Tesi di Laurea Magistrale
The Wolff-Denjoy theorem: old
and new approaches
Candidato
Alessandro Bertellotti
Relatore
Professor Marco Abate
Anno Accademico 2019/2020
23/10/2020
1 Theory in the unit disk 1
1.1 Schwarz lemma and some consequences . . . 1
1.2 Fixed points of automorphisms . . . 4
1.3 The upper half-plane . . . 5
1.4 The Poincar´e distance . . . 7
1.5 The Poincar´e distance on the upper half plane . . . 12
2 The Kobayashi distance on complex manifolds 13 2.1 The Kobayashi distance . . . 13
2.2 Hyperbolic manifolds . . . 18
2.3 Complete hyperbolic manifolds . . . 20
3 Introduction to holomorphic dynamical systems 23 3.1 Some preliminaries on function spaces . . . 23
3.2 Holomorphic dynamical systems . . . 26
4 Dynamics on taut manifolds 29 4.1 Taut manifolds . . . 29
4.2 Dynamics on taut manifolds . . . 31
5 Dynamics on the unit disk 39 5.1 Horocycles and Wolff lemma . . . 39
5.2 The Wolff-Denjoy theorem . . . 44
6 Dynamics on domains 47 6.1 Horospheres and dynamics . . . 47
7 Dynamics on convex domains 53 7.1 Convex domains . . . 53
7.2 Dynamics on convex domains . . . 58
8 Dynamics on pseudoconvex domains 65 8.1 Regular domains . . . 65
8.2 Pseudoconvex domains . . . 66
8.3 The Kobayashi distance on regular and pseudoconvex domains . . . . 67
8.4 Dynamics on strongly pseudoconvex domains . . . 70 i
9 Geometry of metric spaces 73
9.1 Geodesics . . . 73
9.2 Length of a curve . . . 74
9.3 The Hopf-Rinow theorem . . . 79
9.4 The Gromov product . . . 84
10 Dynamics on metric spaces 87 10.1 Topological dynamical systems . . . 87
10.2 The Wolff-Denjoy theorem for metric spaces . . . 91
11 The Kobayashi metric 99 11.1 The Kobayashi metric . . . 99
11.2 Relationship with the Kobayashi distance . . . 102
11.3 The Kobayashi metric on domains . . . 106
11.4 Geodesics in bounded domains . . . 110
12 Dynamics on admissible domains 113 12.1 Admissible domains . . . 113
12.2 m-convex domains . . . 116
12.3 Domains with finite line type . . . 118
12.4 Regular and pseudoconvex domains . . . 120
13 Dynamics on Goldilocks domains 123 13.1 Goldilocks domains . . . 123
13.2 Almost-geodesics . . . 124
13.3 A visibility result . . . 127
13.4 The Wolff-Denjoy theorem for Goldilocks domains . . . 130
13.5 Examples of Goldilocks domains . . . 131
14 Gromov hyperbolic spaces 133 14.1 Gromov hyperbolic spaces . . . 133
14.2 The Gromov boundary . . . 134
14.3 The Wolff-Denjoy theorem for Gromov hyperbolic spaces . . . 137
15 Gromov hyperbolic domains 141 15.1 Some questions about Gromov hyperbolicity . . . 141
Theory in the unit disk
1.1
Schwarz lemma and some consequences
Schwarz lemma is one of the most well known result in basic complex analysis. De-spite its simplicity and apparently innocuous formulation, it has deep consequences. In particular, it is one of the first example of the strong relationship that in the com-plex setting holds between the geometry of a space and the properties if holomorphic functions defined on it. This relationship will appear clearly in the next sections, where we will introduce the Poincar´e distance on the unit disk.
Let ∆ = {z ∈ C : |z| < 1} be the unit disk of C. We also denote with ∆∗ = ∆\{O} the punctured unit disk.
Theorem 1.1.1 (Schwarz lemma). Let f : ∆ → ∆ be a holomorphic function such that f (0) = 0. Then
(1) |f (z)| ≤ |z| for all z ∈ ∆; (2) |f0(0)| ≤ 1.
Moreover, equality holds in (1) for some z 6= 0 or in (2) if and only if f (z) = λz with λ ∈ S1, and in this case equality holds everywhere.
Proof. Let g : ∆∗ → C be the function defined by g(z) = f (z)z . Then g is holomorphic and from f (z) = f0(0)z + O(z2), z → 0 and the Riemann extension theorem we
deduce that g is holomorphic on all ∆ with g(0) = f0(0). Let z ∈ ∆∗ and 0 < r < 1 be such that |z| < r. Then by the maximum principle
|g(z)| ≤ max{|g(ζ)| : |ζ| = r} ≤ 1 r from which follows
|g(z)| ≤ lim
r→1
1 r = 1,
that is |f (z)| ≤ |z|. Arguing in the same way we get |g(0)| ≤ 1 that is |f0(0)| ≤ 1. Now if there exists z0 ∈ ∆∗ such that |f (z0)| = |z0| we have |g(z0)| = 1. So |g| has
a maximum in z0 and hence from the maximum principle follows that g is constant,
that is f (z) = eiϑz.
Analogously, if |f (0)| = 1 then 0 is a maximum point for |g|. Therefore g is constant and f (z) = eiϑz.
Finally if f (z) = eiϑz then we have the equality in (1) for all z and in (2).
Definition 1.1.2. The biholomorphism f : ∆ → ∆, z 7→ λz with λ ∈ S1 is called a rotation. If λ = eiϑ with ϑ ∈ [0, 2π) then ϑ is called the angle of the rotation.
Schwarz lemma Theorem 1.1.1 allows us to completely describe the automor-phisms group of the disk ∆.
Theorem 1.1.3. Aut(∆) is formed by the M¨obius transformations, that is by the functions γ : ∆ → ∆ of the kind
γ(z) = eiϑ z − a
1 − az, (1.1.1)
with a ∈ ∆ and ϑ ∈ R, where a = γ−1(0). Proof. Let
γ(z) = eiϑ z − a 1 − az and let us prove that γ ∈ Aut(∆). First of all
1 − |γ(z)|2 = 1 − |z − a|
2
|1 − az|2 =
(1 − |a|2)(1 − |z|2)
|1 − az|2 > 0
from which we have γ(z) ∈ ∆. Then γ : ∆ → ∆. Now γ is invertible, because it is easy to check that
η(w) = eiϕ w − b 1 − bw
where ϕ = −ϑ and b = −eiϑa is the inverse of γ. Therefore γ ∈ Aut(∆). Define
Γ = {γ ∈ Aut(∆) : γ is a Mobius transformation}.
It is easy to check that Γ is a subgroup. If ϕ ∈ Aut(∆) let a = ϕ(0) and γ as in (1.1.1). Then γ(a) = 0 and ψ = γ ◦ ϕ satisfies ψ(0) = 0. Thanks to Schwarz lemma Theorem 1.1.1, we have |ψ0(0)| ≤ 1 and |(ψ−1)0(0)| ≤ 1 that is |ψ0(0)| = 1. It follows that ψ(z) = λz with λ ∈ S1 and hence ϕ = γ−1◦ ψ ∈ Γ . Therefore Γ = Aut(∆) as wanted.
Thanks to Theorem 1.1.3 every automorphism γ of ∆ extends to a homeomor-phism γ : ∆ → ∆ with γ(∆) = ∆ and γ(∂∆) = ∂∆.
Proposition 1.1.4. Aut(∆) acts transitively on ∆ and doubly transitively on ∂∆. Proof. We prove only the first assertion; for the second one see [Aba89]. Let a, b ∈ ∆. If a = b we are done. Let a 6= b and define
γa(z) =
z − a
1 − az, γb(z) = z − b 1 − bz. Then γa, γb, γb−1◦ γa∈ Aut(∆) and γb−1◦ γa(a) = b.
On the contrary Aut(∆) does not act doubly transitively on ∆. Indeed given z0, z1 ∈ ∆ and γ(z0) = 0 then |γ(z1)| is completely determined by z0 and z1, and
therefore γ(z1) cannot be an arbitrary point of ∆. This means that there is a certain
rigidity in the geometry of the disk, which constitutes an obstruction for the action of Aut(∆).
Theorem 1.1.5 (Schwarz-Pick Lemma). Let f : ∆ → ∆ be a holomorphic function. Then: (1) f (z)−f (w) 1−f (w)f (z) ≤ z−w 1−wz for all z, w ∈ ∆; (2) |f0(z)| 1−|f (z)|2 ≤ 1 1−|z|2 for all z ∈ ∆.
Moreover, equality holds in (1) for some z 6= w or in (2) for some z ∈ ∆ if and only if f ∈ Aut(∆), and in this case holds everywhere.
Proof. Let z, w ∈ ∆ fixed and γ1, γ2 ∈ Aut(∆) given by
γ1(ζ) =
ζ + w
1 + wζ, γ2(ζ) =
ζ − f (w) 1 − f (w)ζ. Let g = γ2◦ f ◦ γ1. Then g ∈ Hol(∆, ∆) and
g(ζ) = f (γ1(ζ)) − f (w) 1 − f (w)f (γ1(ζ))
.
Moreover g(0) = 0 and |g(ζ)| ≤ |ζ| by Schwarz lemma. Therefore, if ζ ∈ ∆ is such that γ1(ζ) = z, we have f (z) − f (w) 1 − f (w)f (z) ≤ z − w 1 − wz and (1) is proved.
Again by Schwarz lemma Theorem 1.1.1 we have |g0(0)| ≤ 1 and with some simple computations we obtain (2).
Finally, equality holds in (1) for some z 6= w if and only if |g(ζ)| = |ζ| with γ(ζ) = z and hence ζ 6= 0, if and only if g is a rotation if and only if f ∈ Aut(∆), and in this case holds everywhere, while equality holds in (2) for some w ∈ ∆ if and only if |g0(0)| = 1 if and only if g is a rotation if and only if f ∈ Aut(∆), and in this case holds everywhere.
Remark 1.1.6. If f ∈ Aut(∆) then f (0) = 0 if and only if f is a rotation. More in general, if f ∈ Aut(∆) has a fixed point then f is conjugated to a rotation, that is there exists γ ∈ Aut(∆) such that (γ ◦ f ◦ γ−1)(z) = eiϑz for all z ∈ ∆. Vice versa if
1.2
Fixed points of automorphisms
We have seen that the Schwarz lemma enables us to characterize the group of automorphisms of the disk. Thanks to this characterization, it is possible to classify them depending on their fixed points in ∆.
Proposition 1.2.1. Let γ ∈ Aut(∆) with γ 6= Id∆. Then either
(1) γ has a unique fixed point in ∆, or (2) γ has a unique fixed point in ∂∆, or (3) γ has exactly two fixed points in ∂∆. Proof. Let γ ∈ Aut(∆) with
γ(z) = λz − a 1 − az,
where a ∈ ∆ and λ ∈ S1. Being γ : ∆ → ∆ continuous it has at least one fixed point.
Now z0 ∈ ∆ is a fixed point for γ if and only if az02+ (λ − 1)z0− λa = 0. Therefore the
fixed points of γ are the roots in ∆ of the polynomial p(z) = az2+(λ−1)z −λa ∈ C[z].
If a = 0 then γ(z) = λz and the unique fixed point is 0. Let a 6= 0. In this case p(z) has degree 2, and therefore it has 2 complex roots z0, z1 in C, counted with
multiplicity. Now −λa = az0z1 and hence 1 = |z0||z1|. Without loss of generality we
can suppose z0 ∈ ∆. If z0 ∈ ∆ then |z1| > 1 and therefore there is a unique fixed
point in ∆, while if z0 ∈ ∂∆ then |z1| = 1, and therefore there are one or two fixed
points in ∆.
Accordingly to the previous proposition, we can classify the automorphisms of ∆ depending on their fixed points.
Definition 1.2.2. An automorphism γ ∈ Aut(∆) with 6= Id∆ is called
• elliptic if it has a unique fixed point in ∆; • parabolic if it has a unique fixed point in ∂∆; • hyperbolic if it has two distinct fixed points in ∂∆.
One other interesting consequence of Schwarz lemma is the following easy result. Proposition 1.2.3. Let f : ∆ → ∆ an holomorphic function. If f has at least two fixed points in ∆ then f = Id∆.
Proof. Up to conjugating with an automorphism of ∆ we can suppose f (0) = 0. Then there exists z0 6= 0 such that f (z0) = z0. Therefore |f (z0)| = |z0| and by Schwarz
lemma Theorem 1.1.1 we have f (z) = λz where λ ∈ S1. Therefore z
0 = λz0, so λ = 1
1.3
The upper half-plane
It is often useful, for example in the study of Aut(∆), to consider different models of the disk, that is complex manifolds biholomorphic to ∆. Classically, the model more used is that of the upper half-plane.
Definition 1.3.1. The upper half-plane is the domain of C given by H+ = {z ∈ C : Im z > 0}.
We have to check that ∆ and H+ are biholomorphic. Let us introduce the Cayley transform
Ψ : ∆ → H+, z 7→ i1 + z 1 − z. We have
Proposition 1.3.2. The Cayley transform is a biholomorphism, with inverse Φ : H+ → ∆, w 7→ w − i
w + i. Proof. Let z ∈ ∆. We have
Im Ψ(z) = Im i1 + z 1 − z = Im i 1 + z |1 − z|2(1 − z) = Im i1 + z − z − zz |1 − z|2 = Im i1 + 2i Im z − |z| 2 |1 − z|2 = Im i1 − |z| 2 |1 − z|2 − 2 Im z |1 − z|2 = 1 − |z| 2 |1 − z|2.
Therefore Ψ(z) ∈ H+ and being z ∈ ∆ arbitrary we have Ψ(∆) ⊆ H+. Now define the map Φ : H+ → C by Φ(w) = w − i w + i. If w ∈ H+ we have 1 − |Φ(w)|2 = 1 − w − i w + i · w + i w − i = 1 − |w|2+ iw − iw + 1 |w|2− iw + iw + 1 = −2iw + 2iw |w|2− iw + iw + 1 = 2i(w − w) |w|2+ i(w − w) + 1 = 4 Im w |w|2+ 2 Im w + 1.
Therefore Φ(w) ∈ ∆ and being w ∈ H+ arbitrary we have Φ(H+) ⊆ ∆. Finally it is
easy to check that Φ is the inverse of Ψ.
In the preceding proof we have found some useful relations, that we list here for future reference: Im Ψ(z) = Im i1 + z 1 − z = 1 − |z| |1 − z|2 (1.3.1) for all z ∈ ∆; 1 − |Φ(w)|2 = 4 Im w |w|2+ 2 Im w + 1 (1.3.2)
for all w ∈ H+.
Now let bC be the one-point compactification of C. Then we can consider the closure and the boundary of H+ in bC:
H+ = {z ∈ C : Im z ≥ 0} ∪ {∞},
∂H+ = {z ∈ C : Im z = 0} ∪ {∞}.
As expected ∆ and H+are homeomorphic. Indeed the Cayley transform Ψ : ∆ → H+
extends continuously to a homeomorphism Ψ∗: ∆ → H+ setting
Ψ∗(z) = (
i1+z1−z if z ∈ ∆ \ {1}, ∞ if z = 1.
Its inverse is Φ∗: H+→ ∆ given by
Φ∗(w) = (
w+i
w−i if z ∈ H+\ {∞},
1 if w = ∞.
In particular Ψ∗(∂∆) = ∂H+ and 1 ∈ ∆ is mapped to ∞.
Now we turn to the study of Aut(H+). Since the Cayley transform is a biholomor-phism and since ∆ and H+ are homeomorphic, it is evident that all the result seen
in the case of ∆ admit an obvious translation in our new vocabulary. For instance, Proposition 1.2.1 translates into the following:
Proposition 1.3.3. Let γ ∈ Aut(H+) with γ 6= IdH+. Then either
(1) γ has a unique fixed point in H+, or
(2) γ has a unique fixed point in ∂H+, or
(3) γ has exactly two fixed points in ∂H+.
Unsurprisingly we can introduce the following terminology.
Definition 1.3.4. An automorphism γ ∈ Aut(H+) with 6= IdH+ is called
• elliptic if it has a unique fixed point in H+;
• parabolic if it has a unique fixed point in ∂H+;
• hyperbolic if it has two distinct fixed points in ∂H+.
The usefulness of the upper half plane resides, for example, in the possibility of give a simpler description of its automorphisms (for a proof see, for instance, [Aba89]). Theorem 1.3.5. The automorphisms of H+ are the linear fractional transforma-tions, that is the holomorphic functions γ : H+→ H+ of the form
γ(w) = aw + b cw + d, with a, b, c, d ∈ R such that ad − bc = 1.
Thanks to the preceding theorem we can explicitly write the hyperbolic and parabolic automorphisms of H+.
Proposition 1.3.6. Let γ ∈ Aut(H+).
(1) If γ is an hyperbolic automorphism with 0 and ∞ as its fixed points then γ(w) = αw
for all w ∈ H+, where α ∈ (0, +∞) with α 6= 1.
(2) If γ is a parabolic automorphism with ∞ as its unique fixed point then γ(w) = w + β
for all w ∈ H+, where β ∈ R∗. Proof. Write
γ(w) = aw + b cw + d with a, b, c, d ∈ R and ad − bc = 1.
(1) We have γ(0) = 0 and therefore b = 0. Now γ(∞) = ∞ and hence c = 0. Therefore we have
γ(w) = a dw.
Now from ad = 1 it follows that a/d > 0. Moreover a/d 6= 1 since γ 6= IdH+ and
therefore
γ(w) = αw with α ∈ (0, +∞) and α 6= 1.
(2) From γ(∞) = ∞ we have c = 0 and therefore γ(w) = a
dw + b d.
Since ∞ is the unique fixed point we have a = d and in particular b/d 6= 0 since γ 6= Id∆. Therefore
γ(w) = w + β where β ∈ R∗.
1.4
The Poincar´
e distance
In Section 1.1 we have recalled the Schwarz lemma and some of its consequences, as the Schwarz-Pick lemma. We have noticed that there is a sort of rigidity in ∆. This is a hint of the existence of some more deep geometric structure on the unit disk, responsible for this behaviour. This structure is the Poincar´e distance, which will be the prototype for further generalizations such as the Kobayashi distance.
Definition 1.4.1. The Poincar´e metric on the disk ∆ is the Hermitian metric given by
K∆(z; ·, ·) =
1
(1 − |z|2)2dz ⊗ dz
for all z ∈ ∆, that is
K∆(z; v, w) =
vw (1 − |z|2)2
for all z ∈ ∆ and v, w ∈ Tz∆ ' C.
The norm of a vector v ∈ Tz∆ ' C is then
K∆(z; v) =
|v| 1 − |z|2.
Since ∆ is a complex manifold of dimension one K∆ is a K¨ahler metric. Moreover K∆
has constant sectional curvature equal to −4. As every Hermitian metric K∆ induces
a distance on ∆.
Definition 1.4.2. The distance κ∆: ∆ × ∆ → [0, +∞) induced by K∆ is called the
Poincar´e distance on ∆.
Now we compute the Riemannian metric g∆induced by K∆. If z = x + iy are real
coordinates on ∆ we have dz = dx + idy and dz = dx − idy. Then
dz ⊗ dz = (dx + idy) ⊗ (dx − idy) = dx ⊗ dx − idx ⊗ dy + idy ⊗ dx + dy ⊗ dy. (1.4.1) Thus we obtain the Riemannian metric
g∆(z; ·, ·) =
1
(1 − (x2+ y2))2(dx ⊗ dx + dy ⊗ dy). (1.4.2)
From (1.4.2) we immediately deduce that g∆ is invariant under rotations and
complex conjugation, which are isometries for the standard Hermitian metric of C. Thus we have
Proposition 1.4.3. Rotations z 7→ eiϑz and complex conjugation z 7→ z are isome-tries for the Poincar´e metric on ∆.
We can easily describe the isometries of ∆.
Theorem 1.4.4. The isometries of K∆are exactly the holomorphic and anti-holomorphic
automorphisms of ∆:
Isom(∆) = Aut(∆) ∪ Aut(∆).
Proof. Let f : ∆ → ∆ be a smooth diffeomorphism. Then f is an isometry if and only if f∗g∆= g∆. Let (x, y) be real coordinates on ∆ and let f = u + iv. Then
f∗g∆= 1 (1 − (u2+ v2))2(du ⊗ du + dv ⊗ dv). (1.4.3) Now du = ∂u ∂xdx + ∂u ∂ydy, dv = ∂v ∂xdx + ∂v ∂ydy. (1.4.4)
Then plugging (1.4.4) in (1.4.3) after some simple computations we obtain that f∗g∆= g∆ if and only if the following equations hold
(∂u∂x)2+ (∂u∂x)2 = 1−(u2+v2) 1−(x2+y2) 2 (∂u∂y)2+ (∂u∂y)2 =1−(u1−(x22+v+y22))
2 ∂u ∂x ∂u ∂y + ∂v ∂x ∂v ∂y = 0. (1.4.5)
If f ∈ Isom(∆) then it satisfies (1.4.5) form which follows ( (∂u∂x)2+ (∂u ∂x) 2 = (∂u ∂y) 2+ (∂u ∂y) 2 ∂u ∂x ∂u ∂y + ∂v ∂x ∂v ∂y = 0. (1.4.6)
Let a = ∂u∂x, b = ∂v∂y, c = ∂v∂x and d = ∂u∂y. Then ad + bc = 0 and therefore there exists λ ∈ R such that a = −λb and c = λd. Now from the equations (1.4.6) we have a2+ c2 = b2+ d2 and hence λ2 = 1. If λ = 1 then f is antiholomorphic while if
λ = −1 then f is holomorphic. Being f a smooth diffeomorphism it follows that f or f is a biholomorphism. This prove that Isom(∆) ⊆ Aut(∆) ∪ Aut(∆).
Finally, if f ∈ Aut(∆) ∪ Aut(∆) the equality case in Schwarz lemma Theorem 1.1.1 shows that f satisfies (1.4.5) and we are done.
From Theorem 1.4.4, recalling that Aut(∆) acts transitively on ∆, it follows that the Riemannian manifold (∆, g∆) is homogeneous. Then ∆ is geodetically complete
and hence is complete by the Hopf-Rinow theorem. In particular every maximal geodesic is defined for all t ∈ R and any two points in ∆ are joined by a geodesic. Theorem 1.4.5. The geodesics originating form 0 ∈ ∆ are the diameters of ∆. Moreover they are the unique (up to reparametrization) geodesics connecting 0 and z ∈ ∆, and thus realize the distance between these two points.
Proof. Let α : R → ∆ be a geodesic with α(0) = 0 and ˙α(0) = v. Consider the real line ` = Rv. Thank to the preceding results, the reflection ρ about ` is an isometry. Thus β = ρ ◦ α is a geodesics, with β(0) = 0 and ˙β(0) = ρ( ˙α(0)) = v. It follows that β = α, that is α is fixed by ρ. Thus we have α(R) ⊆ Fix(ρ) = `. By completeness we have α(R) = ∆ ∩ `, as claimed. Now let z ∈ ∆. If α is a geodesic connecting 0 to z then α is radial and contained in ` = ˙α(0)R ∩ ∆ = zR ∩ ∆. Then we have that
˙
α(0) ∈ zR and hence α in uniquely determined, up to parametrization.
Definition 1.4.6. The geodesics of ∆ originating from 0 are called radial geodesics. The preceding theorem allows us to compute the distance of a point z ∈ ∆ from the origin 0. Since Aut(∆) acts transitively and isometrically on ∆ it is not restrictive to suppose z real and 0 < z = |z| < 1. Let α : [0, a] → ∆ be a piecewise C1 curve
connecting 0 to z. Then κ∆(0, z) = Z a 0 K∆(α(t); ˙α(t))dt = Z a 0 | ˙α(t)| 1 − |α(t)|2dt ≥ Z a 0 Re ˙α(t) 1 − (Re α(t))2dt = Z z 0 1 1 − s2ds = 1 2log 1 + |z| 1 − |z|,
where we have performed the change of variable s = Re α(t). If we take α(t) = tz then we have the equality. Therefore we have
κ∆(0, z) =
1 2log
1 + |z|
1 − |z|. (1.4.7)
From the formula (1.4.7) above it is possible to deduce the general formula for the Poincar´e distance between two points of ∆.
Proposition 1.4.7. Let z, w ∈ ∆. Then the Poincar´e distance between z and w is given by κ∆(z, w) = 1 2log 1 + 1−wzz−w 1 − 1−wzz−w . (1.4.8)
Proof. Let γ ∈ Aut(∆) be the automorphism γ(z) = z − w
1 − wz. Then, being γ an isometry, we have
κ∆(w, z) = κ∆(γ(w), γ(z)) = κ∆(0, γ(z)) =
1 2log
1 + |γ(z)| 1 − |γ(z)|, and the claim follows.
Recall that
tanh−1: (−1, 1) → R, tanh−1(t) = 1 2log
1 + t 1 − t. Therefore (1.4.7) and (1.4.8) become, respectively
κ∆(0, z) = tanh−1(|z|) and κ∆(z, w) = tanh−1 z − w 1 − wz . Moreover tanh−1 is strictly increasing.
Now we can reformulate the Schwarz-Pick lemma using the Poincar´e distance. Theorem 1.4.8. Let f : ∆ → ∆ be a holomorphic function. Then:
(1) κ∆(f (z), f (w)) ≤ κ∆(z, w) for all z, w ∈ ∆;
(2) f∗(K∆) ≤ K∆.
Moreover, equality holds in (1) for some z 6= w or in (2) for some z ∈ ∆ if and only if f ∈ Aut(∆) and in this case equality holds everywhere.
Proof. (1) From the strict monotonicity of tanh−1 we have κ∆(f (z), f (w)) ≤ κ∆(z, w) ⇐⇒ f (z) − f (w) 1 − f (w)f (z) ≤ z − w 1 − wz
Thus the claim follows from Schwarz-Pick lemma Theorem 1.1.5. (2) We have f∗K∆(z; v, w) = K∆(f (z); f0(z)v, f0(z)w) = |f0(z)|2vw 1 − |f (z)|2. Therefore f∗K∆≤ K∆ ⇐⇒ |f (z)|2 1 − |f (z)|2 ≤ 1 1 − |z|2 ∀z ∈ ∆,
and so the claim follows again thanks to Schwarz-Pick lemma Theorem 1.1.5.
We have just discovered that using the Poincar´e distance every family of func-tions F ⊆ Hol(∆, ∆) is equicontinuous, more precisely 1-Lipschitz. This fact will be crucial in the study of holomorphic dynamics and is one of the main features of the Poincar´e distance that we would like to preserve in our future generalizations.
The metric balls are the sets naturally associated to a distance, and therefore we introduce them.
Definition 1.4.9. Let z ∈ ∆, R > 0. The Poincar´e ball of center z and radius R is the set B∆(z, R) = {w ∈ ∆ : κ∆(z, w) < R}. The function tanh−1(|t|) = 1 2log 1 + |t| 1 − |t| is strictly increasing. Thus we have
κ∆(z, w) < R ⇐⇒ z − w 1 − wz < tanh R. Therefore B∆(z, R) = w ∈ ∆ : z − w 1 − wz < tanh R .
It is possible to describe the Poincar´e balls in terms of the Euclidean metric of ∆. In fact, it is possible to prove [Aba89] that
B∆(z, R) = B(z0, ρ0), where z0 = 1 − tanh2R 1 − |z|2tanh2Rz, ρ = (1 − |z|2) tanh R 1 − |z|2tanh2R.
In particular B∆(z, R) ⊂⊂ ∆ that is the Poincar´e balls are relatively compact in ∆,
1.5
The Poincar´
e distance on the upper half plane
Using the Cayley transform we can transfer the Poincar´e metric and distance from ∆ to H+.
Definition 1.5.1. The Poincar´e metric on H+ is the Hermitian metric
KH+ = Ψ∗K∆
or in other words the push-forward of K∆ via Ψ.
We can compute KH+ explicitly. Let w ∈ H+ and u, v ∈ TzH+ ' C. Then
KH+(w; u, v) = K∆(Φ(w); Φ0(z)u, Φ0(w)v) = |Φ0(w)|2 (1 − |Φ(w)|2)2uv. (1.5.1) Now Φ0(w) = 2i (w + i)2, 1 − |Φ(w)| 2 = 4 Im w |w + i|2
and therefore from (1.5.1) with some simple computations we get KH+(w; u, v) = 1 4(Im w)2uv that is KH+ = 1 4(Im w)2dw ⊗ dw. (1.5.2)
With this definition Ψ : ∆ → H+is a holomorphic isometry with inverse given by Φ : H+ → ∆. In particular the distance induced on H+ by K
H+ is the push-forward
of the Poincar´e distance on ∆ via Ψ.
Definition 1.5.2. The distance on H+ induced by K
H+ is called the Poincar´e
dis-tance on H+.
In particular it is straightforward to check that κH+(w1, w2) = tanh−1 w1− w2 w1− w2 (1.5.3) for every w1, w2 ∈ H+.
We can reformulate the Schwarz-Pick lemma Theorem 1.1.5 in this context: Theorem 1.5.3 (Schwarz Pick lemma). Let f ∈ Hol(H+, H+). Then
(1) f (w1)−f (w2) f (w1)−f (w2) ≤ w1−w2 w1−w2 for all w1, w2 ∈ H +; (2) (Im f (w))|f0(w)|2 ≤ 1
(Im w)2 for all w ∈ H+.
Moreover, equality holds in (1) for some w1 6= w2 or in (2) for some w ∈ H+ if and
The Kobayashi distance on
complex manifolds
2.1
The Kobayashi distance
Now we want to generalize what seen so far to the case of complex manifolds, introducing a distance with analogous property to the Poincar´e distance on the unit disk. In particular we would like to preserve Lipschitz continuity of holomorphic functions, that is to obtain a generalized Schwarz-Pick lemma.
We remark that all manifolds are supposed to be connected. Moreover, if X, Y are complex manifolds, Hol(X, Y ) denote the set of holomorphic maps from X to Y . Definition 2.1.1. Let X be a complex manifold. The Lempert function
δX: X × X → [0, +∞]
is the function defined as
δX(z, w) = inf {κ∆(ζ0, ζ1) : ∃ϕ ∈ Hol(∆, X) such that ϕ(ζ0) = z and ϕ(ζ1) = w} ,
(2.1.1) for all z, w ∈ X.
The Lempert function is positive but can be +∞ for some pair of points z, w ∈ X. This happens exactly when does not exist any holomorphic function ϕ : ∆ → X such that z, w ∈ ϕ(∆). In this case, indeed, the set on the right hand side of (2.1.1) is empty and inf ∅ = +∞.
The definition of δX can be simplified as
δX(z, w) = inf{κ∆(0, ζ) : ∃ϕ ∈ Hol(∆, X) such that ϕ(0) = z and ϕ(ζ) = w}.
In fact it is sufficient to remember that Aut(∆) act transitively and isometrically on the unit disk ∆.
In general δX is not a distance on X, not only because it can obtain the value
+∞, but also because it does not satisfy the triangular inequality. Hence we are led to introduce the so called Kobayashi pseudo-distance. First of all, a preliminary definition.
Definition 2.1.2. Let S be a set. A pseudo-distance on S is a function d : S × S → [0, ∞)
such that:
(1) d(x, x) = 0 for all x ∈ S;
(2) d(x, y) = d(y, x) for all x, y ∈ S;
(3) d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z ∈ S.
In this case the pair (S, d) is called a psudo-metric space.
The substantial difference with a distance is that d(x, y) = 0 does not imply x = y. Moreover, if ∼ is the equivalence relation on S defined by x ∼ y if and only if d(x, y) = 0 then (S/∼, d
∼) is a metric space, where
d∼: S/∼×S/∼→ [0, ∞)
is given by d∼([x], [y]) = d(x, y).
After this brief digression let us return to our manifolds.
Definition 2.1.3. Let X be a complex manifold. The Kobayashi pseudo-distance on X is the function κX: X × X → [0, +∞) defined as κX(z, w) = inf ( k X j=1 δX(zj−1, zj) : z0, . . . , zk∈ X, z0 = z, zk= w, k ∈ N ) for all z, w ∈ X.
In some sense, given z and w ∈ X we are joining z to w with a finite number of points, that is with a sort of broken line, of which we compute the length using δX,
and finally we take the infimum of these lengths.
Remark 2.1.4. From the definition follows immediately that κX(z, w) ≤ δX(z, w) for
all z, w ∈ X.
As suggested by the name, one has
Proposition 2.1.5. κX is a pseudo-distance on X.
Proof. First of all we claim that κX(z, w) < +∞ for all z, w ∈ X. Indeed let
z ∈ X fixed. We say that w ∈ X is linkable to z if there exist z0, . . . , zk ∈ X with
z0 = z, zk = w such that for all j = 1, . . . , k the points zj−1, zj are contained in a
chart (Uj, ψj) of X with ψj(zj−1) = O and ψj(Uj) = Bn. It is easy to show that
every point w ∈ X is linkable to z, because X is connected. So let z, w ∈ X and z0, . . . , zk, (U1, ψ1), . . . , (Uk, ψk) be as above. Let xj = ψj(zj) and define ϕj: ∆ → X
as ϕj(ζ) = ψj−1 ζ xj |xj| .
Then ϕj(0) = zj−1, ϕj(|xj|) = zj and hence δX(zj−1, zj) < +∞ from which we have
κX(z, w) < +∞.
It remains to check that κX satisfies the triangular inequality, being the other two
conditions of Definition 2.1.2 easily verified. Let z, w, u ∈ X and set A = ( k X j=1 δX(zj−1, zj) : z0, . . . , zk∈ X, z0 = z, zk= w, k ∈ N ) , B = ( h X j=1 δX(wj−1, wj) : w0, . . . , wh ∈ X, w0 = w, wk= u, h ∈ N ) , C = ( l X j=1 δX(uj−1, uj) : u0, . . . , ul∈ X, u0 = z, ul = u, l ∈ N ) . Then A + B ⊆ C from which follows inf C ≤ inf A + inf B, or in other terms
κX(z, u) ≤ κX(z, w) + κX(w, u)
as wanted.
Remark 2.1.6. It is possible to have κX(z, w) = 0 with z 6= w. Indeed there are cases
in which κX is degenerated, that is κX ≡ 0. See, for instance, Proposition 2.1.10
One of the main property of κX is the following generalization of Schwarz-Pick
lemma.
Proposition 2.1.7. Let f : X → Y be a holomorphic map between complex mani-folds. Then
κY(f (z), f (w)) ≤ κX(z, y) (2.1.2)
for all z, w ∈ X. In particular:
(1) if X is a submanifold of Y , then κY|X×X ≤ κX;
(2) all biholomorphisms between X and Y are isometries for the Kobayashi distance. Proof. Let z, w ∈ X. If there exists ϕ ∈ Hol(∆, X) such that ϕ(0) = z and ϕ(ζ) = w then ψ = f ◦ ϕ ∈ Hol(∆, Y ) satisfies ψ(0) = f (0) and ψ(ζ) = f (w). This shows that
{κ∆(0, ζ) : ∃ϕ ∈ Hol(∆, X) such that ϕ(0) = z and ϕ(ζ) = w}
is contained in
{κ∆(0, ζ) : ∃ψ ∈ Hol(∆, Y ) such that ψ(0) = f (z) and ψ(ζ) = f (w)}
and thus δY(f (z), f (w)) ≤ δX(z, w). Now set
A = ( k X j=1 δX(zj−1, zj) : z0, . . . , zk ∈ X, z0 = z, zk = w, k ∈ N ) , B = ( h X j=1 δY(uj−1, uj) : u0, . . . , uk ∈ Y, u0 = f (z), uh = f (w), h ∈ N ) .
Then it is easy to show that for all a ∈ A there exists b ∈ B such that b ≤ a, and this implies inf B ≤ inf A, that is
κY(f (z), f (w)) ≤ κX(z, w).
Now (1) follows from (2.1.2) applied to the inclusion ι : X ,→ Y .
Finally if f : X → Y is a biholomorphism, with inverse g : Y → X, we have κX(z, w) = κX(g(f (z)), g(f (w))) ≤ κY(f (z), f (w)) ≤ κX(z, w),
and we are done.
In this way we have recovered the 1-Lipschitz continuity of holomorphic maps. Remark 2.1.8. In the proof of Proposition 2.1.7 we have proved something stronger, that is
δY(f (z), f (w)) ≤ δX(z, w)
for all z, w ∈ X, with equality if f is a biholomorphism.
Remark 2.1.9. It is useful to note that if δX is finite and satisfies the triangular
inequality then δX = κX.
In Section 1.4 we have denoted the Poincar´e distance on ∆ by κ∆. It is now time
to check that this notational choice is coherent with the definition of the Kobayashi pseudo-distance. The following proposition shows some interesting examples and answers positively to our doubt.
Proposition 2.1.10. (1) The Kobayashi pseudo-distance on ∆ coincides with the Poincar´e distance.
(2) If X = Cn or X = CPn then δX ≡ κX ≡ 0.
(3) For all z, w ∈ ∆n we have
δ∆n(z, w) = κ∆n(z, w) = max
j=1,...,nκ∆(zj, wj).
Proof. (1) Denote for the moment the Poincar´e distance on the unit disk by ω∆. Let
ϕ ∈ Hol(∆, ∆) with ϕ(ζ1) = z and ϕ(ζ2) = w. Then
ω∆(z, w) = ω∆(ϕ(ζ1), ϕ(ζ2)) ≤ ω∆(ζ1, ζ2)
and therefore ω∆(z, w) ≤ δ∆(z, w). Moreover, taking ϕ = id∆ in (2.1.1) we have
δ∆(z, w) ≤ ω∆(z, w) and thus ω∆(z, w) = δ∆(z, w). In particular δ∆ is a
pseudo-distance on ∆ and we conclude thanks to Remark 2.1.9.
(2) Let X = Cn and let z, w ∈ Cn. Suppose z = O and w = (w
1, . . . , wn) 6= O.
We can assume w1 6= 0. Let ν ∈ N be such that ν > |w1| and define ϕν: ∆ → Cn as
ϕν(ζ) =
ν w1
ζw. Then ϕν(0) = O and ϕν wν1 = w. It follows that
δCn(O, w) ≤ κ∆ 0,w1 ν = tanh−1 |w1| ν → 0 as ν → ∞
which implies δCn(O, w) = 0 and therefore κ
Cn(O, w) = 0.
Now in the general case if z, w ∈ Cn we have
κCn(z, w) ≤ κ
Cn(z, O) + κCn(O, w) = 0
that is the claim.
The case X = CPn easily follows. Indeed let (U
0, ϕ0), . . . , (Un, ϕn) be the affine
charts. Since every Uj is biholomorphic to Cn we have κUj ≡ 0. Moreover from
the inclusion Uj ,→ CPn and using (2.1.2) we get κCPn|Uj×Uj ≡ 0. Let p ∈ Uj and
q ∈ CPn. Then there exists a point u ∈ CPn such that both elements of the pairs
(p, u) and (q, u) are in the same affine chart. Therefore κCPn(p, q) ≤ κ
CPn(p, u) + κCPn(u, q) = 0.
(3) Let z, w ∈ ∆n. Suppose z = O and let ` be such that
|w`| = max j=1,...,n|wj|.
Now define ϕ : ∆ → ∆n with
ϕ(ζ) = ζ w`
w. Then ϕ(0) = O and ϕ(w`) = w. Thus
δ∆n(O, w) ≤ κ∆(0, w`) = max
j=1,...,nκ∆(0, wj).
For the converse consider ϕ : ∆ → ∆n with ϕ(0) = O and ϕ(ζ) = w. Writing
ϕ = (ϕ1, . . . , ϕn) we have
κ∆(0, wj) = κ∆(ϕj(0), ϕj(ζ)) ≤ κ∆(0, ζ)
for every j = 1, . . . , n which implies max j=1,...,nκ∆(0, wj) ≤ δ∆ n(O, w). Then we have max j=1,...,nκ∆(0, wj) = δ∆ n(O, w).
For the general case, given z and w ∈ ∆n, it is sufficient to consider γ ∈ Aut(∆n) such that γ(z) = O. First of all let us see that such a γ exists. Indeed it is sufficient to consider γ = (γ1, . . . , γn) where each γj ∈ Aut(∆) is such that γj(zj) = O. Now
we have
δ∆n(z, w) = δ∆n(O, γ(w)) = max
j=1,...,nκ∆(O, γ`(w)) = maxj=1,...,nκ∆(zj, wj).
Therefore δ∆n is a distance and finally we have
κ∆n(z, w) = δ∆n(z, w) = max
j=1,...,nκ∆(zj, wj)
2.2
Hyperbolic manifolds
We have seen in Section 2.1 that κX is a pseudo-distance and in some cases it is
degenerated. Therefore it makes sense to introduce a specific class of manifolds for which the Kobayashi distance is a real distance.
Definition 2.2.1. A complex manifold X is (Kobayashi) hyperbolic if κX is a distance
on X.
We are therefore interested in understanding when a manifold is hyperbolic. First of all let us explore the relationship between κX and the manifold topology of X.
Proposition 2.2.2. Let X be a complex manifold. Then κX is continuous with
respect to the manifold topology of X. Proof. If z0, w0, z, w ∈ X then
|κX(z0, w0) − κX(z, w)| ≤ κX(z0, z) + κX(w0, w).
Therefore it is sufficient to prove that the function z 7→ κX(z0, z) is continuous in
z0. If (U, ϕ) is a local chart in z0 with ϕ(U ) = ∆n then κU is continuous, being κ∆n
continuous. Hence z 7→ κU(z0, z) is continuous in z0 which together with κX|U ×U ≤
κU implies that z 7→ κX(z0, z) is continuous in z0.
From Proposition 2.2.2 it follows that the manifold topology on X is finer than the one induced by κX. It is clear that when the Kobayashi distance is degenerated it
does not induce the manifold topology. Actually this condition characterizes exactly hyperbolic manifolds. We need the following alternative description of κX.
Lemma 2.2.3. Let X be a complex manifold and let z, w ∈ X. Then κX(z, w) = inf ( k X j=1 κ∆(ζj−1, ζj) : ζ0, . . . , ζk∈ ∆, ϕ1, . . . , ϕk ∈ Hol(∆, X) with ϕ1(ζ0) = z, ϕk(ζk) and ϕj+1(ζj) = ϕj(ζj) ∀j = 1, . . . , k − 1, k ∈ N )
Proof. The inequality ≤ is obvious, so we have only to prove that ≥ holds. Let ε > 0. From the definition of κX there exist k ∈ N, z0, . . . , zk ∈ X with z0 = z, zk = w such
that
k
X
j=1
δX(zj−1, zj) ≤ κX(z, w) + ε/2.
From the definition of δX we have that for all j = 1, . . . , k there exists ϕj ∈ Hol(∆, X)
such that ϕj(ζj−1) = zj−1, ϕj(ζj) = zj and
κ∆(ζj−1, ζj) ≤ δX(zj−1, zj) + ε/2k
for some ζj ∈ ∆. Therefore ϕ1(ζ0) = z, ϕk(ζk) = w, ϕj+1(ζj) = ϕj(ζj) for all
j = 1, . . . , k − 1 and k X j=1 κ∆(ζj−1, ζj) ≤ k X j=1 δX(zj−1, zj) + ε/2 ≤ κX(z, w) + ε.
Proposition 2.2.4. X is hyperbolic if and only if κX induces the manifold topology
on X.
Proof. Suppose that κX induces the manifold topology on X. Now X is an Haussdorff
space and therefore κX(z, w) 6= 0 if z 6= w. Thus κX is a distance and X is a
hyperbolic manifold.
If X is a hyperbolic manifold it is sufficient to prove that for all z ∈ X and for all open neighbourhood U ⊆ X of z there exists a Kobayashi ball BX(z, r) contained
in U . Let z ∈ X and U ⊆ X be as above, where we can suppose U compact. By contradiction, if such a ball BX(z, r) does not exist then we get a sequence (zν) in
X such that zν ∈ U and κ/ X(zν, z) ≤ 1/ν for all ν ∈ N. Given ν ∈ N, thanks to
Lemma 2.2.3, there exist ζ0, . . . , ζkν ∈ ∆ and ϕ1, . . . , ϕkν ∈ Hol(∆, X) with ϕ1(ζ0) =
z, ϕkν(ζkν) = zν and ϕj+1(ζj) = ϕj(ζj) for all j = 1, . . . , kν − 1 such that
k
X
j=1
κ∆(ζj−1, ζj) ≤ 1/ν.
Let σj be the geodesic arc in ∆ connecting ζj−1 to ζj. The arcs ϕ1◦ σ1, . . . , ϕkν◦ σkν
can be connected to form a continuous curve σ in X from z to zν. On the other hand,
since z ∈ U and zν ∈ U there exists w/ ν ∈ ∂U on σ. If w is a point on σ we have
κX(z, w) ≤ 1/ν and therefore
lim
ν→∞κ∆(z, wν) = 0.
Now ∂U is compact and κ∆(z, ·) is continuous and strictly positive on ∂U . Therefore
infw∈∂UκX(z, w) > 0; contradiction.
Let us see how we can obtain hyperbolic manifolds. It will be useful the following result.
Proposition 2.2.5. Let X, Y be complex manifolds. Then
max{κX(z1, z2), κY(z2, w2)} ≤ κX×Y((z1, w1), (z2, w2)) ≤ κX(z1, z2) + κY(w2, w2)
(2.2.1) for all z1, z2 ∈ X and w1, w2 ∈ Y .
Proof. Let πX: X × Y → X and πY : X × Y → Y be the natural projection. Then
by Proposition 2.1.7 we have
κX(z1, z2), κY(w1, w2) ≤ κX×Y((z1, w1), (z2, w2))
and hence the first inequality follows. Let ι1: X → X × Y and ι2: Y × X × Y be
given by ι1(z) = (z, w1) and ι2(w) = (z2, w). Therefore
κX×Y((z1, w1), (z2, w1)) = κX×Y(ι1(z1), ι1(z2)) ≤ κX(z1, z2) (2.2.2)
κX×Y((z2, w1), (z2, w2)) = κX×Y(ι2(w1), ι2(w2)) ≤ κY(w1, w2) (2.2.3)
Now summing (2.2.2) and (2.2.3) and using the triangular inequality we have κX(z1, z2) + κY(w1, w2) ≥ κX×Y((z1, w1), (z2, w1)) + κX×Y((z2, w1), (z2, w2))
≥ κX×Y((z1, w1), (z2, w2))
Proposition 2.2.6. (1) Let X be a hyperbolic manifold. If Y ⊆ X is a submanifold then it is hyperbolic; in particular, if D ⊆ Cn is a bounded domain then it is
hyperbolic;
(2) Let X, Y be complex manifold. Then X × Y is hyperbolic if and only if both X and Y are hyperbolic.
Proof. The first part of (1) and of (2) are easy consequences of 2.1.7 and 2.2.5, respectively.
If D ⊆ Cn is a bounded domain then there exists a polydisk ∆n(O, r) containing D. Being ∆n(O, r) hyperbolic we conclude thanks to the first part of (1).
2.3
Complete hyperbolic manifolds
It is natural to ask whether the Kobayashi distance of a hyperbolic manifold is complete or not. We therefore introduce the following notion.
Definition 2.3.1. An hyperbolic manifold X is complete hyperbolic if κX is a
com-plete distance.
Proposition 2.3.2. (1) Let X be a complete hyperbolic manifold. If Y ⊆ X is a closed submanifold, then Y is a complete hyperbolic manifold.
(2) Let X, Y be hyperbolic manifolds. Then X × Y is complete hyperbolic if and only if both X and Y are.
Proof. The claim (1) follows from κX|Y ×Y ≤ κY and Y being closed, while (2) follows
from Proposition 2.2.5.
In particular from the preceding proposition it follows that ∆nis complete hyper-bolic, being the product of n copies of ∆, which is complete hyperbolic.
To better understand completeness we need to study the metric balls of X. The aim is to formulate a result analogous to Hopf-Rinow theorem in Riemannian geom-etry, which will link completeness and compactness.
Definition 2.3.3. Let X be a complex manifold and let z0 ∈ X, r > 0 and A ⊆ X.
• BX(z0, r) = {z ∈ X : κX(z0, z) < r} is called the (open) Kobayashi ball of center
z0 and radius r;
• BX(z0, r) = {z ∈ X : κX(z0, z) ≤ r} is called the closed Kobayashi ball of center
z0 and radius r;
• BX(A, r) =
S
a∈ABX(a, r) is called the r-neighborhood of A.
The first result is the following useful lemma:
Lemma 2.3.4. Let X be a complex manifold and let z0 ∈ X, r, s > 0. Then
Proof. Let w ∈ BX(BX(z0, r), s). Then w ∈ BX(z, s) for some z ∈ BX(z0, r).
There-fore
κX(z0, w) ≤ κX(z0, z) + κX(z, w) < r + s,
that is w ∈ BX(z0, r + s). This proves the inclusion ⊆.
Now let z ∈ BX(z0, r+s) and put 3ε = r+s−κX(z0, z). Then r+s−2ε > κX(z0, z)
and from Lemma 2.2.3 there exist ζ0, . . . , ζk ∈ ∆ and ϕ1, . . . , ϕk ∈ Hol(∆, X) with
ϕ1(ζ0) = z0, ϕk(ζk) = z, ϕj(ζj) = ϕj+1(ζj) for all j = 1, . . . , k − 1 such that k
X
j=1
κ∆(ζj−1, ζj) < r + s − 2ε.
Let µ ≤ k be the largest integer such that Pµ
j=1κ∆(ζj−1, ζj) < r − ε. If µ = k then we have k X j=1 κ∆(ζj−1, ζj) < r − ε
and therefore κX(z0, z) < r. This says that z ∈ BX(z0, r) ⊆ BX(BX(z0, r), s) and we
are done. Thus we can assume µ ≤ k − 1. Since
µ
X
j=1
κ∆(ζj−1, ζj) + κ∆(ζµ, ζµ+1) ≥ r − ε
there exists a point ηµ on the geodesic arc in ∆ connecting ζµ to ζµ+1 such that µ
X
j=1
κ∆(ζj−1, ζj) + κ∆(ζµ, ηµ) = r − ε.
Let w = ϕµ+1(ηµ) ∈ X. Then κX(z0, w) < r and κX(w, z) < s and thus z ∈ BX(w, s).
Since w ∈ BX(z0, r) then z ∈ BX(BX(z0, r), s). This proves the inclusion ⊇ and hence
the claim.
Thanks to Lemma 2.3.4 it is possible to show that the closure of an open Kobayashi ball is the closed ball with the same center and radius.
Proposition 2.3.5. Let X be a complex manifold and let z0 ∈ X, r > 0. Then
BX(z0, r) = BX(z0, r) in the distance topology. In particular, if X is hyperbolic this
holds in the manifold topology.
Proof. The inclusion BX(z0, r) ⊆ BX(z0, r) is trivial, so we have only to show the
converse.
So let z ∈ BX(z0, r). If κX(z0, z) < r we are done. Therefore suppose κX(z0, z) =
r and let ε > 0. From Lemma 2.3.4 we have
BX(z, ε) ⊆ BX(z0, r + ε) = BX(BX(z0, r), ε).
Then there exists w ∈ BX(z0, r) such that z ∈ BX(w, ε). In particular w ∈ BX(z, ε)
that implies BX(z0, r) ∩ BX(z, ε) 6= ∅. Since this holds for every ε > 0 we get
We need another lemma.
Lemma 2.3.6. Let X be a hyperbolic manifold and z0 ∈ X, r > 0. If there exists
ρ > 0 such that BX(z, ρ) is compact for all z ∈ BX(z0, r) then BX(z0, r) is compact.
Proof. Since X is locally compact and hyperbolic there exists 0 < s < r such that BX(z0, s) is compact. Therefore it is sufficient to show that if BX(z0, s) is
com-pact then even BX(z0, s + ρ/2) is compact. Let (zν) ⊆ BX(z0, s + ρ/2) and choose
(wν) ⊆ BX(z0, s) such that κX(zν, wν) < 3ρ4 . Up to subsequences we can assume (wν)
converging to w ∈ BX(z0, s). Then zν ∈ BX(w, ρ) for ν large enough and thus (zν)
admits a converging subsequence.
Finally we have the announced result.
Theorem 2.3.7. Let X a hyperbolic manifold. Then X is complete hyperbolic if and only if every closed Kobayashi ball is compact.
Proof. Assume that every Kobayashi closed ball in X is compact. If (zν) is a Cauchy
sequence in X then it is bounded, that is it is contained in a closed Kobayashi ball. Therefore it admits a converging subsequence and we are done.
Conversely suppose that X is complete. Thanks to Lemma 2.3.6 it is sufficient to show that there exists ρ > 0 such that BX(z0, ρ) is compact for every z0 ∈ X. By
contradiction assume that such a ρ does not exists. Then there exists z1 ∈ X such that
BX(z1, 1/2) is not compact. Again from Lemma 2.3.6 there exists z2 ∈ BX(z1, 1/2)
such that BX(z2, 1/4) is not compact. Proceeding in this way we construct a sequence
(zν) such that zν ∈ BX(zν−1, 1/2ν−1) and BX(zν, 1/2ν) is not compact for all ν ≥ 2.
Clearly (zν) is a Cauchy sequence and therefore zν → z ∈ X. Being X locally
compact there exists ε > 0 such that BX(z, ε) is compact. If ν is large enough then
BX(zν, 1/2ν) ⊆ BX(z, ε) and therefore BX(zν, 1/2ν) is compact against the definition
of BX(zν, 1/2ν).
This Hopf-Rinow like criterion is useful in a number of situation. In particular it allows us to obtain results analogous to the ones well known in Riemannian geometry. First of all the following obvious fact holds true
Proposition 2.3.8. Let X be a compact hyperbolic manifold. Then X is complete. It is evident that the result is false without the hyperbolicity assumption, as the example of CPn shows.
Definition 2.3.9. A complex manifold X is homogeneous if Aut(X) acts transitively on X.
Proposition 2.3.10. An homogeneous hyperbolic manifold is complete. In particular every bounded and homogeneous domain of Cn is complete hyperbolic.
Proof. Let z0, z ∈ X. Since X is locally compact and hyperbolic there exists s > 0
such that the closed ball B(z0, s) is compact. Now by homogeneity there exists an
automorphism f : X → X such that f (z0) = z. Then
BX(z, s) = f (BX(z0), s)
Introduction to holomorphic
dynamical systems
3.1
Some preliminaries on function spaces
In the study of dynamics we will work with function spaces. So it is useful to recall some basic notions about them.
Let X and Y be topological spaces, and set
C(X, Y ) = {f ∈ YX: f is continuous}
The function space C(X, Y ) is considered with the compact-open topology. If K ⊆ X is compact and U ⊆ Y is open, set
W(K, U ) = {f ∈ C(X, Y ) : f (K) ⊆ U }. Then the family
B = {W(K, U ) : K ⊆ X compact, U ⊆ Y open}
of subsets of C(X, Y ) is a pre-basis for the compact-open topology (see [Mun75]). Moreover if (Y, d) is a metric space then the compact-open topology is the topology of uniform convergence on compact subsets. In particular this says that uniform convergence on compact subsets is independent of the specific distance on Y , or in other terms that topologically equivalent distances on Y induce the same notion of convergence (see [Mun75]). Thus we can put on Y the distance we prefer and that is more suitable for our purposes, provided that it induces the given topology.
Since we shall deal with sequences it is useful to know when the compact-open topology is metrizable (see [Kel75]).
Proposition 3.1.1. Let X be a second countable topological space and Y a metric space. Then the space C(X, Y ) of continuous functions equipped with the compact-open topology is metrizable.
The other result we will need is Ascoli-Arzel`a theorem. First of all let us recall a definition.
Definition 3.1.2. A family of continuous functions F ⊆ C(X, Y ), where (Y, d) is a metric space, is equicontinuous at the point x ∈ X if for every ε > 0 there exists U ⊆ X neighbourhood of x such that d(f (y), f (x)) < ε for all y ∈ U and f ∈ F . The family F is equicontinuous if it is equicontinuous at every point of X.
Theorem 3.1.3 (Ascoli-Arzel`a). Let X be a locally compact metric space and Y a metric space. Let F ⊆ C(X, Y ) be a family of continuous functions. Then F is relatively compact in C(X, Y ) if and only if the following conditions hold
F is equicontinuous;
for all x ∈ X the set F(x) = {f(x): f ∈ F} is relatively compact in Y .
For a proof consult, for example, [Kel75]. The next result shows the power of the Ascoli-Arzel`a Theorem 3.1.3, giving a sufficient condition for uniform convergence. Corollary 3.1.4. Let X be a locally compact separable metric space and Y a metric space. Let (fν) ⊆ C(X, Y ) be a sequence pointwise converging to f ∈ YX. If (fν)
is equicontinuous then f ∈ C(X, Y ) and (fν) converges to f uniformly on compact
subsets.
Proof. Let x ∈ X. Since (fν(x)) converges the set {fν(x)} ⊆ Y is relatively compact
in Y . Thanks to Ascoli-Arzel`a Theorem 3.1.3 the family (fν) is relatively compact in
C(X, Y ). If h ∈ C(X, Y ) is the pointwise limit of a subsequence (fkν) then for every
x ∈ X the sequence (fkν(x)) converges to h(x). This implies that h = f and therefore
(fν) has a unique limit point f . So (fν) converges to f uniformly on compact subsets
and in particular f ∈ C(X, Y ).
A concept strictly related to equicontinuity is that of normality. As we will see, normality will be one of the leading concept in this work. In the following the term “convergence” is to be intended as convergence in the compact-open topology. Definition 3.1.5. A sequence (fν) ⊆ C(X, Y ) is compactly divergent, or divergent
on compact subsets, if for every compact set H ⊆ X and K ⊆ Y there exists ν0 ∈ N
such that fν(H) ∩ K = ∅ for all ν ≥ ν0.
In other terms (fν) is compactly divergent if it escapes from every compact.
Definition 3.1.6. A family F ⊆ C(X, Y ) is normal if every sequence in F admits either a convergent subsequence or a compactly divergent subsequence.
A sequence (fν) ⊆ F can have both converging or compactly divergent
subse-quences, but for the same subsequence the two conditions are mutually exclusive. Proposition 3.1.7. Let X be a separable topological space and Y be a compact metric space. Then F ⊆ C(X, Y ) is normal if and only if is relatively compact.
Proof. First of all observe that being Y compact no sequence in C(X, Y ) can be compactly divergent. Moreover it is sufficient to prove that normality implies relative compactness. Let F be normal. Then every sequence in F admits a convergent subsequence and therefore F is relatively compact in C(X, Y ).
Compact divergence can be interpreted as uniform convergence in an appropriate function space. Let Y∗ be the one-point compactification of Y . We have
Proposition 3.1.8. Let X, Y be topological spaces. A sequence (fν) ⊆ C(X, Y )
is compactly divergent if and only if (fν) converges to the constant function ∞ in
C(X, Y∗) endowed with the compact-open topology.
Proof. Suppose that (fν) converges to ∞ in C(X, Y∗). Let H ⊆ X and K ⊆ Y
be compact subsets. Then U = Y∗ \ K is an open neighbourhood of the point ∞ in Y∗ and W(H, U ) is an open neighbourhood of the constant function ∞ in
C(X, Y∗). Then there exists ν0 ∈ N such that fν ∈ W(H, U ) for all ν ≥ ν0. Therefore
fν(H) ∩ K = ∅ for all ν ≥ ν0 that is (fν) is compactly divergent.
Conversely suppose that (fν) is compactly divergent. If W ⊆ C(X, Y∗) is an open
set containing the constant function ∞ we have to show that fν ∈ W definitively. It
is sufficient to consider the case
W = W(K1, U1) ∩ . . . ∩ W(Km, Um)
where K1, . . . , Km are compact subsets of X and U1, . . . , Um are open subsets of Y∗
containing ∞. In particular there exist H1, . . . , Hm compact subsets of Y such that
Y∗\ Hi ⊆ Ui for all i = 1, . . . , m. Now there exists ν0 ∈ N such that fν(Hi) ∩ Ki = ∅
for all ν ≥ ν0 and i = 1, . . . , m. Therefore fν ∈ W for all ν ≥ ν0 and (fν) converges
to ∞ in C(X, Y∗), since W is arbitrary.
Proposition 3.1.9. Let X, Y be topological spaces and F ⊆ C(X, Y ) be a normal family. If a sequence (fν) ⊆ F is not compactly divergent then it has a converging
subsequence.
Proof. Suppose by contradiction that (fν) has no converging subsequences. Then
every subsequence (fνk) has a compactly divergent subsequence, that is a subsequence
converging to the constant function ∞ in C(X, Y∗). Then (fν) converges to ∞ in
C(X, Y∗) and this is a contradiction.
We shall need a multivariable version of Weierstrass convergence theorem. The proof is similar to that in one variable.
Proposition 3.1.10 (Weierstrass). Let Ω ⊆ Cnbe a domain and let (f
ν) ⊆ Hol(Ω, C)
be a sequence of holomorphic functions. If (fν) converges to f ∈ C(Ω, C) then f is
holomorphic.
Proof. Let z0 ∈ Ω and let D = ∆(z0, r) ⊂⊂ Ω be a polydisk. From Cauchy integral
formula we have fν(z) = 1 2πi nZ dD fν(ζ) (ζ1− z1) · · · (ζn− zn) dζ (3.1.1)
for all z ∈ D and ν ∈ N, where dD ⊆ ∂D is the distinguished boundary of D. Since fν → f uniformly on compact subsets, passing to the limit in (3.1.1) as ν → ∞ we
obtain f (z) = 1 2πi nZ dD f (ζ) (ζ1− z1) · · · (ζn− zn) dζ
for all z ∈ D. Therefore f is holomorphic on D and being z0 ∈ Ω any point we
Remark 3.1.11. The preceding theorem can be generalized to holomorphic maps (fν) ⊆ Hol(Ω, Cm) considering each single component.
Theorem 3.1.12. Let X, Y be complex manifolds and (fν) ⊆ Hol(X, Y ). If (fν)
converges to f ∈ C(X, Y ) then f is holomorphic.
Proof. Let p ∈ X and set q = f (p) ∈ Y . Since f is continuous there exist (U, ϕ) and (V, ψ) local charts of X at p and of Y at q such that U is compact and f (U ) ⊆ V . Then there exists ν0 ∈ N such that fν(U ) ⊆ V for all ν ≥ ν0. Set ef = ψ ◦ f |U ◦ ϕ−1
and efν = ψ ◦ fν|U ◦ ϕ−1. Then efν ∈ Hol(ϕ(U ), ψ(V )) and ef ∈ C(ϕ(U ), ψ(V )). Being
ϕ and ψ biholomorphisms ( efν) converges to ef in C(ϕ(U ), ψ(V )). From Weierstrass
Theorem 3.1.10 and Remark 3.1.11 we have ef ∈ Hol(ϕ(U ), ψ(V )) and therefore f |U ∈
Hol(U, V ). Since this holds for every p ∈ X we have the claim.
3.2
Holomorphic dynamical systems
This short section is devoted to introduce the basic concepts of holomorphic dy-namics and to furnish a glossary for future references.
Definition 3.2.1. Let X be a complex manifold and f ∈ Hol(X, X). The pair (X, f ) is called a holomorphic, or complex, dynamical system.
In the following, we will refer to a holomorphic dynamical system (X, f ) specifying only the map f . Moreover, the function space C(X, Y ), where X and Y are complex manifolds, is equipped with the compact-open topology, as usual.
Given a holomorphic dynamical system (X, f ), it is natural to consider the iterates of f , that is the sequence (fν)ν∈N, where
fν = f ◦ · · · ◦ f
| {z }
ν times
.
Definition 3.2.2. Let f ∈ Hol(X, X). The (positive or forward ) orbit of z0 ∈ X is
the set
O+(z
0) = {fν(z0) : ν ∈ N}.
Given f ∈ Hol(X, X), studying (X, f ) amounts to know the behaviour of all of its orbits. This is often too difficult a task, and it is preferable to study the global asymptotic behaviour of the sequence (fν), which is the approach we will follow.
Definition 3.2.3. Let X Y be complex manifolds and let f ∈ Hol(X, Y ). A holo-morphic map g ∈ Hol(X, Y ) is a limit point of f if there exists a subsequence (fνk)
that converges to f . We will denote by Γ(f ) ⊆ Hol(X, Y ) the set of all limit points of f . If X = Y and IdX ∈ Γ(f ) the map f is called pseudo-periodic.
Remark 3.2.4. Often we shall explicitly write Γ(f ) ⊆ Hol(X, Y ) if X and Y are not clear by the context.
Definition 3.2.5. Let f ∈ Hol(X, X). A point z0 ∈ X is called a fixed point if
f (z0) = z0 and a periodic point if there exists ν ∈ N such that fν(z0) = z0. In this
case, the integer
P (f, z0) = min{ν ∈ N : fν(z0) = z0}
is the period of z0.
Definition 3.2.6. Let f ∈ Hol(X, X). The map f is said periodic if there exists ν ∈ N such that fν = Id
X. In this case, the integer
P (f ) = min{ν ∈ N : fν = IdX}
is the period of f .
Given two holomorphic dynamical systems, we would like to know if they share the same dynamical properties, in some sense. This idea is formalized in the following notion.
Definition 3.2.7. Two holomorphic dynamical systems (X, f ) and (Y, g) are conju-gated if there exists a biholomorphism ψ : Y → X such that g = ψ−1◦ f ◦ ψ.
If (X, f ) and (Y, g) are conjugated via ψ, then (1) gk = ψ−1◦ fk◦ ψ;
(2) O+(ψ(z0)) = ψ(O+(z0)) for all z0 ∈ X;
(3) the subsequences (gνk) and (fνk) have the same behaviour;
(4) Γ(g) = ψ−1Γ(f )ψ.
Example 3.2.8. Let γ ∈ Aut(∆) be an elliptic automorphism. We already know that γ is conjugated to a rotation η(z) = eiϑz with ϑ ∈ R. Therefore the dynamics
Dynamics on taut manifolds
4.1
Taut manifolds
Taut manifolds are a class of manifolds intermediate between hyperbolic and com-plete hyperbolic manifolds. First of all, it is possible to show that the hyperbolicity condition can be characterized in terms of compactness of Hol(∆, X). This will justify the definition of tautness condition.
Proposition 4.1.1. [Aba93] A complex manifold X is hyperbolic if and only if Hol(∆, X) is relatively compact in C(∆, X∗). In particular if X is compact then X is hyperbolic if and only of Hol(∆, X) is compact. Moreover if X is hyperbolic then Hol(Y, X) is relatively compact in C(Y, X∗) for every complex manifold Y .
Let us consider Hol(∆, X) ⊆ C(∆, X∗). If X is hyperbolic but not necessarily compact then there can exist non constant maps f ∈ Hol(∆, X) such that ∞ ∈ f (∆) or, in other terms, such that f (∆) ∩ X 6= ∅ and ∞ ∈ f (∆). Such maps are neither constant or holomorphic, and so we would like to get rid of them. Taut manifolds allow us to exclude these maps.
Definition 4.1.2. A complex manifold X is a taut manifold if it is hyperbolic and every map f ∈ Hol(∆, X∗) ⊆ C(∆, X∗) is either holomorphic or constant equal to ∞.
Using the concept of normality we can give a different formulation of the taut condition, avoiding the use of the compactification X∗.
Proposition 4.1.3. Let X be a complex manifold. Then X is taut if and only if Hol(∆, X) is a normal family.
Proof. Let X be taut. Then X is hyperbolic and Hol(∆, X) is relatively compact in C(∆, X∗) thanks to Proposition 4.1.1. If (fν) ⊆ Hol(∆, X) then there exists a
subsequence (fνk) converging to a function f ∈ C(∆, X
∗). Now f ∈ Hol(∆, X);
hence f ∈ Hol(∆, X) or f ≡ ∞ thanks to tautness condition. Therefore Hol(∆, X) is a normal family.
Conversely suppose that Hol(∆, X) is a normal family. If (fν) ⊆ Hol(∆, X) then
there exists a subsequence (fνk) either converging or compactly divergent. In both
cases (fνk) is converging in C(∆, X
∗). Therefore Hol(∆, X) is relatively compact
in C(∆, X∗) and X is hyperbolic. Moreover if f ∈ Hol(∆, X) then there exists 29
(fν) ⊆ Hol(∆, X) converging to f . Thanks to normality, up to a subsequence (fν) is
either converging or compactly divergent. In the first case f ∈ Hol(∆, X) while in the second f ≡ ∞. This proves that X is taut.
The following result tell us that the role of ∆ is not essential (for a proof see [Aba89]).
Proposition 4.1.4. If X is a taut manifold then Hol(Y, X) is a normal family for every complex manifold Y .
Remark 4.1.5. It is immediate to see that X is taut if and only if Hol(∆, X) ∪ {∞} is compact. In this case we have Hol(∆, X) = Hol(∆, X) ∪ {∞}.
We now show that taut manifolds exist.
Theorem 4.1.6. Let X be a complete hyperbolic manifold. Then X is a taut mani-fold.
Proof. Let (fν) ⊆ Hol(∆, X). If (fν) is compactly divergent we are done. Therefore
we can suppose that (fν) is not compactly divergent. Up to a subsequence there exist
H ⊆ ∆ and K ⊆ X compact subsets such that fν(H) ∩ K 6= ∅ for all ν ∈ N. Let
ζ0 ∈ H and z0 ∈ K. If ζ ∈ ∆ we have
κX(fν(ζ), z0) ≤ κX(fν(ζ), fν(ζ0)) + κX(fν(ζ0), z0) ≤ κ∆(ζ, ζ0) + κX(fν(ζ0), z0).
(4.1.1) Now for every ν ∈ N there exists ζν ∈ H such that fν(ζν) ∈ K. Then
κX(fν(ζ0), z0) ≤ κX(fν(ζ0), fν(ζν)) + κX(fν(ζν), z0) ≤ κ∆(ζ0, ζν) + κX(fν(ζν), z0).
(4.1.2) Set r1 = max{κ∆(ζ0, ζ) : ζ ∈ H} and r2 = max{κX(z0, w) : w ∈ K}. Using (4.1.1)
and (4.1.2) we get
κX(fν(ζ), z0) ≤ κ∆(ζ, ζ0) + κX(fν(ζ0), z0) ≤ κ∆(ζ, ζ0) + r1 + r2 = r(ζ)
for all ν ∈ N. Therefore {fν(ζ) : ν ∈ N} ⊆ X is relatively compact in X for every
ζ ∈ ∆ and we can apply Ascoli-Arzel`a Theorem 3.1.3. Therefore there exists a subsequence (fνk) converging to f ∈ C(∆, X). Finally f ∈ Hol(∆, X) thanks to
Weierstrass Theorem 3.1.10.
Remark 4.1.7. The converse of Theorem 4.1.6 is false. In fact there exist taut man-ifolds which are not hyperbolic complete. For an example see, for instance, [Ros82] .
Proposition 4.1.8. (1) If Y ⊆ X is a closed submanifold of a taut manifold X then Y is taut.
(2) Let X, Y be complex manifolds. Then X × Y is taut if and only if both X and Y are taut.
Proof. (1) First of all Y is hyperbolic thanks to Proposition 2.2.6. Let (fν) ⊆
Hol(∆, Y ). Then (fν) ⊆ Hol(∆, X). Therefore, being X taut, there exists a
sub-sequence (fνk) either compactly divergent or convergent in C(∆, X). In the first case
it is easy to prove that (fνk) is compactly divergent in C(∆, Y
∗). In the second case
if f ∈ C(∆, X) is the limit of (fνk) then f (∆) ⊆ Y by closedness of Y . Moreover
f ∈ Hol(∆, Y ) thanks to Weierstrass theorem. It follows that Y is taut.
(2) We know from Proposition 2.2.6 that X × Y is hyperbolic if and only if X and Y are. If X and Y are taut then X × Y is hyperbolic. Let (ϕν) ⊆ Hol(∆, X × Y ) with
ϕν = (fν, gν), where fν ∈ Hol(∆, X) and gν ∈ Hol(∆, Y ). Now up to a subsequence
(fν) and (gν) converge respectively to f ∈ Hol(∆, X) ∪ {∞} and to g ∈ Hol(∆, Y ) ∪
{∞}. If f and g are both finite then (ϕν) converges to (f, g) ∈ Hol(∆, X × Y ). If
f ≡ ∞ or g ≡ ∞ we claim that ϕν is compactly divergent. Indeed suppose f = ∞
and let H ⊆ ∆ and K ⊆ X × Y be compact subsets. Now eK = πX(K) ⊆ X is
compact and therefore there exists ν0 ∈ N such that fν(H) ∩ eK = ∅ for all ν ≥ ν0.
Hence we have ϕk(H) ∩ K = ∅ for all ν ≥ ν0 and thus (ϕk) is compactly divergent.
This proves that X × Y is taut.
Conversely suppose that X × Y is taut. In this case X and Y are hyperbolic. Let (fν) ⊆ Hol(∆, X) and set ϕν = (fν, w) where w ∈ Y is fixed. Then up to
a subsequence (ϕk) converges to ϕ ∈ Hol(∆, X × Y ) ∪ {∞}. If ϕ is finite then
ϕ = (f, w) with f ∈ Hol(∆, X) and therefore (fν) converges to f . If ϕ ≡ ∞ we have
to prove that (fν) is compactly divergent. In fact if H ⊆ ∆ and K ⊆ X are compact
subsets then L = K × {w} ⊆ X × Y is compact. Therefore there exists ν0 ∈ N such
that ϕν(H) ∩ L = ∅ for all ν ≥ ν0. It follows that fν(H) ∩ K = ∅ for all ν ≥ ν0 and
so (fν) is compactly divergent. This proves that X is taut and in the same manner
we can see that Y is taut.
4.2
Dynamics on taut manifolds
We are now interested in holomorphic dynamical systems (X, f ) where X is a taut manifold. In this case (fν) is a normal family by Proposition 4.1.4 and we can
subdivide our study in two cases: (a) (fν) is not compactly divergent;
(b) (fν) is compactly divergent.
Moreover we are asking for conditions implying (a) and possibly that are equivalent to it. The fundamental idea is that in case (a) the dynamics is well behaved, for example (fν) has converging subsequences. Therefore we have a dichotomy between
convergence and divergence.
Definition 4.2.1. Let X be a complex manifold. A holomorphic retraction is a map ρ ∈ Hol(X, X) such that ρ2 = ρ. In this case, ρ(X) is said a holomorphic retract of X.
If ρ is a holomorphic retraction then ρ(X) = Fix(ρ) and the dynamics of ρ is trivial, being constant after the first step:
Nevertheless holomorphic retractions play a central role in the dynamics on taut manifolds.
Proposition 4.2.2. Let X be a complex manifold and let f ∈ Hol(X, X). If (fν)
converges to ρ ∈ Hol(X, X) then ρ is a holomorphic retraction.
Proof. If (fν) converges to ρ then the subsequence (f2ν) still converges to ρ. Now
f2ν = fν ◦ fν and therefore f2ν converges to ρ2 = ρ ◦ ρ. It follows that ρ2 = ρ.
Remark 4.2.3. If dim X = 1 a holomorphic retraction ρ ∈ Hol(X, X) is either constant or IdX. Indeed if ρ is constant we are done. If ρ is not constant ρ(X) is an open
subset of X. Moreover ρ(X) = Fix(X) and hence ρ(z) = z
for all z ∈ ρ(X). By the identity principle it follows that ρ = IdX and the claim
follows.
Proposition 4.2.4. Let X be a complex manifold and ρ ∈ Hol(X, X) a holomorphic retraction. Then ρ(X) is a closed submanifold of X.
Proof. Let X = D ⊆ Cn be a bounded domain. Let M = ρ(D) and z0 ∈ M . Then
z0 is a fixed point for ρ. Set P = dρz0: C
n→ Cn and let ψ ∈ Hol(D, Cn) be given by
ψ = IdD+ (2P − IdD) ◦ (ρ − P ).
It is straightforward to check that dψz0 = IdCn and therefore ψ is a local
biholomor-phism in z0. Now P2 = P and we have
ψ ◦ ρ = ρ + (2P − IdD) ◦ (ρ2− P ◦ ρ) = ρ + (2P − IdD) ◦ (ρ − P ◦ ρ)
= ρ + 2P ◦ ρ − 2P ◦ P ◦ ρ − ρ + P ◦ ρ = P ◦ ρ;
P ◦ ψ = P + (2P2− P ) ◦ (ρ − P ) = P + 2P ◦ ρ − 2P − P ◦ ρ + P = P ◦ ρ. Therefore ψ ◦ ρ = P ◦ ψ. Let V ⊆ D be an open neighborhood of z0 such that
ψ|V : V → Z is a biholomorphism with Z open subset of Cnand put W = ρ−1(V )∩V .
Then V is an open neighborhood of z0 and we have
ψ|V ◦ ρ|W = P |Z◦ ψ|W,
from which we obtain
ψ|V ◦ ρ|W ◦ (ψ|W) −1
= P |ψ(W ). (4.2.1)
Now ρ(W ) ⊆ W as an easy computation shows. Therefore from (4.2.1) we get ψ|W ◦ ρ|W ◦ (ψ|W)
−1
= P |ψ(W ). (4.2.2)
Let z ∈ W . Then it follows
ρ(z) = z ⇐⇒ ψ|W ◦ ρ|W ◦ (ψ|W) −1
where y = ψ(z). Therefore (4.2.2) yields
ψ(M ∩ W ) = {y ∈ ψ(W ) : P (y) = y} = Ker(P − IdCn) ∩ ψ(W ).
In other terms (W, ψ|W) is a local chart of D adapted to M at z0. Being z0 ∈ M any
and M connected it follows that M is a complex submanifold of D.
Now let X be a complex manifold and let M = ρ(X). Then M = Fix(ρ) and hence is closed in X. Let z0 ∈ M and let (U, ϕ) be a local chart of X in z0 with U
connected and ϕ(U ) ⊆ Cn bounded. If we set Ω = ρ−1(U ) ∩ U then Ω is an open
neighborhood of z0 and ρ(Ω) ⊆ Ω. Without loss of generality we can suppose Ω
connected. Let D = ϕ(Ω) and set ρ = ϕ|e Ω◦ ρ|Ω ◦ (ϕ|Ω) −1
. Then D is a bounded domain of Cn and
e
ρ ∈ Hol(D, D) is a holomorphic retraction. Therefore, from the case seen above, we have that fM = ρ(D) is a submanifold of D. Now fe M = ϕ(M ∩ Ω) and hence M ∩ Ω is a submanifold of Ω. It follows that M is a submanifold of X and we are done.
After these results of general kind, we are ready to deal with the dynamics on taut manifolds.
Theorem 4.2.5. Let X be a taut manifold and f ∈ Hol(X, X). Suppose that (fν) is not compactly divergent. Then there exists a unique holomorphic retraction ρ ∈ Hol(X, X), with ρ ∈ Γ(f ), such that for every h ∈ Γ(f ) there exists γ ∈ Aut(M ) such that h = γ ◦ ρ, where M = ρ(X). Moreover, ϕ = f |M∈ Aut(M ).
Proof. Being (fν) not compactly divergent there exists a subsequence (fνk)
converg-ing to h ∈ Hol(X, X). Without loss of generality we can suppose that pk = νk+1− νk
and qk = pk− νkare monotone and divergent and that (fpk) and (fqk) are converging
or compactly divergent (not necessarily with the same behaviour). Let z ∈ X. Then lim
k→∞f
pk(fνk(z)) = lim
k→∞f
νk+1(z) = h(z).
This implies that (fpk) is not compactly divergent. Indeed let K ⊆ X be a compact
neighborhood of h(z). Then there exists k0 ∈ N such that fνk(z) ∈ K for all k ≥ k0
from which fpk(K) ∩ K 6= ∅ for all k ≥ k
0.
Therefore (fpk) is convergent and let ρ ∈ Hol(X, X) be its limit. Now we have
fpk ◦ fνk → h, fpk → ρ, fνk → h as k → ∞
from which it follows that ρ ◦ h = h. Moreover fνk◦ fpk = fpk ◦ fνk and so
ρ ◦ h = h = h ◦ g. (4.2.3) If z ∈ X we have lim k→∞f qk(fνk(z)) = lim k→∞f pk(z) = ρ(z).
Therefore (fqk) is not compactly divergent and hence converges to g ∈ Hol(X, X).
Now
fqk◦ fνk → ρ, fqk → g, fνk → h as k → ∞
from which we have g ◦ h = ρ. Moreover fνk◦ fqk = fqk ◦ fνk and therefore
