Computational Methods and Function Theory Volume 2 (2002), No. 2, 353–366
Fractional Iteration and Functional Equations for Functions Analytic in the Unit Disk Mark Elin, Victor Goryainov, Simeon Reich, and David Shoikhet (Communicated by Mario Bonk) Abstract. We establish criteria for the embeddability of an analytic function into a semigroup of analytic self-mappings of the open unit disk. We do this by studying properties of solutions to the Abel and Schr¨ oder functional equations. Keywords. Embeddability, fractional iteration, functional equation, infinitesimal generator, Koenigs function, semigroup of analytic functions. 2000 MSC. 30D05, 39B12, 39B32.
One problem concerning the iteration of analytic functions is that of extending the natural iterates f 0 (z) ≡ z, f 1 = f , f n = f ◦ f n−1 , n = 2, 3, . . . , of a function f to the so-called fractional iterates f t , t ≥ 0. The history of this problem is very long. Its origin goes back to the classical papers of Schr¨oder [23] and Koenigs [18]. Note that Schr¨oder related the problem to the solution of a certain functional equation and Koenigs introduced a limit technique for solving Schr¨oder’s equation. Later on, it was pointed out (see [13]) that the fractional iterates form a one-parameter semigroup (or group) which can be described via its infinitesimal generator. The problem of fractional iteration was first studied for functions analytic in a neighborhood of a fixed point. For certain classes of analytic functions (for example, meromorphic functions), the problem was studied in [3, 16]. The question of embedding probability generating functions into continuous semigroups of fractional iterates arises naturally in the theory of stochastic branching processes [15, 2]. This problem was solved in various terms in [8, 9, 12]. Our goal in the present paper is to study the problem of fractional iteration and some related topics (such as the Koenigs function and the functional equations of Schr¨oder and Abel) for the general semigroup of all holomorphic self-mappings Received December 21, 2002. The work of M. Elin was partially supported by the Sakta-Rashi Fund. The work of S. Reich was partially supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities (Grant 592/00), by the Fund for the Promotion of Research at the Technion, and by the Technion VPR Fund. c 2002 Heldermann Verlag ISSN 1617-9447/$ 2.50
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of the open unit disk. Our results could be considered an answer to a question raised by C. C. Cowen on p. 92 of [5].
1. Notation and formulation of the main results The set of all functions f analytic in the open unit disk D = {z ∈ C : |z| < 1} and satisfying the condition f (D) ⊂ D is denoted by P. Obviously, P is a topological semigroup with respect to the operation of composition and the topology of locally uniform convergence in D. Denote by I the set of all the invertible elements of P, i.e., all the automorphisms of D. In other words, the set I consists of M¨obius transformations of the open unit disk D onto itself. Definition 1. Let f be in P. We say that f is embeddable if there exists a family {f t }t≥0 in P such that: (i) f 0 (z) = z and f 1 (z) = f (z); (ii) f t+s (z) = f t ◦ f s (z) for s, t ≥ 0; (iii) f t (z) → z locally uniformly with respect to z ∈ D as t → 0+ . Note that the action t 7→ f t is a continuous homomorphism taking the additive semigroup R+ = {t ∈ R : t ≥ 0} into P. Thus it is a one-parameter continuous semigroup in P. Regarding iterates of an analytic function as a dynamical system, we have to consider the nature of the fixed points of the given function. In this connection an important role is played by a classical result of Denjoy [7] and Wolff [25] which asserts that for each f ∈ P \ I there exists a unique point q, |q| ≤ 1, such that f n (z) → q locally uniformly with respect to z ∈ D as n → ∞. Moreover, if q ∈ D, then f (q) = q. In the case where |q| = 1, there exist the angular limits f (q) := lim f (z) z→q
and
f 0 (q) := lim f 0 (z) z→q
0
with f (q) = q and 0 < f (q) ≤ 1. In the literature (see, for example, [21] and [22]) q is called the Denjoy-Wolff point of the function f . It is easy to see that if q is the Denjoy-Wolff point of a function f ∈ P \ I, then q is also the Denjoy-Wolff point for all its iterates. So it is natural to decompose P into subsemigroups P[q], |q| ≤ 1, where P[q] is the collection of all functions f in P that have q as their Denjoy-Wolff point. Let f be in P and let l ∈ I. Then f is embeddable if and only if g = l ◦ f ◦ l−1 is embeddable. Therefore we can restrict our attention to P[0] and P[1]. In addition, note that each element in I is embeddable in a one-parameter group of automorphisms. Theorem 1. Let f be in P[0] \ I and let f 0 (0) = γ 6= 0. Then f is embeddable if and only if there exists a solution F of the functional equation (1)
F ◦ f (z) = γF (z)
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which is a holomorphic function in D satisfying the following condition: zF 0 (z) p(0) (2) = F (z) p(z) where p is holomorphic in D, Re p(z) > 0 for z ∈ D and (3)
e−p(0) = γ.
Observe that if f 0 (0) = 0, then f is not embeddable. A different form of Theorem 1 was presented in [17]. Theorem 2. Let f be in P[1] \ I. Then f is embeddable if and only if there exists a solution F of the functional equation (4)
F ◦ f (z) = F (z) + 1
which is a holomorphic function in D satisfying the following condition: Re (1 − z)2 F 0 (z) > 0 for z ∈ D.
The functional equation (1) is called Schr¨ oder’s equation while (4) is called Abel’s equation. Koenigs constructed a solution of the Schr¨oder equation in the following way (see [24], §44). Let f be in P[0] \ I and f 0 (0) = γ 6= 0. Then the limit f n (z) (5) K(z) := lim n→∞ γ n exists and is a holomorphic function in D. This function K is known as the Koenigs function and is a solution of Schr¨oder’s equation (1), as well as of the equations K ◦ f n (z) = γ n K(z) for all n = 1, 2, . . . . It is obvious that equation (1) has a unique solution in the class of holomorphic functions in D normalized by F (0) = 0 and F 0 (0) = 1. Therefore we can assert that f ∈ P[0] \ I with f 0 (0) = γ 6= 0 is embeddable if and only if its Koenigs function (5) satisfies hypotheses (2) and (3) of Theorem 1. In the case f ∈ P[1] \ I with f 0 (1) = µ < 1, there exists the limit |1 − f n (0)| (6) Q(z) = lim n→∞ 1 − f n (z) which is a holomorphic function in D and is also sometimes called the Koenigs function of f (cf. [24], §§45, 46). For n = 1, 2, . . . we have Q ◦ f n (z) = M n Q(z), where M = 1/µ. This means, in particular, that Q is a solution of the Schr¨oder equation Q ◦ f (z) = M Q(z).
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Theorem 3. Let f be in P[1] \ I and let f 0 (1) = µ < 1. Then f is embeddable if and only if the Koenigs function (6) satisfies the following condition: 0 2 Q (z) Re (1 − z) >0 Q(z) for z ∈ D. Our approach is based on the infinitesimal description of one-parameter semigroups in P. In this way one can formulate additional embeddability criteria. Theorem 4. Let f be in P[0] \ I. Then f is embeddable if and only if there exists a solution v of the equation (7)
v ◦ f (z) = v(z)f 0 (z)
which is a holomorphic function in D satisfying the following conditions: v(0) = 0,
0
f 0 (0) = ev (0)
and Re
v(z) 0 (1 − z)2 for z ∈ D, and Z f (0) dz > 0. v(z) 0 The last two theorems have been announced in [10].
2. Functional equations and embeddability In this section we prove Theorem 1 and Theorem 2. Before starting the proofs, we recall some results on infinitesimal generators of subsemigroups of the semigroup P. Let {f t }t≥0 be a one-parameter continuous semigroup in P. It is known that the mapping t 7→ f t (z) is infinitely differentiable with respect to t. The derivative ∂ t f (z) = v(z) ∂t t=0
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is a holomorphic function in D and is called the infinitesimal generator of the one-parameter semigroup {f t }t≥0 or an infinitesimal transformation of the semigroup P. The infinitesimal generator v fully determines the one-parameter semigroup via the differential equation ∂ t (8) f (z) = v f t (z) . ∂t Equation (8) and the semigroup property f t+s (z) = f t ◦ f s (z) also yield the equation ∂ ∂ t f (z) = v(z) f t (z). ∂t ∂z In addition, equation (8) and the uniqueness of the solution to the Cauchy problem dw = v(w), w = z, dt t=0 t imply the univalence of the functions f in D. Thus, a necessary condition for the embeddability of f is that f be univalent. In particular, if f is embeddable, then f 0 (z) 6= 0 in D. (9)
It is known that all the fractional iterates f t , t ≥ 0, have the same DenjoyWolff point. Therefore it is natural to study one-parameter semigroups in P[q], |q| ≤ 1. The set of all the infinitesimal generators v of such semigroups forms a convex cone K(P[q]) in the vector space H of all functions holomorphic in D. A description of K(P[q]) is given by the formula (10)
v(z) = (q − z)(1 − qz)p(z),
where p is a holomorphic function with nonnegative real part in D. Note that (10) is known as the formula of Berkson and Porta [4]. In the case q = 0 the formula was obtained by L¨owner in his famous paper [19]. Proof of Theorem 1. Suppose f ∈ P[0] \ I is embedded in a one-parameter semigroup {f t }t≥0 in P[0]. By (10), its infinitesimal generator admits a representation of the form v(z) = −zp(z), where p is a holomorphic function with positive real part in D. We now define the function Z F (z) := z exp 0
z
p(0) − p(ζ) dζ . ζp(ζ)
This function is well defined because p does not vanish in D. Moreover, p(0) zF 0 (z) = . F (z) p(z)
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For fixed z ∈ D, the function ζ(t) = F (f t (z)) is defined on [0, ∞). Note that (8) implies that dζ ∂ = F 0 (f t (z)) f t (z) = −f t (z)p(f t (z))F 0 (f t (z)) dt ∂t = −p(0)F (f t (z)) = −ζp(0). Hence ζ(t) is a solution of the Cauchy problem dζ = −p(0)ζ, ζ = F (z). dt t=0 Solving this problem, we obtain ζ(t) = e−tp(0) F (z). Thus, for z ∈ D and t ≥ 0, we have F (f t (z)) = e−tp(0) F (z). For t = 1 we get F ◦ f (z) = e−p(0) F (z). Differentiating this equality and noting that F (0) = 0, F 0 (0) = 1, we obtain f 0 (0) = e−p(0) . This means that F is the required solution of Schr¨oder’s equation (1). Conversely, let f be in P[0] \ I, f 0 (0) = γ 6= 0, and let F be a solution of (1) satisfying (2) for some function p which is holomorphic in D with Re p(z) > 0, z ∈ D, and e−p(0) = γ. It follows from (2) that zF 0 (z) 1 Re p(0) = Re >0 F (z) p(z) for z ∈ D. Therefore (see, for example, [20], §6.3) F is seen to be a spirallike function. This means that, together with each point w0 , the image domain F (D) contains the logarithmic spiral w(t) = w0 e−tp(0) , t ≥ 0. This geometric property of F allows us to define the family {g t }t≥0 by g t (z) := F −1 e−tp(0) F (z) for all t ≥ 0. Note that {g t }t≥0 is a one-parameter semigroup in P. Indeed, g t+s (z) = F −1 e−(t+s)p(0) F (z) = F −1 e−tp(0) e−sp(0) F (z) = F −1 e−tp(0) F ◦ F −1 e−sp(0) F (z) = F −1 e−tp(0) F (g s (z)) = g t ◦ g s (z).
On the other hand, for t = 1 we have e−p(0) = γ and g 1 (z) = F −1 (γF (z)) = f (z).
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This completes the proof of Theorem 1. Proof of Theorem 2. Let f ∈ P[1] \ I is embedded in a one-parameter semigroup {f t }t≥0 in P[1]. Its infinitesimal generator can be represented in the form v(z) = (1 − z)2 p(z), where p is a holomorphic function with positive real part in D. Define Z z dζ F (z) := . 2 0 (1 − ζ) p(ζ) Since F 0 (z) =
1 , (1 − z)2 p(z)
we have 1 >0 Re (1 − z)2 F 0 (z) = Re p(z) for z ∈ D. For fixed z ∈ D we now consider ζ(t) = F (f t (z)), t ≥ 0. Using (8) and the representation of v, we see that dζ ∂ = F 0 (f t (z)) f t (z) = F 0 (f t (z))(1 − f t (z))2 p(f t (z)) = 1. dt ∂t From this relation and the initial condition ζ(0) = F (z) it follows that F (f t (z)) = F (z) + t. This equality means that F is the required solution of Abel’s equation (4). Conversely, suppose that f is in P[1] \ I and that F is a holomorphic function in D satisfying equation (4) and Re((1−z)2 F 0 (z)) > 0 for z ∈ D. Noting that the function F (z)−F (0) satisfies the same conditions, we can assume that F (0) = 0. Further, the function G(z) = z/(1 − z) is convex and F 0 (z) Re 0 = Re (1 − z)2 F 0 (z) > 0 G (z) for z ∈ D. Therefore F is a close-to-convex univalent function. Moreover, it follows from [11] that, together with each point w0 , the image domain F (D) contains the ray w(t) = w0 + t, t ≥ 0. This geometric property of F allows us to define the family {g t }t≥0 by g t (z) := F −1 (F (z) + t) for t ≥ 0. Also, {g t }t≥0 is a one-parameter semigroup in P and g 1 (z) = F −1 (F (z) + 1) = f (z). The proof of Theorem 2 is complete.
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3. The Koenigs function In the previous section we have seen how the embeddability problem is related to the solution of functional equations. The classical functional equations of Schr¨oder and Abel have been studied by many authors for various classes of functions. In this section we prove Theorem 3 by using the well-known limit technique introduced by Koenigs to solve Schr¨oder’s equation. Proof of Theorem 3. Let f ∈ P[1] \ I with f 0 (1) = µ < 1 be embedded in a one-parameter semigroup {f t }t≥0 in P[1]. As we have seen above, its infinitesimal generator v admits the representation v(z) = (1 − z)2 p(z), where p is a holomorphic function with positive real part in D. We can write the function p in the form 1 + ω(z) p(z) = , 1 − ω(z) where ω is a holomorphic function satisfying the condition |ω(z)| < 1 for z ∈ D. It follows by the Julia-Carath´eodory Theorem that there exist the angular limits v(1) = lim v(z) = 0
and
z→1
v 0 (1) = lim
z→1
v(z) z−1
with v 0 (1) ≤ 0 (see, for example, [1], §1.4). By the continuous analog of the Denjoy-Wolff Theorem [4], f t (z) → 1 as t → ∞, locally uniformly in D. We now want to show that v(f t (z)) (11) lim t = v 0 (1) t→∞ f (z) − 1 for z ∈ D. To verify (11), we just have to prove that f t (z) tends to 1 nontangentially (in a Stolz angle). Indeed, since f 0 (1) = µ < 1, if ∆ is a compact subset of D, then (see [6], p. 83, and [24], §44) the set ∞ [
f n (∆)
n=1
lies in some nontangential approach region |1 − w| ≤β , w∈D: 1 − |w| 1 ≤ β < ∞. Since ∆1 =
[
0≤t≤1
f t (∆)
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is also compact and, by the semigroup property, ∞ [ [ t f (∆) = f n (∆1 ), t≥0
the set
S
n=1
t
t≥0
f (∆) is also contained in such a region. Thus (11) is proved.
Further, equating the right-hand sides of equations (8) and (9), we arrive at the relation 0 v ◦ f t (z) = v(z) f t (z), where (f t )0 (z) = (∂/∂z)f t (z), for all t ≥ 0. From this and (11), it follows that 0
(f t ) (z) 1 v (f t (z)) v 0 (1) = lim = − . t→∞ 1 − f t (z) v(z) t→∞ 1 − f t (z) v(z) On the other hand, by the Weierstraß Theorem, it follows from the definition of the Koenigs function (6) that (f n )0 (z) 1 Q0 (z) lim = − = . n→∞ |1 − f n (0)| Q(z) (Q(z))2 Now we have Q0 (z) |1 − f n (0)| (f n )0 (z) (f n )0 (z) v 0 (1) = lim = lim = − . n→∞ 1 − f n (z) |1 − f n (0)| n→∞ 1 − f n (z) Q(z) v(z) Consequently, 0 (1 − z)2 1 2 Q (z) = −v 0 (1) Re = −v 0 (1) Re >0 Re (1 − z) Q(z) v(z) p(z) for z ∈ D. lim
Conversely, let f be in P[1] with f 0 (1) = µ < 1 and suppose the conditions of the theorem are satisfied for its Koenigs function Q. We recall that the Koenigs function Q satisfies Schr¨oder’s equation 1 Q ◦ f (z) = Q(z). µ Indeed, |1 − f n+1 (0)| Q(z) = lim n→∞ 1 − f n+1 (z) |1 − f n (0)| f (f n (0)) − 1 = lim n→∞ 1 − f n (f (z)) f n (0) − 1 = µQ ◦ f (z). Further, noting that Q takes its values in the right half-plane, we can define a single-valued branch of ln Q(z) and consider the function F (z) = −
ln Q(z) . ln µ
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Schr¨oder’s equation in terms of the function F is then transformed into Abel’s equation F ◦ f (z) = F (z) + 1. Moreover, 0 1 2 0 2 Q (z) Re (1 − z) F (z) = − Re (1 − z) >0 ln µ Q(z) for z ∈ D. Thus F is a solution of Abel’s equation (4) which satisfies the conditions of Theorem 2. From this it follows that f is embeddable and the proof is complete. The following result can be considered a complement of Theorem 1. Corollary 1. Let f be in P[1] \ I with f 0 (1) = µ < 1. Then f is embeddable if and only if there exists a solution Q of Schr¨ oder’s functional equation 1 Q ◦ f (z) = Q(z) µ which is an analytic function in D satisfying the conditions Q(z) 6= 0 and 0 2 Q (z) Re (1 − z) >0 Q(z) for z ∈ D. Proof. In the case where f is embeddable, the required solution of Schr¨oder’s equation is given by the Koenigs function. Conversely, let f be in P[1] with f 0 (1) = µ < 1 and let Q be a solution of Schr¨oder’s equation with the properties described above. Since Q(z) 6= 0 in D, we can define the function ln Q(z) F (z) = − . ln µ It is obvious that F is a solution of Abel’s equation with Re (1 − z)2 F 0 (z) > 0. By Theorem 2 it follows that f is embeddable.
4. The equation v ◦ f (z) = v(z)f 0 (z) Observe first that each element f t of a one-parameter semigroup {f t }t≥0 in P satisfies the equality 0 v ◦ f t (z) = v(z) f t (z), where v is the infinitesimal generator. Thus, in the case where f is embeddable, the infinitesimal generator v of the corresponding one-parameter semigroup is a
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solution of (7). On the other hand, if v is a solution of (7) with f ∈ P (not necessary embeddable), then for n = 2, 3, . . . we have v ◦ f n (z) = v f f n−1 (z) = v f n−1 (z) f 0 f n−1 (z) = v f n−2 (z) f 0 f n−2 (z) f 0 f n−1 (z) = . . . = v(z)f 0 (z)f 0 (f (z)) · · · , f 0 f n−1 (z) = v(z) (f n )0 (z).
In other words, v is simultaneously a solution of (7) for all the iterates f n , n = 2, 3, . . .. We recall that in the study of the embeddability problem, equation (7) was first used by T. E. Harris [14]. Proof of Theorem 4. Assume that f ∈ P[0] \ I and that it can be embedded in a one-parameter semigroup {f t }t≥0 with the infinitesimal generator v. As we already know, v is a solution of (7). Moreover, it follows from (10) that v(z) = − Re p(z) > 0 z for z ∈ D. Further, differentiating (8) with respect to z we obtain 0 0 ∂ f t (z) = v 0 f t (z) f t (z). ∂t t 0 Setting ζ(t) = (f ) (0), we have Re
dζ = ζv 0 (0) dt
and
ζ(0) = 1.
This implies that 0
ζ(t) = etv (0) and 0
f 0 (0) = ζ(1) = ev (0) . Conversely, let f ∈ P\I and let v be a solution of (7) satisfying the conditions of the theorem. In particular, p(z) = −v(z)/z is a holomorphic function with positive real part in D. Once again, it follows by the Berkson-Porta representation that v(z) (= −zp(z)) is the infinitesimal generator of a one-parameter semigroup {g t }t≥0 in P[0]. As we have already seen, for t ≥ 0 the equality 0 0 g t (0) = etv (0) holds. From this and the condition of the theorem we obtain (g 1 )0 (0) = f 0 (0). Since v(0) = 0 and v 0 (0) 6= 0, equation (7) with respect to f has a unique solution with a given value of the derivative at z = 0 (see [8]). Thus, g 1 = f and f is embeddable.
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Proof of Theorem 5. Let f ∈ P[1] \ I be embedded in a one-parameter semigroup {f t }t≥0 with infinitesimal generator v(z) = (1 − z)2 p(z), where p is a holomorphic function with positive real part in D. We can define Z z dζ (12) F (z) = . 0 v(ζ) As we have seen in the proof of Theorem 2, for t ≥ 0 the equality F f t (z) = F (z) + t holds. Putting t = 1 and z = 0, we obtain F (f (0)) = 1. On the other hand, F (f (0)) =
Z
f (0) 0
F (z) dz =
0
Z 0
f (0)
dz . v(z)
Thus, v is a solution of (7) satisfying the conditions of the theorem. Conversely, suppose that f ∈ P[1] \ I and that v is a solution of (7) such that Re
v(z) >0 (1 − z)2
for z ∈ D and Z 0
f (0)
dz = τ > 0. v(z)
The first inequality implies that v is an infinitesimal transformation of the semigroup P[1]. Let {g t }t≥0 be the one-parameter semigroup with infinitesimal generator v. Now we can, once again, define the function F by (12). It follows from the previous argument that Z f (0) Z f (0) dz τ = F 0 (z) dz = F (f (0)). F (g (0)) = τ = v(z) 0 0 Since F is univalent, we obtain f (0) = g τ (0). It remains to be observed that equation (7) with respect to f has in this case a unique solution with a given value f (0). Thus f = g τ and the theorem is proved. Acknowledgement. Part of this research was carried out when V. Goryainov was visiting the Departments of Mathematics at Braude College and at the Technion. He thanks both Departments for their hospitality.
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Mark Elin E-mail: elin
[email protected] Address: Department of Mathematics, Braude College, 21982 Karmiel, Israel Victor Goryainov E-mail: goryainov
[email protected] Address: Department of Economics and Mathematics, Volzhsky Institute of Humanities of Volgograd State University, 404133 Volzhsky, Russia Simeon Reich E-mail:
[email protected] Address: Department of Mathematics, The Technion — Israel Institute of Technology, 32000 Haifa, Israel David Shoikhet E-mail:
[email protected] Address: Department of Mathematics, Braude College, 21982 Karmiel, Israel
