Exercises for XII.2
1. Show that e1/ z + e- 1/ z omits only the value 00 at z = O.
2. Show that the meromorphic function e1/ z + 1/(1 – e1/ z ) on C\{O}
does not omit any value at z = O.
3. Show that for a> 0 and 0 < (3 < 1, there are constants C(a,(3) > 0 with the following property. If F is a family of analytic functions on the open unit disk that omits the values 0 and 1, and if IJ(O)I < a for all J E F, then IJ(z)1 ::; C(a,(3) for all J E F and Izl ::; (3. Remark. This is Schottky’s theorem.
4. Let J(z) be analytic on the punctured disk {O < Izl < 1}, and define In(z) = J(z/n), n ~ 1. Show that {fn(z)} is a normal family on the punctured disk if and only if the singularity of J(z) at 0 is removable or a pole.
5. Let Eo, Ell E2 be three disjoint compact subsets of the Riemann sphere, and let F be a family of meromorphic functions on a do- main D such that each J E F omits at least one point of each of the three sets. Show that F is a normal family.
6. Let 9 be the family of univalent analytic functions on a fixed domain
D. (a) Show that 9 is not normal. (b) Show that the family of functions in 9 that omit 0 is normal. (c) Show that the family of derivatives of functions in 9 is normal. (d) Show that for fixed Zo E D and M > 0 the family of functions J E 9 satisfying IJ'(zo)1 ::; M
is normal.
7. Let S denote the family of univalent functions J(z) on the open unit disk 1Ol, normalized by J(O) = 0 and IJ'(O)I = 1. (a) Show that there exists K, > 0 such that the image of the open unit disk IOl under any J E S includes the open disk centered at 0 of radius K,.
(b) Show that the function J(z) = z/(1 – z)2 belongs to Sand maps IOl onto the complex plane slit along the negative real axis from -~ to -00. Conclude that K, ::; ~. Remark. The theorem in (a) was first proved by P. Koebe. L. Bieberbach showed that the estimate holds with K, = ~, and this estimate is sharp. The theorem is known as Koebe’s one-quarter theorem, and the function in (b) is referred to as the Bieberbach function.
8. Show that there is a constant (3 > 0 with the following property. If f(z) is an analytic function on the open unit disk II} such that f(O) = 0 and 1′(0) = 1, there is a subdisk DelI} such that f(z) is one-to-one on D and f(D) contains a disk of radius (3. Remark. This is Bloch’s theorem, and the optimal (largest) constant (3 is
Bloch’s constant.
9. Give a proof of Royden’s theorem (Exercise 1.9) based on the Zalcman lemma.
10. A family F of meromorphic functions on a domain D is normal at Zo ED if F is normal on some open disk centered at Zoo Show that F is normal at Zo if and only if whenever {zn} is a sequence in D that converges to Zo and Pn ~ 0, then every sequence in F has a subsequence {fn(z)} for which the corresponding scaled functions
gn(() = fn(zn +Pn() converge normally to a constant (possibly 00).
11. Let F be a family of meromorphic functions on a domain D that is not normal at Zo E D, and suppose that fn E F, Zn ~ zo, and Pn ~ 0 are such that the scaled functions gn(() = fn(zn +Pn() converge normally to a nonconstant meromorphic function g((). Let (0 E C, Wo = g((o).
(a) Show that there is a sequence ~n ~ Zo such that fn(~n) = Woo
(b) Show that if (0 is not a critical point of g((), then f~(~n) ~ 00.
(c) Show that if’lj;(w) is a meromorphic function defined near Wo such that ‘lj;(wo) = zo, then there is a sequence ‘TJn ~ Zo such that ‘TJn is a fixed point of’lj; 0 fn, that is, ‘lj;(fn(‘T/n)) = ‘T/n· (d) Show that if (0 is not a critical point of g(), and if Wo is not a critical point of ‘lj;(w) , then (‘lj; 0 fn)~(‘TJn) ~ 00.
One of the early applications of Montel’s theorem was to complex dynamics, the study of the behavior of the iterates of an analytic or meromorphic function. Julia and Fatou used Montel’s theorem as a key tool for studying the iterates of a rational function. One of their main ideas was to understand the dynamical behavior of the iterates by splitting the extended complex plane C* into two invariant subsets, on one of which (the Fatou set) the iterates are well behaved, and on the other of which (the Julia set)
their behavior is chaotic. We will derive some basic facts about Fatou and Julia sets in this section and the next.
Let U be a domain in the extended complex plane C*, and let f(z) be an analytic map from U to U. In other words, f (z) is an analytic function on U (meromorphic if 00 E U) whose range is contained in U. For the remainder of this chapter it will be convenient to denote the nth iterate f(f(··· (f(z))···)) (n times) of fez) by r(z). Danger! This should not be confused with the nth power f(z)n of fez)
Exercises for XII.3
1. Show that the Julia set .:1 and the Fatou set F of a rational function J(z) satisfy J(.:1) = .J and J(F) = F.
2. Show that the rational function J(z) = z2/(z2 + 1) is conjugate to the quadratic polynomial P( z) = Z2 + 2.
3. For )., J.L =1= 0, the two maps J(z) = )’z and g(z) = J.LZ of C* are conjugate if and only if ). = J.L or ). = 1/J.L. Hint. A conjugation maps fixed points to fixed points.
4. Let Zo be a fixed point of J(z). Define the multiplier of the fixed point Zo to be ). = J'(zo). Show that the multiplier at a fixed point is a conjugation invariant, that is, if ( = cp(z) conjugates J(z) to g((), then the multiplier of g(() at the fixed point cp(zo) is equal to the multiplier of J(z) at zoo
5. A fixed point Zo of J(z) is a repelling fixed point if 1f'(zo)1 > 1.Show that the Julia set of a rational function J(z) contains all its repelling fixed points.
6. Show that the Julia set of a fractional linear transformation is either empty or consists of one fixed point.
7. A fixed point Zo of J(z) is an attracting fixed point if 1f'(zo)1 < 1.
The basin of attraction of zo, denoted by A(zo), is the set of z whose iterates r(z) converge to Zo as n —> 00. Show that if J(z) is a rational function, then A(zo) is an open subset of C* containing Zo whose boundary coincides with the Julia set.
8. Show that if J(z) is a rational function of degree d, then the mth iterate Jm (z) is a rational function of degree dm.
9. Show that the Julia set of a rational function J(z) coincides with the Julia set of its mth iterate Jm(z).
10. A point Zo is a periodic point of J(z) if it is a fixed point of Jm(z) for some m ~ 1. For such a zo, set Zl = J(zo), Z2 = J(Zl), … ,Zm-l = J(Zm-2). Show that each Zj is a fixed point of Jm(z) with the same multiplier A = f'(zo)··· f'(zm-d. Remark. Assuming
that the Zj’S are distinct, we define the multiplier of the cycle {zo, Zl, … ,Zm-l} to be the multiplier of Jm(z) at any of the points of the cycle. The cycle is an attracting cycle if its multiplier A satisfies IAI < 1, and it is a repelling cycle if IAI > 1. The integer m
is the period of the periodic point, or the length of the cycle.
11. Find all repelling cycles of the polynomial J(z) = z2.
12. Show that all repelling cycles of a rational function are contained in its Julia set.
13. Find all attracting cycles of length two of the quadratic polynomial z2 + c. Show that the values of the complex parameter c for which there is an attracting cycle of length two form an open disk.
14. Let J(z) be a rational function. We define the basin of attraction of an attracting cycleof J(z) to consist of the points z E C* whose iterates In(z) accumulate on the cycle as n —> 00. Show that the basin of attraction of an attracting cycle is an open subset of C* whose boundary coincides with the Julia set. Show that different points of an attracting cycle lie in different components of the basin of attraction.
15. We define the multiplicity of a fixed point Zo of J(z) to be the order of the zero of J(z) – z at z = zoo Show that a fixed point has nmultiplicity m ~ 2 if and only if its multiplier is 1. Show that the multiplicity of a fixed point is a conjugation invariant.
16. Show that a fixed point of a rational function fez) of multiplicity m ~ 2 belongs to the Julia set.
17. Show that a rational function of degree d has d + 1 fixed points,counting multiplicity.
18. Define the analytic index of a fixed point Zo f 00 of fez) to be the residue of 1/(z – fez)) at zoo (a) Show that if Zo is a fixed point of fez) with multiplier>. f 1, then the analytic index of fez) at Zo is 1/(1 – >.). (b) Show that if fez) has a fixed point of multiplicity m at ZO, then for any small e > 0, fe(z) = fez) – e has m fixed points
near zo, each of multiplicity one, for which the sum of the analytic indices of fe(z) tends to the analytic index of fez) at zo as e -+ O.
(c) Show that the analytic index of a fixed point is a conjugation invariant.
19. Find the fixed points and their analytic indices for the rational function fez) = (3z2+1)/(z2+3). Determine the Julia set and the Fatou set of fez).
20. Suppose that fez) = z_zm+1+Az2m+1+0(z2m+2) for some integer m ~ 1. Show that the analytic index of fez) at the fixed point z = 0 is A.
21. Suppose that g(z) = z – zm+l + O(zm+2) for some integer m ~ 1.
Show that g(z) can be conjugated near 0 to fez) = z – Zm+l + AZ2m+l +O(z2m+2), where A is the analytic index of fez) at z = O. Hint. ‘Thy conjugating by cp(z) = z(1 + az k ).
22. If 00 is a fixed point for fez), we define the analytic index at 00 of fez) to be the analytic index of 1/f(I/() at ( = O. (Danger! This is not the residue of z – fez) at z = 00.) Show that the sum of the analytic indices of a rational function f (z) at its fixed points in C*is +1.
23. Show that if fez) is a rational function of degree d ~ 2, and if all the fixed points of fez) have multiplicity one, then fez) has at least one repelling fixed point. Hint. Use Exercise 18a, and sum the real parts of the analytic indices at the d + 1 fixed points.
24. Let fez) be a rational function of degree d ~ 2. (a) Show that .1 is nonempty. (b) Show that either :F is empty or :F is dense in C*. (c) Show that .1 has no isolated points.
25. Let I(z) be a rational function of degree d ~ 2. Show that the repelling periodic points of I (z) are dense in the Julia set of I (z). Remark. This theorem was proved independently by Fatou and Julia. It can be regarded as the first substantial theorem in rational iteration theory. For the proof, fill in the details of the following
argument. Using the fact that .:T is a compact set with no isolated points, show that the points Zo E .:T that are not in the forward orbit of a critical point are dense in .:T. For such a point zo, refer to the entire function g(() from Exercise 2.11. Show that there is a point (0 E C such that (0 is not a critical point of g(() and Wo = g((o) satisfies 1m (wo) = Zo for some m. Show that the points ‘fin from Exercise 2.11 are repelling periodic points of I(z) that converge to zoo
