Iteration in the disk and the ball

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1 Noname manuscript No. (will be inserted by the editor) Iteration in the disk and the ball A survey of the role of hyperbolic geometry Pietro Poggi-Corradini Received: date / Accepted: date Abstract We review how the hyperbolic geometry of the unit disk in the complex plane and of the unit ball in several complex dimensions comes into play in the theory of iteration of analytic maps. Keywords Self-maps Conjugations Orbits Mathematics Subject Classification (2000) MSC 32H50 MSC 32A10 MSC 30D05 1 Iteration in one variable 1.1 Introduction Let D := {z C : z < 1} be the unit disk in the complex plane. A self-map of the disk is an analytic map φ defined on D such that φ(d) D. Thus iteration arises naturally and we define the n-th iterate of φ to be the composition φ n = φ φ, n times Schwarz s Lemma Given an analytic self-map of the unit disk D such that f (0) = 0 one can factor out one power of z and consider the function f (z)/z, which by the maximum principle is still a self-map of D. This gives rise to the following fundamental result. Theorem 1 (Schwarz s Lemma) If f is analytic in D, with f (z) < 1 for z < 1, and f (0) = 0, then Moreover, if equality holds for some z 0 in (1), then f is a rotation. f (z) z, z D, (1) In this note we will see how Schwarz s Lemma gets used over and over again. Pietro Poggi-Corradini Department of Mathematics, Kansas State University, Manhattan, KS Tel.: Fax: pietro@math.ksu.edu

2 2 Pietro Poggi-Corradini Automorphisms The group of analytic automorphisms of D, i.e., analytic one-to-one (or conformal) and onto self-maps of D, has a very simple description. It consists of the linear fractional transformations of the form: φ a,c (z) := c z a 1 az, (2) with c = 1 and a < 1. Clearly, each φ a,c is an automorphism. Moreover, given any point a D the map φ a,1 sends a to the origin. This fact is extremely useful in practice, as we will see. Conversely, given an automorphism φ. We let a := φ(0). Then postcomposing φ with φ a,1 we get an automorphism ψ = φ a,1 φ with ψ(0) = 0. Applying Schwarz s Lemma to ψ and its inverse ψ 1 we find that ψ must be a rotation and hence the original map φ must be a linear fractional transformation of the form (2) Hyperbolic geometry In the unit disk D one defines the pseudo-hyperbolic distance between two points z 1, z 2 as follows: pick any automorphism of D that sends one of the two points, say z 1, to the origin. For instance, consider φ z1,1(z), and look at the image point φ z1,1(z 2 ) and consider its distance to the origin. We define d(z 1, z 2 ) := φz1,1(z 2 ) z 1 z 2 = 1 z 1 z 2. (3) This is well-defined and satisfies the triangular inequality in fact the following stronger form holds: d(z 1, z 3 ) d(z 1, z 2 ) + d(z 2, z 3 ) 1 + d(z 1, z 3 )d(z 2, z 3 ) d(z 1, z 2 ) + d(z 2, z 3 ); as well as the other properties of distances. By construction, the pseudo-hyperbolic distance is invariant under conformal automorphisms of the unit disk. And Schwarz s Lemma implies that every analytic self-map f of D is a contraction for d: d( f (z 1 ), f (z 2 )) d(z 1, z 2 ) z 1, z 2 D. Moreover, for distinct z 1 and z 2 the contraction is always strict unless f is an automorphism. Geometrically, a pseudo-hyperbolic disk (z 0, r) centered at a point z 0 D of radius r is the image of the Euclidean disk { z < r} under the automorphism φ z0,1 = (z+z 0 )/(1+z 0 z). In particular, r 1 and since linear fractional transformations map Euclidean disks to Euclidean disks, (z 0, r) must have the shape of a Euclidean disk. As a result the geometry of (z 0, r) is well-known. The Euclidean center is closer to the origin than z 0. Also the Euclidean diameter of (z 0, r) and its Euclidean distance to the boundary of D are comparable to the Euclidean distance of z 0 to D, i.e., to 1 z 0. In practice the following identity is very useful when relating the pseudo-hyperbolic distance to the Euclidean distance: 1 d(z 1, z 2 ) 2 = (1 z 1 2 )(1 z 2 2 ) 1 z 2 z 1 2, (4) so much so that this is sometimes known as the World s Greatest Identity. Also very useful is the following observation which we leave to the reader to verify: If a sequence p n D converges to a boundary point ζ D and another sequence q n is within a fixed pseudo-hyperbolic distance r < 1 from p n, i.e., d(q n, p n ) < r < 1, then the sequence q n also converges to ζ.

3 Iteration in the disk and the ball Self-maps with no fixed points in the disk interior If an analytic self-map f fixes two or more points in D, then it s not a strict contraction in the pseudohyperbolic distance, and must thus be an automorphism, in fact it must be the identity, i.e., it fixes every z D. In all other cases analytic self-maps may fix at most one point in D. Maps with exactly one fixed point in D are called elliptic and a full discussion of this case would take us too far astray. For simplicity of exposition we will restrict our discussion to self-maps that have no fixed points in D and that are not automorphisms, unless we explicitly state otherwise Forward orbits tend to the ideal boundary The first concern of iteration theory is to study forward orbits: given a starting point z 0 interested in the sequence z n = φ n (z 0 ) for n = 1, 2, 3... We will proceed in steps. D we are Proposition 1 (Orbits tend to the boundary of the disk) For every z 0 D, the orbit z n = φ n (z 0 ) tends to D, i.e., z n 1. Proof Suppose to the contrary that a subsequence z nk of z n converge to a point p D. Consider the sequence of hyperbolic steps d n := d(z n, z n+1 ), which by Schwarz s Lemma is strictly decreasing and hence converges to a limit d 0. Note that φ(z nk ) converges to φ(p) (by continuity). Thus d nk d = d(p, φ(p)), and d = d(p, φ(p)) > 0 since φ does not fix the point p. By the same token, d nk +1 tends to d(φ(p), φ 2 (p)). Hence, d(p, φ(p)) = d(φ(p), φ 2 (p)). But this contradicts the strict contractivity of φ Julia s Lemma (Schwarz s Lemma at the boundary) The next step is to show that there is a special point on D, known as the Denjoy-Wolff point, which is the limit-point of every orbit. But before we can prove this fact we need to discuss Julia s Lemma, which is a version of Schwarz s Lemma at the boundary, and whose proof (see below) is an application of Schwarz s Lemma to large pseudo-hyperbolic disks with radius very close to 1. Definition 1 For ζ D and t > 0, the horodisk with vertex at ζ and size t is the set H ζ (t) = {z D : 1 z 2 ζ z > 1 }. 2 t One may notice that H ζ (t) is a level-set of the Poisson kernel with pole at ζ D. Straightforward calculations show the following. Lemma 1 The horodisk H ζ (t) is a Euclidean disk internally tangent to D at ζ of diameter 2t/(1 + t), so t H ζ (t) = and t H ζ (t) = D. By letting p D tend to ζ D and by letting the radius r grow appropriately, one can approximate the horodisk H ζ (t) using the pseudo-hyperbolic disks (p, r), as the following lemma shows. Lemma 2 Given p D and 0 < r < 1, consider the pseudo-hyperbolic disk (p, r) = {z D : d(z, p) < r}. Then, for p 0, (p, r) = z D : 1 z 2 > 1 r 2 1 p z 2 1 p 2 p 2.

4 4 Pietro Poggi-Corradini Proof Use the World s Greatest Identity (4). Applying Schwarz s Lemma to the psuedo-hyperbolic disks that approximate a horodisk, we get a version of Schwarz s lemma on the boundary called Julia s Lemma. Theorem 2 (Julia s Lemma) Suppose φ is analytic on D and φ(d) D. Suppose there is a point ζ D, and a sequence of points {p n } n=0 D such that p n ζ, φ(p n ) ζ. And assume that there is δ [0, ) such that 1 φ(p n ) δ (as n ). 1 p n Then, δ > 0 and φ maps horodisks at ζ into horodisks at ζ; more precisely, for every t > 0, φ ( H ζ (t) ) H ζ (δt). (5) Remark 1 The hypothesis of Theorem 2 can be understood as a very weak notion of derivative for the map φ at the fixed boundary point ζ. These properties are only required along a sequence p n and one of the strengths of this theorem is that there are no restrictions on the way the p n s need to approach ζ. Also, if instead φ(p n ) η for some η ζ, η D, then φ ( H ζ (t) ) H η (δt) (just apply Julia s Lemma to the function ζηφ). Proof Consider p D and set r = p. Then d(p, 0) = r. By Schwarz s Lemma, d(φ(p), φ(0)) r. By the World s Greatest Identity (4), So for every p D, 1 r 2 (1 φ(p) 2 )(1 φ(0) 2 ) (1 φ(0) φ(p) ) 2 (1 φ(p) 2 ) 1 + φ(0) 1 φ(0). 1 φ(p) 2 1 p 2 1 φ(0) 1 + φ(0) > 0. This shows that δ > 0. Now fix t (0, ). Without loss of generality, 1 p n < t for all n, so that r n := 1 1 p n t (0, 1). Note that 1 r n = (1 p n )/t. By Lemma 2, z (p n, r n ) if and only if 1 z 2 z > 2 1 p n 1 t 1 + r n 1 + p n p n 2. Fix z H ζ (t) and let ɛ := 1 2 ( 1 z ζ z 1 ). 2 t Since 1/p n 1/ζ = ζ, r n 1, p n 1, and ɛ > 0, there exists n 0 such that for all n n 0 : 1 z 2 z > 2 1 p n 1 z ζ z 2 ɛ > 1 t 1 + r n 1 + p n p n 2.

5 Iteration in the disk and the ball 5 In other words, for n n 0, z (p n, r n ) and therefore by Schwarz s Lemma φ(z) (φ(p n ), r n ). Hence, for n n 0, 1 φ(z) 2 > 1 rn 2 1 p n 2 1 φ(z) 2 1 p n 2 1 φ(p n ) φ(p n) 2. 2 φ(p n ) Letting n, we get 1 φ(z) 2 ζ φ(z) 2 1 tδ, i.e., φ(z) is in the closure of H ζ (δt). However, φ is an open map, so φ(z) H ζ (δt) Convergence to the Denjoy-Wolff point We now return to self-maps φ of the disk with no fixed points in D. Theorem 3 (Denjoy-Wolff) Assume that φ is analytic on D, φ(d) D, and φ(z) z for every z D. Then there is a point ζ D (called the Denjoy-Wolff point of φ) such that for every z 0 D, the orbit z n = φ n (z 0 ) tends to ζ, and the convergence is uniform if z 0 ranges in a compact subset of D. Proof Pick z 0 D and consider the orbit z n = φ n (z 0 ). We saw in Proposition 1 that it must tend to D. So we can choose a subsequence z nk such that z nk +1 > z nk for all k (note z nk +1 = φ(z nk )). Since D is compact we can assume (upon extracting a subsequence and relabeling) that z nk ζ for some ζ D and that for some δ 1, 1 z nk +1 δ. 1 z nk By Schwarz s Lemma, d(z nk +1, z nk ) d(z 0, z 1 ) < 1. So, by the remark following (4), z nk +1 = φ(z nk ) also tends to ζ. Hence, by Julia s Lemma (Theorem 2), φ ( H ζ (t) ) H ζ (δt), for every t > 0. In particular, all the horodisks with vertex at ζ are fixed by φ and therefore every orbit must tend to ζ. We leave the rest of the claims to the reader. Corollary 1 (Coefficient of Dilatation) Assume that φ is analytic on D, φ(d) D, and φ(z) z for every z D. Let ζ D be the Denjoy-Wolff point of φ and define α := lim inf D z ζ 1 φ(z). (6) 1 z Then 0 < α 1 and α is the smallest number such that φ ( H ζ (t) ) H ζ (αt), for every t > 0. Proof In the proof of the Denjoy-Wolff Theorem 3 we constructed δ so that α δ 1 and φ ( H ζ (t) ) H ζ (δt), for every t > 0. Let p n D be a sequence such that p n ζ and 1 φ(p n ) 1 p n Then φ(p n ) 1 and we can assume, upon extracting a subsequence, that φ(p n ) η for some η D. By Remark 1, this implies that φ maps horodisks with vertex at ζ into horodisks with vertex at η which is impossible, by the properties of δ above, unless η = ζ. So φ(p n ) ζ and, by Julia s Lemma, we have α > 0 and φ ( H ζ (t) ) H ζ (αt), for every t > 0. α.

6 6 Pietro Poggi-Corradini Now assume there is α that satisfies φ ( H ζ (t) ) H ζ (α t), for every t > 0, and 0 < α < α. Let p = rζ, then p H ζ (t) with t = (1 r)/(1 + r), and φ(p) must be in H ζ (α t), i.e., by Lemma 1, 1 φ(p) 1 1 α t 1 + α t = 2α (1 + α ) + (1 α (1 r). )r Thus, since r = p, as p 1. So α α, which is a contradiction. 1 φ(p) 1 p α Definition 2 We say α is the coefficient of dilatation of φ at its Denjoy-Wolff point The half-plane model Since the boundary Denjoy-Wolff point plays such a crucial role, it is often useful to change coordinates and switch from D to the right half-plane H := {z : Re z > 0} using the Cayley transform C(z) := ζ + z ζ z which sends the Denjoy-Wolff point ζ to infinity. We then get a self-map Φ = C φ C 1 of H, with the property that for every z 0 H the orbit z n := Φ n (z 0 ) tends to infinity. Of particular relevance is the subgroup of automorphisms of H that fix infinity, which is generated by two simple maps: vertical translations z z + ib for b R; and positive dilations z tz for t > 0. The pseudo-hyperbolic distance between points in H is defined by first pulling the points back to D via the inverse of Cayley transform and then measuring the distance there. We get, for a, b H, d(a, b) = a b a + b. (7) In particular, pseudo-hyperbolic disks in H are still Euclidean disks because the Cayley transform is a linear fractional transformation. Moreover, one finds that the horodisks with vertex at infinity in the right half-plane are shifted half-planes of the form C(H ζ (t)) = {z H : Re z > 1/t}. We leave it as an exercise for the reader to show that (6) translates into saying that A := 1/α 1 is the largest number for which Re Φ(z) A Re z z H. Put another way, the function p(z) := Φ(z) Az has positive real part on H and Re p(z) inf = 0. z H Re z We have now come to a juncture where it is convenient to separate the case when A > 1, the hyperbolic case, from the case when A = 1, the parabolic case.

7 Iteration in the disk and the ball The hyperbolic case In this section we assume that Φ is a self-map of the right half-plane H, with no fixed points in H, which has Denjoy-Wolff point at infinity and its coefficient of dilatation A satisfies A > 1, i.e., Φ is of hyperbolic type. As we saw at the end of the previous chapter, such self-maps can be written as Φ(z) = Az + p(z) (8) where Re p(z) 0 and inf z H Re p(z)/ Re z = 0. So p(z) could be viewed as a small error and the model for the orbits of Φ are going to be the orbits of the automorphism z Az, i.e., the sequences A n w 0 given w 0 H. We first introduce some notations. Given z 0 H we write z n = Φ n (z 0 ) = +iy n for the ensuing orbit. We let d n = d(z n, z n+1 ) be the hyperbolic step of this orbit. Note that by Schwarz s Lemma (Theorem 1), the sequence d n is strictly decreasing unless Φ is an automorphism. Also we let τ n (z) = z iy n (9) be the automorphism of H that fixes infinity and sends z n back to 1. These renormalizing automorphisms will be important in the sequel Non-tangential approach to the Denjoy-Wolff point We begin with a lemma which shows that the orbits of Φ tend to infinity non-tangentially, i.e., they stay in sectors centered on the real axis of opening strictly less than π. Given 0 < θ < π, let S [θ] = These sectors are called non-tangential approach regions to infinity in H. { z H : Arg z < θ }. (10) 2 Remark 2 Notice that S [θ] is a pseudo-hyperbolic ɛ-neighborhood of the positive real axis R +, i.e., there is ɛ < 1 such that S [θ] = {z H : d(z, R + ) < ɛ}. Lemma 3 (Orbits tend to infinity non-tangentially) Suppose Φ is an analytic self-map of H of hyperbolic type. Write z n = + iy n = Φ n (z 0 ), for n = 0, 1, 2,.... Then for every z 0 H there is a δ(z 0 ) > 0 such that Arg z n < π 2 δ(z 0), for n = 0, 1, 2, 3,.... Or, equivalently, there is L 0 (z 0 ) > 0 such that y n L 0 (z 0 ) for n = 0, 1, 2,.... Proof The statement is clear if Φ is an automorphism. In the general case, by Schwarz s Lemma (Theorem 1), d n < d 0, i.e., z n+1 belongs to the hyperbolic disk (z n, d 0 ). Also, by Julia s Lemma (Theorem 2), Re z n+1 > A Re z n. We transfer z n back to 1 via the automorphisms τ n defined in (9) and let q n = τ n (z n+1 ) = +1 + i y n+1 y n. (11)

8 8 Pietro Poggi-Corradini By conformal invariance, q n (1, d 0 ). Moreover, by (11), Re q n A > 1. Thus, since the compact set {z H : d(z, 1) d 0, Re z A} is disjoint from the line {Re z = 1}, there is δ > 0 such that Arg(q n 1) π 2 δ, i.e., q n 1 + S 0 where S 0 is the sector S [π 2δ]. Notice that τ 1 n (z) = z + iy n consists of a dilation followed by a translation. So we also have z n+1 z n +S 0, for every n = 0, 1, 2, 3,.... However, since S 0 +S 0 S 0, by induction, we obtain that z n z 0 +S 0 for all n. The lemma follows easily Julia-Carathéodory s Theorem Definition 3 We say that a function f defined on H has non-tangential limit a C at infinity and write n.t. lim z f (z) = a, if for every θ (0, π) and for every ɛ > 0 there is R = R(ɛ, θ) > 0 such that f (z) a < ɛ whenever z belongs to the truncated sector S [θ] { z > r}. Or equivalently, n.t. lim z f (z) = a if for every θ (0, π) we have lim n f (z n ) = a for any sequence z n tending to infinity while staying in S [θ]. Recall the self-maps p that were obtained in (8) and which we wanted to treat as small. The next result makes the notion of smallness more precise. Theorem 4 (Julia-Carathéodory Theorem) Suppose p is analytic on H and p(h) H. If then Re p(z) inf = 0, z H Re z n.t. lim z p(z) z = 0. Corollary 2 Suppose F is analytic on H and F(H) H. Then exists and is a number in [0, ). c := n.t. lim z Proof (Proof of Corollary 2) Let c = inf z H Re F(z)/ Re z. Then c [0, ). Let p(z) = F(z) cz. Then Re p(z) 0, so p(h) H and inf z H Re p(z)/ Re z = 0, and Theorem 4 implies the corollary. Proof (Proof of Theorem 4) Fix θ (0, π) and fix ɛ > 0. Also choose α with θ < α < π. By hypothesis, there is z 0 H with Re p(z 0 ) < ɛ Re z 0. Choose R 1 > 0 so large that the truncated sector S 1 := S [θ] { z > R 1 } is contained in the sector z 0 + S [α]. As w tends to infinity in S 1 the rate of growth of Re w is controlled by the rate at which r := d(z 0, w) tends to 1. To see this note that, on one hand, the World s Greatest Identity (7) translates in H to 1 d(a, b) 2 = F(z) z 4 Re a Re b 4 Re a Re b = a + b 2 (Re a + Re b) 2 + (Im a Im b) 4Re b 2 Re a. (12) Therefore, since d(z 0, w) = r, using a = w and b = z 0 in (12), we get Re w 4 Re z 0 1 r 2. (13)

9 Iteration in the disk and the ball 9 On the other hand, since w S 1, w z 0 S [α], and so we have Re w > Re(w z 0 ) > C(α) w z 0 for some constant C(α) > 0. Moreover, the World s Greatest Identity can also be written as follows: Therefore, using a = w and b = z 0 in (14), we get i.e., 1 d(a, b) Re a Re b = d(a, b). (14) a b 2 (Re w) 2 C(α) 2 r 2 4 Re z 0 Re w 1 r 2, Re w C 1 (α) 4 Re z 0 1 r. (15) 2 Together (13) and (15) give precise upper and lower bounds for Re w when w S 1. We will now use Schwarz s Lemma to get an estimate on p(w), since d(p(z 0 ), p(w)) d(z 0, w) = r. By (12) again Re p(w) 4 Re p(z 0), 1 r 2 and by (14) p(w) p(z 0 ) 2 4r2 1 r Re p(w) Re p(z 0) 16 (Re p(z 0)) 2 2 (1 r 2 ). 2 By our choice of z 0 and by (15) we therefore get that In other words, So p(w) p(z 0 ) ɛ 4 Re z 0 1 r 2 C 2(α)ɛ Re w C 2 (α)ɛ w. p(w) w p(z 0 ) w + C 2(α)ɛ. lim sup p(w) w C 2(α)ɛ, S 1 w and the claim follows by letting ɛ tend to zero Renormalizing Iterates Julia-Carathéodory applies to sequences that tend to infinity non-tangentially and we have shown in Lemma 3 that orbits tend to infinity non-tangentially. Therefore, an immediate corollary is that we can now apply Julia-Carathéodory to orbits in the hyperbolic case. Corollary 3 Suppose Φ is a self-map of H of hyperbolic type, which thus has the form (8) for some A > 1. Then for every z 0 H, the corresponding orbit z n = Φ n (z 0 ) has the property that z n+1 z n A as n. (16) Or, equivalently z n+1 = Az n + o(z n ) as n. Proof Since z n tends to infinity non-tangentially and z n+1 = Φ(z n ) = Az n + p(z n ), we just need to combine Lemma 3 with Theorem 4.

10 10 Pietro Poggi-Corradini Taking arguments in (16) we see that Arg z n+1 Arg z n 0 as n (17) In fact, as the following proposition shows, Julia-Carathéodory s theorem implies that the arguments of the iterates have an actual limit. Proposition 2 (Orbits arguments converge) Suppose Φ is an analytic self-map of H of hyperbolic type. Then for every z 0 H, the corresponding orbit z n = + iy n = Φ n (z 0 ) has the property that: Arg z n Θ(z 0 ) as n, for some Θ(z 0 ) ( π/2, π/2). Or, equivalently, if z n = + iy n, then y n L(z 0 ) as n, for some L(z 0 ) R. Proof From the definition of τ n and q n in (9) and (11) we see that: q n = z n+1 z n + 1. (18) Moreover, from Corollary 3, z n+1 = Az n + o(1)z n, as n. Therefore, dividing both sides by and taking real parts, we find that Re q n = x ( n+1 = A + o(1) 1 + y ) n as n. So Re q n A, by Lemma 3. Also recall that the hyperbolic step d n = d(z n, z n+1 ) = d(1, q n ) is a strictly decreasing sequence by Schwarz s Lemma. And, Re q n > A by Julia s Lemma. However, the pseudo-hyperbolic distance of 1 to the horocycle {z : Re z = A} is attained at z = A. Thus, d n d(1, A) = A 1 > 0 for all n. A + 1 Hence, d n d for some d d(1, A) > 0. The circle C := (1, d ), where (1, d ) is the pseudo-hyperbolic disk centered at 1 of radius d, intersects the line L = {z : Re z = A} in two points q + and q, which are therefore the only possible accumulation points for the sequence q n. If q + = q, which is equivalent to d = d(1, A), then q n A. In general, we will have q + q. In the sequel we assume then that Im q + > 0 and Im q < 0, i.e., that d(1, A) < d. We will show that the sequence q n converges (either to q + or q ). First note that Im q n 0 for all n large, because otherwise there is a subsequence n k so that d(1, A) = lim k d(1, Re q nk ) = lim k d(1, q nk ) = d, which is a case already considered. So, if Im q n doesn t converge, then there is a subsequence n k, such that Im q nk > 0 and Im q nk +1 < 0, so that q nk tends to q +, while q nk +1 tends to q. Combining (18) and (16) we see that Arg(q n 1) = Arg z n + o(1) as n. So Arg z nk Arg(q + 1), while Arg z nk +1 Arg(q 1), but this contradicts (17).

11 Iteration in the disk and the ball Valiron s Theorem The precise knowledge we have built about the behavior of each orbit as it approaches the Denjoy-Wolff point can be used to renormalize the iterate of the function itself and get what is known as Valiron s Theorem, which we state without proof. Theorem 5 (Valiron s Theorem) Suppose Φ is a self-map of H of hyperbolic type with Denjoy-Wolff point at infinity and with dilatation coefficient A > 1. Then there exists an analytic map σ : H σ(h) H such that σ Φ = Aσ. (19) More specifically, given z 0 H, let := Re Φ n (z 0 ). Then Φ n / converges uniformly on compact subsets of H to a function σ that satisfies (19). Valiron renormalized the iterates of Φ dividing Φ n by Φ n (z 0 ). Ch. Pommerenke instead constructed solutions σ to Valiron s equation (19) as limits σ = lim n Φ n /, where = Re Φ n (z 0 ) for an arbitrary point z 0 H. Yet another way to renormalize is to consider the automorphisms τ n and then study the sequence of self-maps of H given by σ n := τ n Φ n. 1.4 The parabolic case Suppose that Φ is a self-map of the right half-plane H, with no fixed points in H, which has Denjoy-Wolff point at infinity and coefficient of dilatation equal to 1, i.e., Φ is of parabolic type. Such self-maps can be written as Φ(z) = z + p(z) (20) where Re p(z) 0 and inf z H Re p(z)/ Re z = 0. Again, by Julia-Carathéodory Theorem we also have n.t. lim z p(z)/z = 0. A major breakthrough in this case was obtained by Ch. Pommerenke in [Pommerenke (1979)]. The convergence of an orbit {z n } n=0 to the Denjoy-Wolff point at infinity is more complicated now and a crucial role is played by the hyperbolic step d n := d(z n, z n+1 ). By Schwarz s Lemma, the sequence of hyperbolic steps is decreasing. Recall that in the hyperbolic case d n has a positive lower-bound (see the proof of Proposition 2). In the parabolic case, there are two subcases: either the step has a positive lower bound, or it tends to zero. There is no agreement in the literature on how to label these two cases. We ve used the terminology zero-step vs. non-zero step before. The zero-step case is arguably the hardest case. It follows from Julia s Lemma, see also (20) that the sequence of heights h n := Re z n is increasing. So in [Poggi-Corradini(2003)] we introduced two more cases: the infinite height case and the finite heigth case. Later we realized that Schwarz s lemma prevents having finite height and zero step. However in the non-zero-step case both finite and infinite height can arise. Pommerenke finds several estimates for the convergence of orbits in the parabolic case, the more important one being that the ratio of heights always tends to one: h n+1 h n 1 as n. This leads to a conjugation result, at least in the non-zero-step case. Renormalizing the iterates Φ n with the automorphism of the right halfplane that brings z n back to 1, namely setting M n = z i Im z n Re z n, and forming the sequence σ n := M n Φ n, he proves the following.

12 12 Pietro Poggi-Corradini Theorem 6 ([Pommerenke (1979)]) In the non-zero step case, σ n tends to a conjugation σ of the right half-plane to itself such that σ Φ = σ + ib with b R \ {0}. It turns out that b = lim n Im z n+1 Im z n Re z n. So in particular the orbits tend to infinity tangentially in the parabolic non-zero-step case,i.e., the argument Arg z n tends to π/2 or π/2. Note also that a simple change of variables produces a map of the right half-plane onto the upper half-plane that conjugates Φ to translation by 1. As mentioned already, the zero-step case is more complicated. But in a subsequent paper Baker and Pommerenke [Baker and Pommerenke(1979)] were able to conjugate Φ to translation by one in the plane, instead of the half-plane. The convergence of the orbits in this case is not necessarily tangential anymore and in fact it can even have the property that Arg z n tends to zero for every orbit (!). It is an open problem to describe the possible behavior of orbits in the parabolic zero-step case. For instance it is not known if one can find a self-map whose orbits have the property that lim sup Arg z n = π n 2 and lim inf n Arg z n = π 2. However, the existence of these conjugations allows one to conclude that the properties of zero-step, nonzero-step, finite-height, and infinite-height are properties of the self-map Φ, i.e., they do not depend on the chosen orbit. 1.5 Backward iteration So far we have been studying forward iterates of a point under the self-map and we ve seen that the orbits this process generates tend to a special point (the Denjoy-Wolff point), that is either in the interior or on the boundary, and that behaves as an attracting fixed-point. There is a similar relationship between backward iteration orbits and special repelling fixed points on the boundary. However, some care has to be taken to describe this relationship exactly. The self-maps of the unit disk or right half-plane that we have been considering are not necessarily one-to-one, nor onto, and do not necessarily extend to the boundary, even continously. Still the notions of boundary repelling fixed-point and backward orbits were explored in [Poggi-Corradini(2000)] and [Poggi-Corradini(2003)]. To describe this topic, assume once again that φ is an analytic self-map of the disk D. We define a backward iteration sequence for φ to be an infinite sequence {w n } n=1 such that φ(w n+1) = w n. In other words, we assume from the beginning that we are able to find a preimage of w 1, call it w 2, a preimage of w 2, call it w 3, etc... infinitely many times. This is clearly not always possible, but if it is, then we are ready to study the sequence thus generated. For instance, Schwarz s Lemma implies that, if d is the pseudo-hyperbolic metric defined in (3), then the hyperbolic step s n := d(w n+1, w n ) must be strictly increasing: s n+1 > s n (we do not consider the trivial case of automorphisms of the disk). In order to obtain a relationship with boundary fixed-points it turns out that there is an extra key property that needs to be required. Namely, we assume that there is a constant a < 1 depending only on φ and on the whole sequence {w n } n=1, such that s n < a < 1 for all n = 1,.... We call such sequences backward iteration sequences of bounded hyperbolic step. Theorem 7 ([Poggi-Corradini(2003)]) Assume that {w n } n=1 is a backward iteration sequence of bounded hyperbolic step for φ.

13 Iteration in the disk and the ball 13 Then there is a boundary point ζ D depending on the sequence {w n } n=1 such that w n ζ as n, and two cases arise: either ζ is the Denjoy-Wolff point of φ in which case the map φ is necessarily of parabolic type; or ζ is not the Denjoy-Wolff point and we then call ζ a boundary repelling fixed point, as will be explained below. If a backward iteration sequence with bounded step w n converges to the Denjoy-Wolff point, it does so tangentially; and if the map is parabolic non-zero-step, then the forward orbits converge tangentially to the Denjoy-Wolff point as well. However, say in the right half-plane, if the forward orbits satisfy Arg z n π/2, then backward orbits must satisfy Arg w n π/2, and vice versa. Furthermore, by Julia s Lemma the heights Re w n are now decreasing and hence have a limit which can either be zero or positive. Again, assuming that φ is parabolic non-zero step, and that a backward sequence w n converges to the Denjoy-Wolff point at infinity, we can show that Re w n must stay bounded away from zero. The model to keep in mind is the automorphism z z + i. In fact, we are able to produce some conjugations even in this case, using the backward iteration sequence w n to renormalize the iterates of φ. On the other hand, when the limit point ζ is not the Denjoy-Wolff point, then the convergence of w n to ζ is non-tangential and Julia s Lemma holds at ζ, see (5). Namely, if H ζ (t) is a horodisk at ζ then there is a constant c > 1 such that φ(h ζ (t)) H ζ (ct) for all t > 0. This geometric property reflects the notion that ζ can be considered as a repelling fixed point. In fact, the theorem of Carathéodory (Theorem 4) also shows that c is the non-tangential limit of the derivative φ (z) as z tends to ζ. We will say that ζ is a boundary repelling fixed point (in the sense of Julia s Lemma). Conversely, given a point ζ on D which is not the Denjoy-Wolff point and such that Julia s lemma holds there for some 1 < c <, so that ζ is a boundary repelling fixed point, we were able to construct another special backward iteration sequence converging to ζ radially and use it to renormalize the iterates of φ to form a conjugation at ζ. Notice that the renormalization is now done as a pre-composition rather than a post-composition, i.e. we let τ n be the automorphism that sends w n back to w 1 and then show that the sequence φ n τ n converges to a conjugation ψ such that φ ψ = ψ η for some automorphism η (η being the hyperbolic automorphism with a repelling fixed point at ζ and multiplier c there). In particular, once this conjugation is established we can construct infinitely many backward iteration sequences with bounded hyperbolic step converging to ζ and we can even prescribe the angle at which such sequences will converge to ζ. In summary, we have shown that there is a correspondence between backward iteration sequences with bounded hyperolic steps (not tending to the Denjoy-Wolff point) and boundary repelling fixed points in the sense of Julia s Lemma. 1.6 Uniqueness of conjugations As we have seen there are many conjugations that arise in this context. Also, historically different authors have obtain these conjugations by some times fundamentally, different methods. Especially if the conjugation is obtained by renormalizing the iterates using an orbit, it is natural to ask how the resulting conjugation depends on the choice of orbit, if at all. For these reasons, we found it necessary to study the notion of uniqueness or at least canonicity of conjugating maps. Describing these results would lead us a bit astray so instead we refer the interested reader to the paper [Poggi-Corradini (2006)].

14 14 Pietro Poggi-Corradini 2 Higher dimensions 2.1 Preliminaries In C N, N = 2, 3,..., we let π j : C N C, j = 1,..., N, be the coordinate mappings; the usual inner product is z 1, z 2 := N j=1 z 1, jz 2, j, where z n, j = π j (z n ); the norm is z 2 := z, z. The unit ball B N is {z C N : z 2 < 1}. A mapping f : B N C N is analytic if each coordinate mapping π j f is analytic in each variable separately. We will be interested in analytic self-maps of the ball, i.e. mappings f such that f (B N ) B N, and their iteration theory. Schwarz s lemma holds verbatim as in Theorem 1, except for the Moreover part. To see this one only needs to restrict f to one-dimensional subspace (or slices ) and apply the regular Schwarz lemma. In particular, it can very well happen that in some directions f is a strict contraction, while in other directions it s a simple rotation. That s why the uniqueness statment fails. Also the group of automorphisms of the ball is well-understood and a description similar to (2) is available (see Abate or Rudin). However, for our purposes this exact description won t be necessary, as I will now explain. In order to compute the pseudo-hyperbolic distance between two points in the ball, it will again be enough to apply an automorphism and send one of the two points to the origin and then measure the Euclidean distance of the other image to 0. However, as described in Section 1.2.4, in the one-variable case, the natural setting for us is the right half-plane and the subgroup of automorphisms that fix infinity. Namely the linear maps of the form az + b with a > 0 and b purely imaginary. Using these linear maps, any point in the right half-plane can be brought back to 1 which plays the role of the origin under the Cayley transform. This is exactly how we proceed in the higher-dimensional case as well. The role of the right half-plane is played by the Siegel domain: H N := {(z, w) C C N 1 : Re z > w 2 } (21) which is biholomorphic to B N via the Cayley transform C : B N H N defined as ( 1 + C(z 1, z z1 z ) ) :=,. (22) 1 z 1 1 z 1 Note that C sends the point (1, 0) (0 C N 1 ) to inifinity. A similar formula can be written down so that a given point ζ B N gets mapped to infinity. We will write self-maps of the Siegel domain as follows φ = (φ 1, φ 2 ) so that Re(φ 1 ) φ 2 2. The following two basic automorphisms fix infinity in the Siegel domain: 1. Dilations: (z, w) (tz, tw), for any given t > Translations: (z, w) (z + ib + w w, w 0, w + w 0 ), for b R and w 0 C. The role of tangential approach regions (or Stolz angles) is played by the so-called Koranyi regions. In the particular case of infinity in the Siegel domain the Koranyi region there can be written as a hyperbolic neighborhood of the positive real axis. Koranyi region at infinity: {(z, w) H N : d((z, w), (x, 0)) < a < 1}. Moreover, if instead of the real axis one considers the geodesic through infinity and demands the distance to vanish then one gets the so-called special approach: Special approach: d((z, w), (z, 0)) 0 as (z, w).

15 Iteration in the disk and the ball 15 Note that the distance between a point and its projection onto the geodesic through infinity {(z, 0) : Re z > 0} can be computed using dilations and translations. After a short manipulation one obtains: Projection: d((z, w), (z, 0)) = w Re z < 1 Going back to self-maps of the ball, we must note that while maps don t necessarily have to fix at most one point in the ball (as they do in dimension 1), maps with no interior fixed points retain many of the same properties. Their orbits tend to the ideal boundary as before, and a Julia s lemma can be established so that a special Denjoy-Wolff point ζ B N will again attract every orbit. Also a coefficient of dilatation can be defined to distinguish between the hyperbolic case (α < 1) and the parabolic case (α = 1): α := lim inf z ζ 1 f (z) 1 z. Then f preserves certain ellipsoids internally tangent to B N at ζ (which we call horospheres ). Define } E(t) := {z B N 1 z, ζ 2 : < t, (23) 1 z 2 then f (E(t)) E(αt). In formulas, 1 f (z), ζ 2 1 z, ζ 2 c, (24) 1 f (z) 2 1 z 2 for every z B N. Switching to the Siegel domain and sending the Denjoy-Wolf point ζ to infinity, we find that these horospheres become halfplanes of the form H(t) := {(z, w) H N : Re z w 2 > t}. Therefore, a self-map φ = (φ 1, φ 2 ) with no internal fixed points (and Denjoy-Wolf point at infinity) has the property that for some minimal A 1: Re φ 1 (z, w) φ 2 (z, w) A(Re z w 2 ). Here A = 1/α, thus the map is hyperbolic if A > 1 and parabolic if A = Koranyi approach of forward iterates The first result in this geometric vein was obtained in [Bracci and Poggi-Corradini(2003)] and is a generalization of Lemma 3. Theorem 8 ([Bracci and Poggi-Corradini(2003)]) Consider a self-map φ of the Siegel domain with no internal fixed points, Denjoy-Wolff point at infinity, and of hyperbolic type. Then its forward orbits tend to infinity while staying inside a Koranyi region, which may depend on the chosen orbit. The proof follows the renormalization technique used in the one-dimensional case (see Picture 2.2) but needs some extra care. Next one might wish to refine this statement and get a more precise description of the behavior of forward orbits. In the one-dimensional case this is done by first establishing the Julia-Caratheodory theorem. The bad news is that unlike in the one-dimensional case, self-maps of the Siegel domain cannot be written as a sum of a linear term and an error term that is also a self-map. Recall that in the right half-plane we could write φ(z) = Az + p(z), with p(z) a self-map of the half-plane as well. As a result, the full Julia-Charatheodory Theorem does not extend to higher dimension. Only the following weaker statement survives, see [Rudin(1980)] Theorem 8.5.6, p. 177.

16 16 Pietro Poggi-Corradini Fig. 1 Renormalization and Koranyi approach Theorem 9 (Rudin-Julia-Caratheodory) Suppose f = ( f 1, f 2 ) is an analytic self-map of H N (N 2) and Re f 1 (z, w) f 2 (z, w) 2 inf = A > 1. (z,w) H N Re z w 2 If (z, w) along a special approach (see (2.1)), then f 1 (z, w) z A. 2.3 A Valiron-type result In [Bracci, Gentili, and Poggi-Corradini(2010)], the Koranyi approach property of Theorem 8 was used to establish a Valiron-type conjugation result for hyperbolic self-maps. We had, however, to assume the full conclusion of the Julia-Caratheodory Theorem in higher dimension and an extra condition that will be discussed below. Theorem 10 ([Bracci, Gentili, and Poggi-Corradini(2010)]) Let φ = (φ 1, φ 2 ) : H N H N be holomorphic, with Denjoy-Wolff point and of hyperbolic type with multiplier A > 1. Assume that 1. (z 0, w 0 ) H N s.t. orbit {( + iy n, w n ) = φ n (z 0, w 0 )} is special. 2. K-lim H N (z,w) φ 1(z,w) z exists. Then the Valiron-like sequence, σ n (z, w) := π 1 φ n (z, w) converges to a non-constant holomorphic map σ : H N H such that σ φ = Aσ. Note that the range of the conjugating map σ is one-dimensional. This cannot be improved in general because some maps may have a one-dimensional range as in the following interesting example. Consider φ : H 2 (z, w) (Az + Aw 2 ψ(z), 0)

17 Iteration in the disk and the ball 17 where ψ : H D is arbitrary and A > 1. Then φ(h 2 ) H 2, is the Denjoy-Wolff point, and the multiplier is A > 1. Note that {φ n (1, 0)} = {(A n, 0)} is special, but (2) fails. Namely, only the weaker version of that Julia- Caratheodory theorem can be assumed. On the other hand, the Valiron construction still works in this case: if π 1 is projection onto the first coordinate, π 1 φ n (z, w) = A n z + A n w 2 ψ(z), hence σ n (z, w) := π 1 φ n (z, w) = z + w 2 ψ(z). At this point we don t know whether the two hypothesis (1) and (2) are necessary for convergence of the Valiron sequence. 2.4 Recent advances In a recent paper [Ostapyuk(2011)], O. Ostapyuk established some results in higher dimension for backward orbits. These are orbits z n such that φ(z n+1 ) = z n. Ostapyuk shows that when the hyperbolic step d(z n, z n+1 ) is uniformly bounded in n, then the orbit converges to a point on the boundary different from the Denjoy-Wolff point (assuming the map is of hyperbolic type), the convergence to this point is Koranyi, and this point has some special properties which justify the name of Boundary Repelling Fixed Point. For instance, a Julia s Lemma will hold at such boundary points. Further, Ostapyuk establishes some semi-conjugations at such points. These results generalize some of the statements in one dimension that were obtained in [Poggi-Corradini(2003)], see Section 1.4. In her thesis, Ostapyuk also obtained some preliminary geometric descriptions of orbits in the parabolic case in higher dimension. This case is currently completely open with many interesting unanswered questions. For instance, even though the notion of zero-step and non-zero-step can easily be defined for a forward orbit. It is not known at the moment whether these are property of the map, i.e. if they are properties independent of the chosen orbit. References [Baker and Pommerenke(1979)] I. N. Baker and Ch. Pommerenke. On the iteration of analytic functions in a halfplane. II. J. London Math. Soc. (2), 20(2): , (1979). [Bracci and Poggi-Corradini(2003)] F.Bracci, P. Poggi-Corradini. On Valiron s theorem. In: Future trends in geometric function theory, RNC Workshop Jyväskylä, University of Jyväskylä, Department of Mathematics and Statistics, volume 92, pages (2003). [Bracci, Gentili, and Poggi-Corradini(2010)] F. Bracci, G. Gentili, P. Poggi-Corradini. Valiron s construction in higher dimension. Rev. Mat. Iberoamericana 26(1):57 76 (2010). [Ostapyuk(2011)] O. Ostapyuk. Backward iteration in the unit ball. To appear in Illinois Journal of Mathematics. [Poggi-Corradini(2000)] P. Poggi-Corradini. Canonical conjugations at fixed points other than the Denjoy-Wolff point. Ann. Acad. Sci. Fenn. Math., 25(2): , (2000). [Poggi-Corradini(2003)] P. Poggi-Corradini. Backward-iteration sequences with bounded hyperbolic steps for analytic self-maps of the disk. Rev. Mat. Iberoamericana, 19(3): (2003). [Poggi-Corradini (2006)] P. Poggi-Corradini. On the uniqueness of classical semiconjugations for self-maps of the disk. Comput. Methods Funct. Theory, 6(2): , (2006). [Pommerenke (1979)] Ch. Pommerenke. On the iteration of analytic functions in a halfplane. J. London Math. Soc. (2), 19(3): , (1979). [Rudin(1980)] W. Rudin. Function theory in the unit ball of C n. Classics in Mathematics. Springer-Verlag, Berlin, Reprint of the 1980 edition.