Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 19, 1994, 3–10

ZETA FUNCTIONS OF RATIONAL FUNCTIONS ARE RATIONAL

A. Hinkkanen

University of Illinois at Urbana–Champaign, Department of Mathematics Urbana, Illinois 61801, U.S.A.; aimo@symcom.math.uiuc.edu

Abstract. We prove that if f is a rational function of degree at least 2 in the Riemann sphere then its zeta function exp {∑_{n=1}^{∞} N_{n}t^{n}/n} is a rational function. Here N_{n} denotes the number of distinct fixed points of the n^{th} iterate of f. Furthermore we show that all the zeros of the zeta function, if any, have modulus 1.

1. Introduction and results

1.1. Zeta functions have been defined and studied for a great variety of algebraic structures and dynamical systems. Suppose that N_{n} is a sequence of complex numbers, usually integers, defined for n ≥ 1 and associated with an algebraic or geometric object. The corresponding zeta function is defined first as a formal power series

(1.1)  Z(t) = exp\{∑_{n=1}^{∞} \frac{N_{n}t^{n}}{n} \}.

Then the question arises if this series has a positive radius of convergence, if it can be continued to a meromorphic function in the plane, if it perhaps represents a rational function, and what can be said about the location of the zeros and poles of Z. One also describes the case when Z is a rational function by saying that the numbers N_{n} can be determined from a finite amount of information, since a rational function is determined by finitely many parameters.

To give an example of a zeta function arising from an algebraic structure, let q be a power of a prime and let F_{q} be the finite field with q elements. Let f(x_{1}, …, x_{n}) be a homogeneous polynomial of degree d with coefficients in F_{q}, such that the partial derivatives of f with respect to the x_{i} have no common projective zeros in any algebraic extension of F_{q}. Then Z(t) is defined by taking N_{s} to be the number of solutions to f = 0 in the projective space P^{n}(F_{q^{s}}). Dwork [3] proved that then Z(t) is a rational function of t. In fact,

1991 Mathematics Subject Classification: Primary 30D05; Secondary 58F20.

Research partially supported by the Alfred P. Sloan Foundation and by the U.S. National Science Foundation.

This research was completed while the author was visiting University College, London, U.K.

The author wishes to thank the Department of Mathematics for its hospitality.

Z(t) = \frac{P (t)^{(−1)^{n}}}{(1 − t)(1 − qt) ⋯ (1 − q^{n−1}t)}

where P is a polynomial whose degree can be given explicitly in terms of d and n. Deligne [2] proved that all the zeros of P have modulus q^{−(n−1)/2} , which was conjectured by A. Weil [12] and which is the statement corresponding to Riemann’s hypothesis in this connection. One may consider the function Z(q^{−s}) for a complex variable s to get a closer analogy with Riemann’s zeta function, and we see that all the zeros of P (q^{−s}) have real part (n−1)/2. For a more detailed survey of the above results, see e.g. [6, pp. 151–163]. Deligne [2] has obtained similar results, conjectured by Weil [12], for more general varieties over finite fields.

In the theory of dynamical systems, zeta functions have been considered for at least individual functions and flows. We shall only consider individual functions. Let X be a compact manifold, and let f: X → X be a continuous function. A point x ∈ X is called a fixed point of f if f(x) = x. We denote the iterates of f by f^{1} = f and f^{n} = f ∘ f^{n−1} for n ≥ 2. The (geometric) zeta function Z_{f}(t) of f is defined by the right hand side of (1.1), taking N_{n} to be the number of distinct isolated fixed points of f^{n} (cf. [11, p. 764], [13, p. 335]). The algebraic zeta function, obtained by taking N_{n} to be the sum of the Lefschetz indices of the fixed points of f^{n} (when X and f are such that these indices are defined), is sometimes simpler. Properties of such zeta functions have been considered particularly for diffeomorphisms and other homeomorphisms, and it has been proved in many cases that the zeta function has a positive radius of convergence or is a rational function or a root of a rational function. We cannot provide a complete list of references to such results here, and merely refer to the survey [11] and to [8]. Noninvertible functions satisfying suitable side-conditions have been considered by some authors. Zhang [14] has proved that the zeta functions of expanding maps are rational.

It seems to me that zeta functions for rational functions mapping the Riemann sphere onto itself have not been previously studied, and I have not been able to find in the literature a more general result that would cover all rational functions as a special case. Smale’s survey [11] on diffeomorphisms makes passing references to Julia’s work [7] on the iteration of rational functions [11, pp. 792, 807], but not in connection with zeta functions.

It is the purpose of this paper to note that zeta functions of rational functions are rational, with all the zeros, if any, on the unit circle. In fact, we shall find a formula for such zeta functions. Furthermore, we raise the question of how to define zeta functions for transcendental entire or meromorphic functions.

1.2. Let f be a rational function of degree d ≥ 2 defined in the extended complex plane or Riemann sphere \overline{C} = C ∪ \{∞\} where C denotes the complex plane. Thus f^{n} is a rational function of degree d^{n}, and so f^{n} has d^{n} + 1 fixed points in \overline{C}, with due count of multiplicity. We say that z_{0} ∈ C is a fixed point of multiplicity m of a rational function g if g has the expansion

g(z) − z = a_{m}(z − z_{0})^{m} + O((z − z_{0})^{m+1})

as z → z_{0} , where m ≥ 1 and a_{m} ≠ 0. We say that g has the multiplier λ = g′(z_{0}) at the fixed point z_{0} . It follows that we have m ≥ 2 if and only if λ = 1. If g(∞) = ∞, we consider 1/g(1/z) at the origin to determine the multiplicity and the multiplier at infinity. A fixed point of f^{n} whose multiplier is equal to 1 is called a parabolic fixed point. Thus the multiplicity of a fixed point is greater than 1 exactly for parabolic fixed points. A cycle of distinct points z, f(z), …, f^{p−1}(z) that are parabolic fixed points of some f^{n}, with f^{p}(z) = z, is called a parabolic cycle of f of length p. Then (f^{p})′(z) is a root of unity.

Let N_{n} be the number of distinct fixed points of f^{n} in \overline{C}, so that 1 ≤ N_{n} ≤ d^{n} + 1. As in the general theory of dynamical systems, the (geometric) zeta function Z_{f} of f is first defined as the formal power series

Z_{f}(t) = exp\{∑_{n=1}^{∞} \frac{N_{n}t^{n}}{n} \}.

Since N_{n} ≤ d^{n} +1, it follows that the radius of convergence of ∑_{n=1}^{∞} N_{n}t^{n}/n and, hence, that of Z_{f}, is at least 1/d. If N_{n} = d^{n} + 1 for all n ≥ 1 then

(1.2)  Z_{f}(t) = (1 − dt)^{−1}(1 − t)^{−1}.

One could define an algebraic zeta function of f by taking N_{n} to be the number of fixed points of f^{n} with due count of multiplicity. However, then N_{n} = d^{n} + 1 for all n ≥ 1 so that this results in a rather trivial zeta function that is equal to (1 − dt)^{−1}(1 − t)^{−1} for every rational function f of degree d ≥ 2.

Theorem 1. Let f be a rational function of degree d where d ≥ 2. Then the zeta function of f is a rational function given by

(1.3) Z_{f}(t) ≡ exp\{∑_{n=1}^{∞} \frac{N_{n}t^{n}}{n} \} = (1 − dt)^{−1}(1 − t)^{−1} ∏_{parabolic cycles}(1 − t^{pq})^{ℓ}

where the product is taken over all the (finitely many) parabolic cycles of f, and where p, q, and ℓ denote certain positive integers associated with and depending on these cycles. The integer p is the length of the cycle, and the multiplier of f^{p} at each point of the cycle is a primitive q^{th} root of unity. An empty product is taken to be equal to 1.

Consequently, all the zeros of Z_{f}, if any, occur at certain roots of unity and hence lie on the unit circle. The zeta function of f has a simple pole at the point 1/d and, possibly, at the point 1, and no other poles. There is a pole at the point 1 if, and only if, f has no parabolic cycles.

Thus the numbers N_{n} for a rational f can be deduced from a finite amount of information. Of course, this is to be expected, not only because f itself depends on finitely many parameters, but because N_{n} can differ from d^{n} + 1 only if there are parabolic cycles, and because there can only be finitely many parabolic cycles. One may also note that an expanding rational function has no parabolic cycles so that its zeta function is given by (1.2).

2. Proof of Theorem 1

2.1. Let the assumptions of Theorem 1 be satisfied. We make use of the Fatou–Julia theory of iteration of rational functions. We define the set of normality N(f) of f as the set of all z ∈ \overline{C} that have a neighbourhood U such that the family { f^{n} | U : n ≥ 1 } of the iterates of f restricted to U is a normal family. The Julia set J(f) of f is defined by J(f) = \overline{C} \ N(f). Clearly N(f) is open, while J(f) is known to be a nonempty perfect set which coincides with \overline{C} or is nowhere dense in \overline{C}. For the basic results in the iteration theory of rational functions based on these concepts we refer to the papers of Fatou [4, 5] and of Julia [7], and to the book of Beardon [1]. First we formulate two lemmas.

Lemma 1. If α is a fixed point of f^{p} of multiplicity ν where p ≥ 1 and ν ≥ 2, then for each n ≥ 1, the point α is a fixed point of f^{np} of the same multiplicity ν. The same applies even if ν = 1 provided that the multiplier λ = (f^{p})′(α) is not a root of unity.

If ν = 1 then λ ≠ 1. Now if λ is a primitive q^{th} root of unity, where q ≥ 2, and if q does not divide n, then α is a fixed point of f^{np} of multiplicity 1.

Proof of Lemma 1. We provide the simple proof only for completeness. Let α be a fixed point of f^{p} of multiplicity ν. We may assume without loss of generality that p = 1 and that α = 0. If ν ≥ 2 then f(z) = z + Az^{ν} + O(z^{ν+1}) as z → 0, where A ≠ 0, and so f^{n}(z) = z + nAz^{ν} + O(z^{ν+1}) as z → 0. Thus α = 0 is a fixed point of f^{n} of multiplicity ν.

If ν = 1 and f′(0) = λ then f(z) = λz+O(z^{2}). Therefore we have f(z)−z = (λ − 1)z + O(z^{2}) while f^{n}(z) − z = (λ^{n} − 1)z + O(z^{2}) as z → 0. If λ is not a root of unity then λ^{n} − 1 ≠ 0 for all n ≥ 1 and therefore α = 0 is a fixed point of f^{n} of multiplicity 1, for each n ≥ 1. Next if ν = 1 (for f), we have λ ≠ 1. Now if λ^{q} = 1 while λ^{j} ≠ 1 for 1 ≤ j ≤ q − 1 then λ^{n} = 1 if, and only if, q divides n. So if q does not divide n then α = 0 is a fixed point of f^{n} of multiplicity 1. This proves Lemma 1.

Lemma 2. Suppose that f^{p}(α) = α and (f^{p})′(α) = λ where p ≥ 1 and λ is a primitive q^{th} root of unity for some q ≥ 1. Define α_{j} = f^{j}(α) and suppose that the α_{j} are distinct and that α_{j} ∈ C for 0 ≤ j ≤ p − 1. Then

(2.1)  f^{pq}(z) = α_{j} + (z − α_{j}) + b_{m,j}(z − α_{j})^{m} + O((z − α_{j})^{m+1})

for z in a neighbourhood of α_{j} where b_{m,j} ≠ 0, m ≥ 2, the number m is independent of j, and m = 1 + ℓq for some integer ℓ ≥ 1. There are pairwise disjoint components U_{i,j} of N(f) for 1 ≤ i ≤ ℓq and 0 ≤ j ≤ p − 1 such that α_{j} ∈ ∂U_{i,j} and

(i) f^{pq}(U_{i,j}) ⊂ U_{i,j}; (ii) f^{np}(z) → α_{j} locally uniformly for z ∈ U_{i,j} as n → ∞.

For any z in any U_{i,j}, the values f^{np}(z) visit q of the U_{r,j} cyclically, namely those with r ≡ i (mod ℓ) provided that the U_{i,j} are properly labelled. Thus for each j, the domains U_{i,j} are divided into ℓ cycles, each consisting of q domains. The function f maps such a cycle of domains U_{i,j} into another cycle of domains U_{i,j+1} where U_{i,p} = U_{i,0} . In each cycle of domains there is a component of N(f) that contains a critical point of f, that is, a point at which f is not locally homeomorphic.

Furthermore, all these domains U_{i,j} are disjoint, that is, U_{i,j} ∩U_{r,s} = ∅ unless (i, j) = (r, s) where 1 ≤ i, r ≤ ℓq and 0 ≤ j, s ≤ p − 1.

Proof of Lemma 2. The statements in Lemma 2 other than the one involving critical points follow from the discussion in the paper of Fatou [4, pp. 217–220] (cf. [1, Theorems 6.5.4 and 6.5.8, pp. 116, 124]). In particular, the fact that m − 1 is divisible by q is obtained by considering the identity f^{pq}∘f^{p} = f^{p}∘f^{pq} for power series at α_{j}. The fact that m is independent of j, can be seen by considering the set N(f) close to the points α_{j}, as Fatou does. We mention that alternatively, the independence of m can be proved by purely formal calculations based on the chain rule, without referring to N(f), but we omit the details. As was proved by Fatou and by Julia (cf. [1, Theorem 9.3.2, p. 194]), in each cycle of domains there is a component of N(f) that contains a critical point of f. This proves Lemma 2.

2.2. We continue with the proof of Theorem 1. Suppose that w is a fixed point of f^{p} where p ≥ 1. We may assume that w is finite and that z is finite whenever z = f^{n}(w) or f^{n}(z) = w for some positive integer n, as the analysis is the same in the general case. Choosing p to be as small as possible, we see that then w is a fixed point of f^{n} if, and only if, n = kp for some positive integer k. The point w has multiplicity 1 as a fixed point of f^{kp} unless the multiplier of f^{kp} at w equals 1. By Lemma 1, w has multiplicity 1 as a fixed point of f^{kp} for all k ≥ 1 unless the multiplier λ of f^{p} at w is a root of unity, say λ = e^{2πiK/q} where the positive integers K and q are relatively prime. Then w has multiplicity at least 2 as a fixed point of f^{kp} exactly when k = mq for some positive integer m.

We have

f^{pq}(z) − w = (z − w) + a_{s}(z − w)^{s} + O((z − w)^{s+1})

as z → w, where s ≥ 2 and a_{s} ≠ 0. Here s is the multiplicity of w as a fixed point of f^{pq}. Furthermore, by Lemma 2, s − 1 is divisible by q, so s = 1 + ℓq for some positive integer ℓ. By Lemma 1, w is a fixed point of multiplicity s of f^{mpq}, for each m ≥ 1.

The function f^{n} has d^{n} + 1 fixed points, with due count of multiplicity. The number of distinct fixed points of f^{n} is therefore d^{n} + 1 − N where N is the sum of the numbers ν −1 for the multiple fixed points of f^{n} where ν is the multiplicity of such a point as a fixed point of f^{n}. Thus, corresponding to the fixed point w of f^{mpq}, we have to subtract s − 1, that is, ℓq, from d^{mpq} + 1.

Consider the points w_{j} = f^{j}(w) for j ≥ 0. The points w = w_{0}, w_{1}, …, w_{p−1} are distinct while w_{p} = w. Suppose that 0 ≤ j ≤ p − 1. Then w_{j} is a fixed point of f^{p} with the same multiplier as w since by the chain rule,

(f^{p})′(w) = ∏_{i=0}^{p−1} f′(w_{i}).

Hence each w_{j} is also a fixed point of f^{kp} for all k ≥ 1, and is a fixed point of multiplicity at least 2 if, and only if, k is an integral multiple of q. By Lemma 2, each w_{j} has the same multiplicity s as w as a fixed point of f^{mpq}, for all m ≥ 1. We deduce that corresponding to the cycle w = w_{0}, w_{1}, …, w_{p−1} of fixed points of f^{p}, we need to subtract altogether pqℓ from d^{mpq} + 1 when dealing with the fixed points of f^{mpq}.

2.3. The cycle w = w_{0}, w_{1}, …, w_{p−1} of fixed points of f^{p} determines a parabolic cycle since each of these points is a parabolic fixed point of f^{pq}. By Lemma 2, for each j with 0 ≤ j ≤ p − 1, there are ℓq components U_{i,j} of N(f) with w_{j} ∈ ∂U_{i,j} where 1 ≤ i ≤ ℓq, such that f^{mpq}(z) → w_{j} as m → ∞, locally uniformly for z ∈ U_{i,j}. The action of f divides these components into ℓ cycles, each consisting of pq components permuted by f. Further by Lemma 2, at least one component in each cycle contains a critical point of f. Since f has at most 2d − 2 distinct critical points, it follows that there are only finitely many (at most 2d − 2) parabolic cycles.

So we see that

∑_{n=1}^{∞} N_{n} \frac{t^{n}}{n} = ∑_{n=1}^{∞}(d^{n} + 1)\frac{t^{n}}{n} − ∑_{par. cycles} ∑_{m=1}^{∞} pqℓ\frac{t^{mpq}}{mpq}

= ∑_{n=1}^{∞} \frac{(dt)^{n}}{n} + ∑_{n=1}^{∞} \frac{t^{n}}{n} − ∑_{par. cycles} ℓ ∑_{m=1}^{∞} \frac{(t^{pq})^{m}}{m}

= − log(1 − dt) − log(1 − t) + ∑_{par. cycles} ℓ log(1 − t^{pq}).

If there are no parabolic cycles, we consider the resulting empty sum to be equal to zero. Similarly, we shall consider an empty product to be equal to 1. It follows that

Z_{f}(t) = exp\{∑_{n=1}^{∞} \frac{N_{n}t^{n}}{n} \} = (1 − dt)^{−1}(1 − t)^{−1} ∏_{par. cycles}(1 − t^{pq})^{ℓ}.

This gives (1.3), and this expression obviously proves all the remaining statements of Theorem 1. Thus the proof of Theorem 1 is complete.

Remark 1. Let f be a continuous, open, discrete mapping of \overline{C} onto itself. By Stoïlow’s theorem, we may write f = φ ∘ h where φ is a rational function and h is a homeomorphism. Every iterate of f is clearly also of this type. We define N_{n} as before, counting only isolated fixed points of f^{n}. Note that already the set of fixed points of f could be quite complicated since h is an arbitrary homeomorphism. One may ask to what extent the conclusions of Theorem 1 remain valid. If f is topologically conjugate to a rational function g, that is, if f = H ∘ g ∘ H^{−1} where H is a homeomorphism, then clearly N_{n} is the same for f and g so that Z_{f} ≡ Z_{g} and the conclusions of Theorem 1 hold for f. More generally, one may ask, in particular, what the situation is if it is assumed that for every n ≥ 1, the fixed points of f^{n} are isolated.

Remark 2. One can ask if there is any reasonable way to define a zeta function for a transcendental entire or meromorphic function in the plane. The problem is to define a formal power series whose properties could then be studied. Of course, now f^{n} has infinitely many fixed points except possibly for n = 1. If one tries to approach the question by approximation, one might first consider finitely many f^{n} in a disk centred at the origin of radius r and then let the number of iterates as well as r tend to infinity. Or one could consider a sequence f_{n} of rational functions such that for 1 ≤ k ≤ N = N(n), the function f_{n}^{k} has the same fixed points as f^{k} in a disk centred at the origin of radius r_{n}, with due count of multiplicity. We assume that N(n) and r_{n} tend to infinity with n. One can ask if there is any way to choose the f_{n} and then normalize the zeta functions Z_{f_{n}}(t), possibly after substituting for t a suitable expression depending on n, so that the sequence of normalized functions tends to a limit that one could define to be the zeta function of f.

Another possibility is to set, for example,

Z_{f}(t) = exp\{∑_{n=1}^{∞} \frac{t^{n}}{n} ∑_{f^{n}(z)=z} \frac{1}{max\{1, |(f^{n})′(z)|\}}\}

or possibly

Z_{f}(t) = exp\{∑_{n=1}^{∞} \frac{t^{n}}{n} ∑_{f^{n}(z)=z} \frac{1}{((f^{n})′(z))^{+}} \}

10 A. Hinkkanen

where for a complex number c, we write c^{+} = c if |c| ≥ 1 and c^{+} = 1 otherwise, provided that the sum over the distinct fixed points of f^{n} converges for each n ≥ 1. More generally, if ψ is a suitable function defined in the complex plane, such as ψ(z) = max{1, |z|^{α}} for some α > 0, we could set

(2.2)  Z_{f}(t) = exp\{∑_{n=1}^{∞} \frac{t^{n}}{n} ∑_{f^{n}(z)=z} \frac{1}{ψ((f^{n})′(z))}\}

or perhaps

Z_{f}(t) = exp\{∑_{n=1}^{∞} \frac{t^{n}}{n} ∑_{f^{n}(z)=z} ∏_{j=0}^{n−1} \frac{1}{ψ(f′(f^{j}(z)))}\}.

It might be of some interest to determine if such sums converge for all f in a reasonable class of functions. In particular, one may ask if (2.2) makes sense for f(z) = e^{z} whenever ψ(z) = max{1, |z|^{α}} and α > 1. In general, one might expect that some choice for ψ depending on (the growth of) f might yield a well defined zeta function for that f. These formulas are somewhat analogous to Ruelle’s definition of a zeta function for expanding maps on compact manifolds [9, 10].

References

[1] Beardon, A.F.: Iteration of rational functions. - Springer, New York, 1991.

[2] Deligne, P.: La conjecture de Weil. - Publ. Math. IHES 43, 1974, 273–307.

[3] Dwork, B.: On the rationality of the zeta function. - Amer. J. Math. 82, 1960, 631–648.

[4] Fatou, P.: Sur les équations fonctionnelles. - Bull. Soc. Math. France 47, 1919, 161–271.

[5] Fatou, P.: Sur les équations fonctionnelles. - Bull. Soc. Math. France 48, 1920, 33–94, 208–314.

[6] Ireland, K., and M. Rosen: A classical introduction to modern number theory, 2nd ed. - Springer, New York, 1990.

[7] Julia, G.: Mémoire sur l’itération des fonctions rationnelles. - J. Math. Pures Appl. 8, 1918, 47–245.

[8] Manning, A.: Axiom A diffeomorphisms have rational zeta functions. - Bull. London Math. Soc. 3, 1971, 215–220.

[9] Ruelle, D.: Generalized zeta-functions for Axiom A basic sets. - Bull. Amer. Math. Soc. 82, 1976, 153–156.

[10] Ruelle, D.: Zeta-functions for expanding maps and Anosov flows. - Invent. Math. 34, 1976, 231–242.

[11] Smale, S.: Differentiable dynamical systems. - Bull. Amer. Math. Soc. 73, 1967, 747–817.

[12] Weil, A.: Number of solutions of equations in a finite field. - Bull. Amer. Math. Soc. 55, 1949, 497–508.

[13] Williams, R.F.: Zeta function in global analysis. - In Global Analysis (Proc. Sympos. Pure Math. XIV, Berkeley, CA, 1968), Amer. Math. Soc., Providence, R.I., 1970, 335–339.

[14] Zhu Sheng Zhang: On the zeta functions of expanding mappings. - Sci. China Ser. A 28, 1985, 1036–1048.

Received 25 August 1992