ϕ′_{j} is a nonzero constant, this differentiation yields

(9)  (f′ ∘ ϕ_{j})ϕ′_{j} = (P′_{j} ∘ f)f′ = Q_{j1}(f, f′) ; (f′′ ∘ ϕ_{j})(ϕ′_{j})^{2} = (P′′_{j} ∘ f)(f′)^{2} + (P′_{j} ∘ f)f′′ = Q_{j2}(f, f′, f′′) ; ⋮ ; (f^{(k)} ∘ ϕ_{j})(ϕ′_{j})^{k} = Q_{jk}(f, f′, f′′, …, f^{(k)})

where each Q_{jν} is a polynomial of ν + 1 variables whose coefficients are linear expressions of the coefficients of P_{j}. Therefore, together with (8) the equations (9) constitute a system which is linear with respect to a_{j0}, a_{j1}, …, a_{jk} and admits a unique solution. More precisely, there is a unique set of constants a_{j0}, a_{j1}, …, a_{jk} such that (8) and (9) are satisfied at each z ∈ D.

We show next that these constants depend holomorphically on ϕ_{j} and z. The proof will depend on the following lemma.

Lemma 3. The Wronskian determinant W (1, f, f^{2}, …, f^{k}) is equal to
k!! (f′)^{k(k+1)/2}

where k!! = 1! · 2! ⋯ k!.

Proof. For the proof of this lemma I am indebted to Kari Katajamäki and Ilpo Laine.

By definition,

W (1, f, f^{2}, …, f^{k}) = det (\frac{d^{i}f^{j}}{dz^{i}} ) (0 ≤ i, j ≤ k)

so that W (1, f, f^{2}, …, f^{k}) is also the determinant of the linear system (8)–(9).

Successive expansion of the determinant in terms of the elements of the first column shows that (10)

\begin{aligned}W (1, f, f^{2}, …, f^{k}) & = ν!!(ν!)^{(k−ν)}(f′)^{ν(2k−ν+1)/2} \\ & × W (\binom{ν + 0}{0}f^{0}, \binom{ν + 1}{1}f^{1}, …, \binom{ν + (k − ν)}{k − ν}f^{k−ν}) \end{aligned}

for each ν = 1, …, k. The assertion follows from (10) when ν = k.

Lemma 3 implies that the determinant of the linear system (8)–(9) is nonzero at each z ∈ D. In particular, the system admits a unique solution of the form

a_{j0} = a_{0}(ϕ_{j}, z), …, a_{jk} = a_{k}(ϕ_{j}, z)

where a_{0}, …, a_{k} are holomorphic functions defined in a subdomain of Aff(C)×D. Moreover, since the expressions a_{0}(ϕ_{j}, z), …, a_{k}(ϕ_{j}, z) are constant with respect to z for each j, we have

(11)  \frac{∂^{μ}}{∂z^{μ}} a_{ν}(ϕ, z_{0}) = 0  (0 ≤ ν ≤ k; μ = 1, 2, …)

whenever z_{0} ∈ D and ϕ = ϕ_{j} for some j. As in [3] we conclude that given z_{0} ∈ D there is a subdomain N of Aff(C) containing ψ such that the set M of all ϕ ∈ N satisfying (11) is a non-discrete complex analytic subset of N. Since the complex dimension of M at ψ is positive, it follows from the structure theory of one-dimensional complex analytic sets [1, p. 68] that there exists a nonconstant holomorphic map η from the open unit disk U into M such that η(0) = ψ.

From the definition of M it follows that for each t ∈ U the polynomial P_{t} with coefficients a_{0}(η(t), z_{0}), …, a_{k}(η(t), z_{0}) satisfies

(12)  f(η(t)(z)) = P_{t}(f(z))

for each z in a neighborhood of z_{0} . Moreover, because the function t ↦ deg P_{t} is semicontinuous and deg P_{0} = k, we may assume that deg P_{t} ≥ k for each t ∈ U.

From (12) we see that each element of η(U) is a pseudopolymorphism of f and, if we choose N small enough as in [3] the composite of two elements of η(U) is again a pseudopolymorphism of f. Since each abelian subgroup of Aff(C) contains at most countably many of such pseudopolymorphisms corresponding to nonlinear polynomials [3, Lemma 6], by repeating the argument presented in [3, p. 76–77] we conclude that there exists a constant a ≠ 1 and a nonconstant holomorphic function b: U → C such that

η(t)(z) = az + b(t)

for each (t, z) ∈ U × C.

Let us first consider the case when D is the whole complex plane C. Then (12) holds by analytic continuation for each (t, z) ∈ U ×C, and we see that Π_{k}(f) is uncountable. Theorem 2 of [3] implies that f is conjugate to the exponential function e^{z}.

In the remaining case D is a half plane, and by conjugation we may assume that D is the upper half plane H. Even in this case it suffices to prove that f is the restriction of an entire function.

Since ϕ_{j}(H) = H for each j, ϕ_{j}(0) is real and ϕ′_{j} > 0 for each j. Thus by continuity b(0) = lim_{j→∞} ϕ_{j}(0) is real and

a = \frac{d}{dz} η(0) = ψ′ = lim_{j→∞} ϕ′_{j}

is real and positive.

Since b is holomorphic and nonconstant and since b(0) is real, there is t_{0} ∈ U such that Im b(t_{0}) < 0. Then (12) defines a holomorphic extension of f to the domain η(t_{0})(H). In particular, f has a holomorphic extension to the real axis, because Im b(t_{0}) < 0.

There are two cases to consider, according as a < 1 or a > 1.

The case a > 1 is easy. In fact, iteration of (12) yields

f ∘ η(t_{0})^{∘n} = P_{t_{0}}^{∘n} ∘ f

in a subdomain of H, and this equation defines a holomorphic extension of f to the domain η(t_{0})^{∘n}(H) for each n ≥ 1. Thus f is the restriction of an entire function, and by Theorem 2 of [3] f is conjugate to the exponential function e^{z}.

If a < 1, there exist t_{1} , t_{2} ∈ U such that b(t_{1}) and b(t_{2}) are different real numbers. Then η(t_{1}) and η(t_{2}) are contracting polymorphisms of f which do not commute, and f has a holomorphic extension to the boundary of H. By Theorem 2 f must be the restriction of a linear polynomial. Hence f is the restriction of an entire function. This completes the proof of Theorem 1.

For a given domain D it may be of interest to determine the semigroup Pol (D) of all holomorphic polynomials mapping D one-to-one onto itself. If D is a strip, a disk or a half plane, it is known that Pol (D) = Aff(D). In these simple cases Pol (D) does not contain any nonlinear polynomials. On the other hand, if D is a Siegel disk of a nonlinear polynomial, then Pol (D) contains a non-discrete subset of nonlinear polynomials, all of which are irrational rotations of the Siegel disk. We do not know any examples of a situation where Pol (D) would contain two nonlinear polynomials which are not permutable. However, by using Theorem 1 we can prove the following general result.

Theorem 3. For each k ≥ 2 the set {P ∈ Pol (D) | deg P = k} is a discrete subset of the Lie group of conformal automorphisms of D.

In particular, it follows from Theorem 3 that the number of nonlinear polynomials in the group of conformal automorphisms of D is at most countable. A proof will appear in a different article.

References

[1] Chirka, E.M.: Complex Analytic Sets. - Kluwer, 1989.

[2] Erkama, T.: Möbius automorphisms of plane domains. - Ann. Acad. Sci. Fenn. Ser. A I Math. 10, 1985, 155–162.

[3] Erkama, T.: The exponential function and linearization of quadratic polynomials. - In: Analysis, Algebra and Computers in Mathematical Research, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 1994, 71–80.

[4] Fatou, P.: Sur les fonctions qui admettent plusieurs théorèmes de multiplication. - C. R. Acad. Sci. Paris Sér. I Math. 173, 1921, 571–573.

[5] Jacobsthal, E.: Über vertauschbare Polynome. - Math. Z. 63, 1955, 243–276.

[6] Julia, G.: Mémoire sur la permutabilité des fractions rationnelles. - Ann. Sci. École Norm. Sup. 39, 1922, 131–215.

[7] Pommerenke, Ch.: Polymorphic functions for groups of divergence type. - Math. Ann. 258, 1982 , 353–366.

Received 4 October 1995