Iteration of meromorphic functions

Theorem 10 or Corollaries 1 or 2 are certainly known to those who work in the field, but they do not seem to have been stated explicitly before. As already mentioned, there are beautiful computer graphics related to the the- ory, and there are many places (besides [112]) where such pictures can be found for rational functions. Although Julia sets (and bifurcation diagrams) of transcenden- tal functions can compete in their beauty and complexity very well with those of rational functions, this art

Lemma 1. If f is rational, f ∈P, or f ∈E, then F(f) = F(f n) and J(f) = J(f n) for all n ≥2. Here we have to exclude f ∈M, because then f n is not meromorphic in C so that F(f n) and J(f n) are not defined. (There is, of course, a natural way to define F(f n) for f ∈M and n ≥2, or, more generally, to define F(f) for functions f meromorphic in C except for countably many points. Then the conclusion of

Lemma 1 holds for such functions.)

Lemma 2. F and J are completely invariant. Here, by definition, a set S is called completely invariant if z ∈S implies that f(z) ∈S, unless f(z) is undefined, and that w ∈S for all w satisfying f(w) = z.

Lemma 3. Either J = bC or J has empty interior. We note that the case J = bC is actually possible. Examples of rational functions with this property are given by the rational functions that come from the multipli- cation theorems of elliptic functions. Usually, Latt`es ([98], see also [90]) is credited with having introduced them into the subject, but it should perhaps be mentioned that already B¨ottcher [38, p. 63] was aware of these examples. For other examples of rational functions satisfying

Lemma 4. If z0 ∈J is not exceptional, then J = O−(z0). Similar to the backward orbit O−(z0), we define the forward orbit O+(z0) of z0 ∈bC by O+(z0) = S n≥0 f n(z0). Of course, here the union is taken only over those n ≥0 for which f n(z0) is defined. The orbit O(z0) of z0 is defined by O(z0) = O+(z0) ∪O−(z0). For a subset S of bC, we put O±(S) = S z∈S O±(z) and O(S) = S z∈S O(z). With this terminology, Lemma 2 may be written in the form O(F) ⊂F and O(J) ⊂J. Another simple consequence of Montel’s th

Theorem 1. An entire transcendental function has infinitely many periodic points of period n for all n ≥2. The idea of the proof is similar. Instead of Picard’s theorem, however, Rosen- bloom used something stronger, namely, Nevanlinna’s theory on the distribution of values, which may be considered as a quantitative version of Picard’s theorem. Since this is the only place in this paper where we use Nevanlinna theory, we do not give an introduction to it but refer to [79, 87, 111] for notation an

Theorem 2. If f is a transcendental meromorphic function and n ≥2, then f has infinitely many periodic points of minimal period n. As already mentioned, this also holds for n = 1 if f ∈P, but not in general if f ∈E or f ∈M.

Theorem 2 was proved in [29] if f ∈E and in [36, Chapter 5.2] if f ∈P. A proof for f ∈M, using the ideas of [30], is as follows. Suppose first that f n−1 has at least three poles p1, p2, p3. Define g = f n−1 and denote by mj the order of the pole pj. There exist functions hj, defined and analytic in a neighborhood of 0, such that g(pj + hj(z)) = z−mj. Define k1(z) = p1+h1(zm2m3), k2(z) = p2+h2(zm1m3), and k3(z) = p3+h3(zm1m2) so that g(kj(z)) = z−m1m2m3. Suppose now that f does not have periodic poi

Theorem 2 in §3.4. Its proof, however, will be less elementary (but still short). 3.3. The Julia set is perfect. The results concerning the existence of periodic points may be used to prove that J ̸= ∅. More generally, we have the following result.

Theorem 3. Let f be a meromorphic function. Then J(f) is perfect. Recall that a set is called perfect if it is closed, nonempty, and does not contain isolated points. We first prove that J is not empty and in fact an infinite set. There are essentially two ways to do this if f is rational. One method is to assume that f nj →φ uniformly in bC. Then φ must also be rational and deg(φ) = limj→∞deg(f nj). But deg(f nj) = (deg(f))nj →∞as j →∞, provided deg(f) ≥2, a contradiction. The other method (which

Theorem 6 in §4.2.) The proof that J is perfect can now be carried out as in the rational case. 3.4. Julia’s approach. So far our development of the theory has followed Fatou’s ideas. Julia based his theory on the closure of the set of repelling periodic points. One of the basic results of the theory is that these two sets are actually equal.

Theorem 4. Let f be a meromorphic function. Then J(f) is the closure of the set of repelling periodic points of f. For rational f, this result was obtained by both Fatou [71, §30, p. 69] and Julia [89, p. 99, p. 118]. Their proofs, however, were different. (A good exposition of both proofs can be found in [108, §11].) Fatou proved first that any point in J is the limit point of periodic points and then that there are only finitely many nonrepelling periodic points, which together implies the result

Lemma 5. Let f be a transcendental meromorphic function, and let D1, D2, . . . , D5 be five simply connected domains in C with disjoint closures. Then there exists j ∈

Lemma 6. Suppose that f ∈M and that z1, z2, . . . , z5 ∈O−(∞)\{∞} are distinct. Define nj by f nj(zj) = ∞. Then there exists j ∈{1, 2, . . ., 5} such that zj is a limit point of repelling periodic points of minimal period nj + 1. If f has only finitely many poles, then “five ” may be replaced by “three”. To deduce Lemma 6 from Lemma 5, we choose the Dj as discs around zj where the radii are chosen so small that the Dj do not contain critical points of f and that their closures are pairwise disjoint

Lemma 6 yields immediately that the conclusion of Theorem 4 holds for f ∈M because J = O−(∞) is perfect. Another interesting consequence of Lemma 6 is that if f −n+1(∞) contains more than four elements, then f has infinitely many repelling periodic points of minimal period n. In particular, this is the case if f ∈M and n ≥4. We also see that f has infinitely many repelling periodic points of minimal period 2 and 3 if f has more than two poles. On the other hand, it was proved in [29] that if f is

Theorem 5. If f is a transcendental meromorphic function and n ≥2, then f has infinitely many repelling periodic points of minimal period n. We remark that in view of Lemma 1, Julia’s method can be used to obtain

Theorem 4 from Theorem 5. This does not, however, constitute a new proof of

Theorem 4, because the argument in [29] also uses Ahlfors’s theorem. The fact that all known proofs of the existence of repelling periodic points are based on this deep result makes Julia’s approach to start with the closure of the set of repelling periodic points inadequate for transcendental functions, because it is difficult to see that this set is not empty. It would be of interest to give a more elementary proof of Theorem 4. Question 2. Is there a proof of Theorem 4 which does not use Ahlfor

Theorem 6. Let U be a periodic component of period p. Then we have one of the following possibilities: • U contains an attracting periodic point z0 of period p. Then f np(z) →z0 for z ∈U as n →∞, and U is called the immediate attractive basin of z0. • ∂U contains a periodic point z0 of period p and f np(z) →z0 for z ∈U as n →∞. Then (f p)′(z0) = 1 if z0 ∈C. (For z0 = ∞we have (gp)′(0) = 1 where g(z) = 1/f(1/z).) In this case, U is called a Leau domain. • There exists an analytic homeomorphism φ

Theorem 7. Let f be a meromorphic function, and let C = {U0, U1, . . . , Up−1} be a periodic cycle of components of F. • If C is a cycle of immediate attractive basins or Leau domains, then Uj ∩ sing(f −1) ̸= ∅for some j ∈{0, 1, . . ., p −1}. More precisely, there exists

Theorem 8. Let f be a meromorphic function, and let U be an invariant compo- nent of F. Then the connectivity of U has one of the values 1, 2, or ∞. Here 2 occurs only when U is a Herman ring.

Theorem 9. If f ∈E, then any preperiodic component of F is simply connected. In other words, multiply connected components of F are necessarily wandering if f ∈E.

Theorem 9 is an immediate consequence of a result of Baker [13, Theorem 1], who proved that multiply connected components of F(f) are bounded if f ∈E. In order to give a proof of Theorem 9 (following Baker’s argument), we start with the following lemma. With further applications in mind, this lemma is stated in a form more general than needed for the proof of Theorem 9. The results contained in it can be found in [20, Lemmas 1 and 2; 23, Lemma 4.1] (see also [15, Theorem 6; 103, Proposition A.1]

Lemma 7. Let G be an unbounded open set in C with at least two finite boundary points, and let g be analytic in G. Let D be a domain contained in G, and suppose that gn(D) ⊂G for all n and that gn|D →∞as n →∞. Then, for any compact subset K of D, there exist constants C and n0 such that |gn(z′)| ≤|gn(z)|C (2) for all z, z′ ∈K and n ≥n0. If, in addition, g(D) ⊂D, then we also have log log |gn(z)| = O(n) (3) for all z ∈D as n →∞, and there exist a constant A > 1 and a curve γ ⊂D tending to ∞which s

Theorem 10. Suppose that f ∈E and that for all ε > 0 there exists a curve γ tending to ∞such that |f(z)| ≤M(|z|ε, f) for z ∈γ. Then all components of F are simply connected. In particular, this is the case if log |f(z)| = O(log |z|) as z →∞ through some path. On the other hand, examples of entire functions with multiply connected compo- nents of the Fatou set are known. The first example was constructed in [7]; further examples can be found in [14, 17, 84]. Baker [17] gave an example of a transce

Theorem 11. Rational functions do not have wandering domains. Together with Theorems 6 and 7, this leads to a fairly complete description of the iterative behavior of rational functions on the Fatou set. Sullivan’s theorem has been extended to various classes of transcendental func- tions. We mention the following classes: • S = {f : f has only finitely many critical and asymptotic values}; • F = {f : f has a representation of the form f(z) = z + r(z)ep(z) where r is rational and p is a polynomia

Theorem 12. Functions in S, F, N, and R do not have wandering domains. We note that all these classes contain the class of rational functions so that

Theorem 12 may be considered as a generalization of Theorem 11. The result that meromorphic functions in S do not have wandering domains was proved by Baker, Kotus, and L¨u [24]. This result had been obtained earlier by Eremenko and Lyubich [67, 70] and Goldberg and Keen [75] for S ∩E and by Keen [91], Kotus [94], and Makienko [104] for S ∩P (and, in fact, for the corresponding class of analytic self- maps of the punctured plane). For other subclasses of S, this had been proved by Baker [16, The

Theorem 13. Let f be a meromorphic function, and let {U0, U1, . . . , Up−1} be a periodic cycle of Baker domains of f. Denote by zj the limit corresponding to Uj, and define zp = z0. Then zj ∈Sp−1 n=0 f −n(∞) for all j ∈{0, 1, . . ., p −1}, and zj = ∞for at least one j ∈{0, 1, . . ., p −1}. If zj = ∞, then zj+1 is an asymptotic value of f.

Corollary 1. If f has a cycle {U0, U1, . . . , Up−1} of Baker domains such that f n|U0 →∞, then ∞is an asymptotic value of f. In particular, this is the case if f has an invariant Baker domain.

Corollary 2. If f has a cycle {U0, U1, . . . , Up−1} of Baker domains such that f n|U0 ̸→∞, then f has a finite asymptotic value.

Corollary 1 can be found in [59, p. 75] for maps with polynomial Schwarzian derivative.

Lemma 7 gives additional information about the asymptotic paths γj and also answers the question how fast f np(z) approaches zj for z ∈Uj. In fact, if Uj, zj, and γj are as above, then |z|1/A ≤|f p(z)| ≤|z|A for z ∈γj and log log |f pn(z)| = O(n) for z ∈D if zj = ∞. If Uj is simply connected and zj = ∞, then we even have |z|/A ≤|f p(z)| ≤A|z| for z ∈γj and log |f pn(z)| = O(n) for z ∈D. Similar results may be obtained if zj ̸= ∞. As already mentioned after Theorem 7, periodic cycles of Baker dom

Theorem 14. If f ∈N or f ∈F, then any periodic cycle of Baker domains contains a point of sing(f −1). This result was proved in [31] for f ∈N, but the proof extends to the case that f ∈F. The proof of Theorem 14 is fairly analogous to the proof that functions in N (and F) do not have wandering domains. Therefore, it seems likely that the conclusion of Theorem 14 remains valid for functions in R. (This is certainly so for cycles of simply connected Baker domains, but in the multiply connected cas

Theorem 2] has shown that in certain cases this argument may also be used to prove that Baker domains contain singularities of f −1. 4.8. Classes of functions without Baker domains. Eremenko and Lyubich [70] considered the class B = {f : sing(f −1) is bounded} and proved the following result.

Theorem 15. If f ∈E ∩B, then there does not exist a component U of F(f) such that f n|U →∞as n →∞.

Corollary 3. If f ∈E ∩B, then f does not have Baker domains. We note that the conclusion of Corollary 3 does not hold in general for f ∈M∩B. As an example, consider f(z) = 1/z −ez. As already mentioned above, Baker, Kotus, and L¨u [23, p. 606] proved that f has a Baker domain of period 2, and it is easy to check that f ∈M ∩B. In this example, the critical values of f accumulate at 0, which is also one of the limits corresponding to the cycle of Baker domains. The following result is a generaliza

Theorem 16. Let f be a meromorphic function, and let {U0, U1, . . . , Up−1} be a periodic cycle of Baker domains of f. Then ∞is in the derived set of p−1 [ j=0 f j(sing(f −1)).

Corollary 4. Functions in S do not have Baker domains. Combining Corollary 4 with Theorem 12, we see that the iteration of functions in S is in many ways analogous to that of rational functions and may thus be analyzed in a similar way. For example, these results allow us to prove that the functions λzez, λez/z, and λ tan z satisfy J = bC for certain values of λ, as mentioned in §2.2. In fact, all these functions are in S and hence do not have wandering or Baker domains by Theorem 12 and Corolla

Lemma 8. Suppose f ∈B, p ≥1, and 0 /∈O−(∞). Then there exist a positive constant R and a curve Γ connecting 0 and ∞such that |f p(z)| ≤R for z ∈Γ. We show first that if r is sufficiently large, then there exists a curve Γ connecting ∞with some point in C such that |f(z)| = r for z ∈Γ. In fact, otherwise the components of f −1(D(0, r)) are bounded for arbitrarily large r. Hence we can find r1 and r2 satisfying 0 < r1 < r2 and sing(f −1) ⊂D(0, r1) such that f −1(D(0, r2)) has a bounded component which

Theorem 17. If f ∈E, then f has at most one completely invariant domain. It is easy to find transcendental entire functions which have a completely invari- ant domain, for example, f(z) = λez has this property if 0 < λ < 1/e. We mention the following question of Baker. Question 12. Suppose f ∈E has a completely invariant domain U. Do we have F(f) = U? Some results supporting the conjecture that the answer is “yes” can be found in [13, Theorem 2] and [70, §6]. In particular, it is shown in [70, Th

Theorem 2; 70, Lemma 11]. Less is known about completely invariant domains of meromorphic functions. The example f(z) = tan z where J = R∪{∞} and where the upper and lower half- plane are completely invariant shows that there may be two completely invariant domains. Question 13. Let f be a meromorphic function. Can f have more than two completely invariant domains? A partial result was obtained by Baker, Kotus, and L¨u [23, Theorem 4.5].

Theorem 18. If f ∈S, then f has at most two completely invariant domains. If the answer to Question 13 is “no”, one may also ask the following question. Question 14. Suppose a meromorphic function f has two completely invariant domains U1 and U2. Do we have F(f) = U1 ∪U2?

Theorem 19. If f ∈E, then J(f) contains nondegenerate continua. For rational functions it is possible that the Julia set is a circle or a straight line; for example, J( 1 2(z −1/z)) = R ∪{∞}. This may also happen for f ∈M. In fact, we have J(λ tan z) = R ∪{∞} if λ ≥1; see [59, pp. 60–61]. For f ∈E this is impossible, as shown by the following result of T¨opfer [133, §3].

Theorem 20. If f ∈E, then J(f) does not contain isolated Jordan arcs. Here, by definition, a Jordan arc is called isolated (in J) if there exists an open set which contains the arc except for its endpoints but no other point of J. To prove Theorem 20, we suppose that such an arc exists and is parametrized by γ : [0, 1] →C. By Theorem 4, the repelling periodic points are dense in this arc. In view of Lemma 1 we may suppose that it contains a fixed point, say, f(z1) = z1 where z1 = γ(t1), t1 ∈(0, 1)

Theorem 21. If f ∈E, then I(f) ̸= ∅. Eremenko also shows that I(f) ∩J(f) ̸= ∅. The proof of Theorem 21 is based on the Wiman-Valiron theory about the behavior of entire functions near points of maximum modulus; see for example [80, 135]. Once Theorem 21 is known, it is not difficult to prove that (10) holds for f ∈E as well. In particular, if an entire function f does not have Baker domains (for example, if f ∈B ∩E), then we have J(f) = I(f). We mention two questions asked by Eremenko [66, pp. 343

Theorem 22. Let g be a polynomial, and let f be defined by (11). Denote by z1, z2, . . . , zm the zeros of g′′ that are not zeros of g′. If f n(zj) converges for all j ∈{1, 2, . . ., m}, then f n(z) converges to zeros of g for all z ∈F(f). The proof of Theorem 22 we have sketched above depends on Theorem 11. It is possible, however, to give a more elementary proof of Theorem 22. In fact, this result can be deduced from the work of Fatou [71, §30–31] and Julia [89, §59] (see also Smale [123, pp. 9

Theorem 23. If g has the form (12) but is not of the form g(z) = eaz+b where a and b are constant, then the conclusion of Theorem 22 holds. The case g(z) = eaz+b has to be excluded because then f(z) = z −1/a, but we always assumed that f is nonlinear. In fact, the conclusion of Theorem 23 is false in this case. Another class of entire functions where Newton’s method does not lead to wan- dering domains are solutions of differential equations of the form g′′ +pg = 0 where p is a polynomial. In thi

Theorem 24. Let g be a meromorphic function. Suppose that sing(g−1) is a dis- crete subset of C and that 0 is not an asymptotic value of g. Then (15) and (16) hold. It seems likely that the conclusion of Theorem 24 remains valid for more general classes of functions. Question 18. Is the hypothesis on the discreteness of sing(g−1) necessary in The- orem 24?