Hyperbolicity in unbounded convex domains
HYPERBOLICITY IN UNBOUNDED CONVEX DOMAINS
FILIPPO BRACCI AND ALBERTO SARACCO
ABSTRACT. We provide several equivalent characterizations of Kobayashi hyperbolicity in un-
bounded convex domains in terms of peak and anti-peak functions at infinity, affine lines, Bergman
metric and iteration theory.
1. INTRODUCTION
Despite the fact that linear convexity is not an invariant property in complex analysis, bounded
convex domains in CN have been very much studied as prototypes for the general situation.
In particular, by Harris’ theorem [6] (see also, [1], [9]) it is known that bounded convex
domains are always Kobayashi complete hyperbolic (and thus by Royden’s theorem, they are
also taut and hyperbolic). Moreover, by Lempert’s theorem [10], [11], the Kobayashi distance
can be realized by means of extremal discs. These are the basic cornerstone for many useful
results, especially in pluripotential theory and iteration theory.
On the other hand, not much is known about unbounded domains. Clearly, the geometry at
infinity must play some important role. In this direction, Gaussier [5] gave some conditions
in terms of existence of peak and anti-peak functions at infinity for an unbounded domain to
be hyperbolic, taut or complete hyperbolic. Recently, Nikolov and Pflug [14] deeply studied
conditions at infinity which guarantee hyperbolicity, up to a characterization of hyperbolicity in
terms of the asymptotic behavior of the Lempert function.
In these notes we restrict ourselves to the case of unbounded convex domains, where, strange
enough, many open questions in the previous directions seem to be still open. In particular an
unbounded convex domain needs not to be hyperbolic, as the example of Ck shows. Some esti-
mates on the Caratheodory and Bergman metrics in convex domains were obtained by Nikolov
and Pflug in [12], [13]. The question is whether one can understand easily hyperbolicity of
unbounded convex domains in terms of geometric or analytic properties. A result in this direc-
tion was obtained by Barth [3], who proved the equivalence of properties (1), (2) and (6) in the
theorem below.
2000 Mathematics Subject Classification. Primary 32Q45 Secondary 32A25; 52A20.
Key words and phrases. Kobayashi hyperbolicity; convex domains; taut; peak functions.
The second author was partially supported by PRIN project “Proprietà geometriche delle varietà reali e
complesse”.
http://arxiv.org/abs/0704.0751v3
2 F. BRACCI AND A. SARACCO
The aim of the present paper is to show that actually for unbounded convex domains, hyper-
bolicity can be characterized in many different ways and can be easily inferred just looking at a
single boundary point.
The dichotomy we discovered for unbounded convex domains is rather stringent: either the
domain behaves like a bounded convex domain or it behaves like Ck. In particular, this pro-
vides examples of unbounded domains which admit the Bergman metric and are complete with
respect to it.
The main result of these notes is the following (notations and terminology are standard and
will be recalled in the next section):
Theorem 1.1. Let D ⊂ CN be a (possibly unbounded) convex domain. The following are
equivalent:
(1) D is biholomorphic to a bounded domain;
(2) D is (Kobayashi) hyperbolic;
(3) D is taut;
(4) D is complete (Kobayashi) hyperbolic;
(5) D does not contain nonconstant entire curves;
(6) D does not contain complex affine lines;
(7) D has N linearly independent separating real hyperplanes;
(8) D has peak and antipeak functions (in the sense of Gaussier) at infinity;
(9) D admits the Bergman metric bD.
(10) D is complete with respect to the Bergman metric bD.
(11) for any f : D → D holomorphic such that the sequence of its iterates {f ◦k} is not
compactly divergent there exists z0 ∈ D such that f(z0) = z0.
The first implications of the theorem allow to obtain the following canonical complete hyper-
bolic decomposition for unbounded convex domains, which is used in the final part of the proof
of the theorem itself.
Proposition 1.2. Let D ⊂ CN be a (possibly unbounded) convex domain. Then there exist a
unique k (0 ≤ k ≤ N) and a unique complete hyperbolic convex domain D′ ⊂ Ck, such that,
up to a linear change of coordinates, D = D′ × CN−k.
By using such a canonical complete hyperbolic decomposition, one sees for instance that the
“geometry at infinity” of an unbounded convex domain can be inferred from the geometry of
any finite point of its boundary (see the last section for precise statements). For example, as an
application of Corollary 4.3 and Theorem 1.1, existence of peak and anti-peak functions (in the
sense of Gaussier) for an unbounded convex domain equals the absence of complex line in the
“CR-part” of the boundary of the domain itself. This answers a question in Gaussier’s paper
(see [5, pag. 115]) about geometric conditions for the existence in convex domains of peak and
anti-peak plurisubharmonic functions at infinity.
The authors want to sincerely thank prof. Nikolov for helpful conversations, and in particular
for sharing his idea of constructing antipeak functions.
HYPERBOLICITY IN UNBOUNDED CONVEX DOMAINS 3
2. PRELIMINARY
A convex domain D ⊂ CN is a domain such that for any couple z0, z1 ∈ D the real segment
joining z0 and z1 is contained in D. It is well known that for any point p ∈ ∂D there exists
(at least) one real separating hyperplane Hp = {z ∈ C
n : ReL(z) = a}, with L a complex
linear functional and a ∈ R such that p ∈ Hp and D ∩ Hp = ∅. Such a hyperplane Hp is
sometimes also called a tangent hyperplane to D at p. We say that k separating hyperplanes
Hj = {ReLj(z) = aj}, j = 1, . . . , k, are linearly independent if L1, . . . , Lk are linearly
independent linear functionals.
Let D := {ζ ∈ C : |ζ | < 1} be the unit disc. Let D ⊂ CN be a domain. The Kobayashi
pseudo-metric for the point z ∈ D and vector v ∈ CN is defined as
κD(z; v) := inf{λ > 0|∃ϕ : D
−→ D,ϕ(0) = z, ϕ′(0) = v/λ}.
If κD(z; v) > 0 for all v 6= 0 then D is said to be (Kobayashi) hyperbolic. The pseudo-
distance kD obtained by integrating κD is called the Kobayashi pseudodistance. The domain D
is (Kobayashi) complete hyperbolic if kD is complete.
The Carathéodory pseudo-distance cD is defined by
cD(z, w) = sup{kD(f(z), f(w)) : f : D → D holomorphic}.
In general, cD ≤ kD.
We refer the reader to the book of Kobayashi [9] for properties of Kobayashi and Carathéodory
metrics and distances.
Another (pseudo)distance that can be introduced on the domain D is the Bergman (pseudo)
distance (see, e.g., [9, Sect. 10, Ch. 4]). Let {ej} be a orthonormal complete basis of the space
of square-integrable holomorphic functions on D. Then let
lD(z, w) :=
ej(z)ej(w).
If lD(z, z) > 0 one can define a symmetric form bD := 2
hjkdzj ⊗ dzk, with hjk =
∂2 log bD(z,z)
∂zj ,∂zk
, which is a positive semi-definite Hermitian form, called the Bergman pseudo-metric
of D. If bD is positive definite everywhere, one says that D admits the Bergman metric bD.
For instance, Ck, k ≥ 1 does not support square-integrable holomorphic functions, therefore
lCk ≡ 0 and C
k does not admit the Bergman metric.
For the next result, see [9, Corollaries 4.10.19, 4.10.20]:
Proposition 2.1. Let D ⊂ CN be a domain.
(1) Assume bD(z, z) > 0 for all z ∈ D. If cD is a distance, if it induces the topology of D
and if the cD-balls are compact, thenD admits the Bergman metric bD and it is complete
with respect to bD.
(2) If D is a bounded convex domain then it (admits the Bergman metric and it) is complete
with respect to the Bergman metric.
4 F. BRACCI AND A. SARACCO
Let G be another domain. We recall that if {ϕk} is a sequence of holomorphic mappings
from G to D, then the sequence is said to be compactly divergent if for any two compact sets
K1 ⊂ G and K2 ⊂ D it follows that ♯{k ∈ N : ϕk(K1) ∩K2 6= ∅} < +∞.
A family F of holomorphic mappings from G to D is said to be normal if each sequence
of F admits a subsequence which is either compactly divergent or uniformly convergent on
compacta.
If the family of all holomorphic mappings from the unit disc D to D is normal, then D is said
to be taut. It is known:
Theorem 2.2. Let D ⊂ CN be a domain.
(1) (Royden) D complete hyperbolic ⇒ D taut ⇒ D hyperbolic.
(2) (Kiernan) If D is bounded then D is hyperbolic.
(3) (Harris) If D is a bounded convex domain then D is complete hyperbolic.
The notion of (complete) hyperbolicity is pretty much related to existence of peak functions
at each boundary point. In case D is an unbounded domain, H. Gaussier [5] introduced the
following concepts of “peak and antipeak functions” at infinity, which we use in the sequel:
Definition 2.3. A function ϕ : D → R ∪ {−∞} is called a global peak plurisubharmonic
function at infinity if it is plurisubharmonic on D, continuous up to D (closure in CN ) and
ϕ(z) = 0,
ϕ(z) < 0 ∀z ∈ D.
A function ϕ : D → R∪{−∞} is called a global antipeak plurisubharmonic function at infinity
if it is plurisubharmonic on D, continuous up to D and
ϕ(z) = −∞,
ϕ(z) > −∞ ∀z ∈ D.
For short we will simply call them peak and antipeak functions (in the sense of Gaussier) at
infinity.
Gaussier proved the following result:
Theorem 2.4 (Gaussier). Let D ⊂ CN be an unbounded domain. Assume that D is locally taut
at each point of ∂D and there exist peak and antipeak functions (in the sense of Gaussier) at
infinity. Then D is taut.
Obviously a convex domain is locally taut at each boundary point, thus tautness follows from
existence of peak and antipeak functions (in the sense of Gaussier) at infinity.
Finally, if f : D → D is a holomorphic function, the sequence of its iterates {f ◦k} is
defined by induction as f ◦k := f ◦(k−1) ◦ f . If f has a fixed point z0 ∈ D, then {f
◦k} is not
compactly divergent. On the other hand, depending on the geometry of D, there exist examples
HYPERBOLICITY IN UNBOUNDED CONVEX DOMAINS 5
of holomorphic maps f such that {f ◦k} is not compactly divergent but f has no fixed points in
D. It is known (see [2]) that
Theorem 2.5 (Abate). Let D ⊂ CN be a taut domain. Assume that Hj(D;Q) = 0 for all j > 0
and let f : X → X holomorphic. Then the sequence of iterates {f ◦k} is compactly divergent if
and only if f has no periodic points in D.
If D is a bounded convex domain then the sequence of iterates {f ◦k} is compactly divergent
if and only if f has no fixed points in D.
3. THE PROOF OF THEOREM 1.1
The proof of Theorem 1.1 is obtained in several steps, which might be of some interest by
their own.
For a domain D ⊂ CN let us denote by δD the Lempert function given by
δD(z, w) = inf{ω(0, t) : t ∈ (0, 1), ∃ϕ ∈ Hol(D, D) : ϕ(0) = z, ϕ(t) = w}.
The Lempert function is not a pseudodistance in general because it does not enjoy the triangle
inequality. The Kobayashi pseudodistance is the largest minorant of δD which satisfies the
triangle inequality. The following lemma (known as Lempert’s theorem in case of bounded
convex domains) is probably known, but we provide its simple proof due to the lack of reference.
Lemma 3.1. Let D ⊂ CN be a (possibly unbounded) convex domain. Then kD = δD = cD.
Proof. The result is due to Lempert [11] in case D is bounded. Assume D is unbounded. Let
DR be the intersection of D with a ball of center the origin and radius R > 0. For R >> 1
the set DR is a nonempty convex bounded domain. Therefore kDR = δDR = cDR . Now
{DR} is an increasing sequence of domains whose union is D. Hence, limR→∞ kDR = kD,
limR→∞ cDR = cD and limR→∞ δDR = δD (see, e.g., [7, Prop. 2.5.1] and [7, Prop. 3.3.5]).
Thus kD = δD = cD. �
Proposition 3.2. Let D ⊂ CN be a (possibly unbounded) convex domain. Then the Kobayashi
balls in D are convex.
Proof. For the bounded case, see [1, Proposition 2.3.46]. For the unbounded case, let Bǫ be the
Kobayashi ball of radius ǫ and center z0 ∈ D, let DR be the intersection of D with an Euclidean
ball of center the origin and radius R > 0, and let BRǫ be the Kobayashi ball in DR of radius ǫ
and center z0. Then the convex sets B
ǫ ⊂ B
ǫ ⊂ Bǫ for all R >> 1, δ > 0, and their convex
increasing union ∪RB
ǫ = Bǫ, since limR→∞ kDR = kD. �
Lemma 3.3. Let D ⊂ CN be a (possibly unbounded) taut convex domain. Then for any couple
z, w ∈ D there exists ϕ ∈ Hol(D, D) such that ϕ(0) = z, ϕ(t) = w t ∈ [0, 1) and kD(z, w) =
ω(0, t).
6 F. BRACCI AND A. SARACCO
Proof. By Lemma 3.1, kD = δD, so there exists a sequence {ϕk} of holomorphic discs and
tk ∈ (0, 1) such that ϕk(0) = z and ϕk(tk) = w and
kD(z, w) = lim
ω(0, tk).
Since D is taut and ϕk(0) = z for all k, we can assume that {ϕk} converges uniformly on
compacta to a (holomorphic) map ϕ : D → D. Then ϕ(0) = z. Moreover, since kD(z, w) <∞,
there exists t0 < 1 such that tk ≤ t0 for all k. We can assume (up to subsequences) that
tk → t ≤ t0. Then
kD(z, w) = lim
ω(0, tk) = ω(0, t).
Moreover, ϕ(t) = limk→∞ ϕk(tk) = w and we are done. �
Proposition 3.4. Let D ⊂ CN be a (possibly unbounded) convex domain. Then D is taut if and
only if it is complete hyperbolic.
Proof. One direction is contained in Royden’s theorem. Conversely, assume that D is taut.
We are going to prove that every closed Kobayashi balls is compact (which is equivalent to be
complete hyperbolic, see [9] or [1, Proposition 2.3.17]).
Let R > 0, z ∈ D and let B(z, R) = {w ∈ D : kD(z, w) ≤ R}. If B(z, R) is not compact
then there exists a sequence {wk} such that wk → p ∈ ∂D ∪ {∞} and kD(z, wk) ≤ R. For any
k, let ϕk ∈ Hol(D, D) be the extremal disc given by Lemma 3.3 such that ϕk(0) = z, ϕk(tk) =
wk for some tk ∈ (0, 1) and kD(z, wk) = ω(0, tk).
Notice that, since kD(z, wk) ≤ R, then there exists t0 < 1 such that tk ≤ t0 for all k. We
can assume up to subsequences that tk → t with t < 1. Since D is taut and ϕk(0) = z,
up to extracting subsequences, the sequence {ϕk} is converging uniformly on compacta to a
holomorphic disc ϕ : D → D such that ϕ(0) = z. However,
ϕ(t) = lim
ϕk(tk) = lim
wk = p,
a contradiction. Therefore B(z, R) is compact and D is complete hyperbolic. �
For the next proposition, cfr. [4, Lemma 3].
Proposition 3.5. Let D ⊂ CN be a convex domain, which does not contain complex affine
lines. Then there exist {L1 = 0}, . . . , {LN = 0} linearly independent hyperplanes containing
the origin and a1, . . . , aN ∈ R such that
D ⊂ {Re L1 > a1, . . . ,Re LN > aN}.
Proof. Without loss of generality we can assume thatO ∈ D. SinceD does not contain complex
affine lines, ∂D is not empty. Take a point p1 ∈ ∂D and a tangent real hyperplane through p1,
given by {Re L1 = a1} (if the boundary is smooth there is only one tangent hyperplane), where
L1 is defined so that D ⊂ {Re L1 > a1}.
Suppose that L1, . . . , Lk, k < N , are already defined, they are linearly independent and
D ⊂ {Re L1 > a1, . . . ,Re Lk > ak}.
HYPERBOLICITY IN UNBOUNDED CONVEX DOMAINS 7
The intersection hk = ∩
1{Li = 0} is a complex (N − k)-dimensional plane through the origin
O (which is also contained in D by hypothesis). Since D does not contain complex affine lines,
∂D ∩ hk is not empty. Take a point pk+1 ∈ ∂D ∩ hk and consider a tangent real hyperplane
through pk+1, {Re Lk+1 = ak+1}, where Lk+1 is defined so that D ⊂ {Re Lk+1 > ak+1}. By
construction Lk+1 is linearly independent from L1, . . . , Lk and
D ⊂ {Re L1 > a1, . . . ,Re Lk+1 > ak+1}.
Continuing this way, the proof is concluded. �
Now we are in a good shape to prove part of Theorem 1.1:
Proposition 3.6. Let D ⊂ CN be a convex domain. The following are equivalent:
(1) D is biholomorphic to a bounded domain;
(2) D is (Kobayashi) hyperbolic;
(3) D is taut;
(4) D is complete (Kobayashi) hyperbolic;
(5) D does not contain nonconstant entire curves;
(6) D does not contain complex affine lines;
(7) D has N linearly independent separating real hyperplanes;
(8) D has peak and antipeak functions (in the sense of Gaussier) at infinity;
Proof. (1) ⇒ (2): every bounded domain in CN is hyperbolic by [8] (see, also, [1, Thm. 2.3.14])
(2) ⇒ (5) ⇒ (6): obvious.
(6) ⇒ (7): it is Proposition 3.5.
(7) ⇒ (1): letL1, . . . , LN be linearly independent complex linear functionals and let a1, . . . , aN ∈
R be such that {ReLj = aj} for j = 1, . . . , N are real separating hyperplanes for D. Up to
sign changes, we can assume that D ⊂ {ReLj > aj}. Then the map
F (z1, . . . , zN) :=
L1(z)− a1 + 1
, . . . ,
LN(z)− aN + 1
maps D biholomorphically on a bounded convex domain of CN .
(6) ⇒ (8): let L1, . . . , LN be as in Proposition 3.5. Up to a linear change of coordinates, we
can suppose that zj = Lj for all 1 ≤ j ≤ N . A peak function is given by
zj − aj + 1
Let Dj := {ReLj > aj}. Then D ⊂
j=1Dj , and Dj is biholomorphic to D for each j.
In particular C \ Dj is not a polar set. We may assume that 0 6∈ Dj. Let Gj be the image
of Dj under the transformation z → 1/z. Since C \ Gj is not a polar set, there exists ε > 0
such that C \ Gεj is not polar, too, where G
j = Gj ∪ εD. Denote by g
j the Green function of
Gεj . Then hj = g
j (0; ·) is a negative harmonic function on Gj with limz→0 hj(z) = −∞ and
8 F. BRACCI AND A. SARACCO
infGj\rD hj > −∞ for any r > 0. Then ψj(z) = hj(1/z) is an antipeak function of Dj at ∞
and hence ψ =
j=1 ψj is an antipeak function for D at ∞.
(8) ⇒ (3): it is Gaussier’s theorem [5, Prop. 2].
(3) ⇒ (4): it is Proposition 3.4.
(4) ⇒ (3) ⇒ (2): it is Royden’s theorem [15, Prop. 5, pag. 135 and Corollary p.136]. �
As a consequence we have Proposition 1.2, which gives a canonical complete hyperbolic
decomposition of a convex domain as the product of a complete hyperbolic domain and a copy
of Ck.
Proof of Proposition 1.2. We prove the result by induction on N . If N = 1 then either D = C
or D is biholomorphic to the disc and hence (complete) hyperbolic.
Assume the result is true for N , we prove it holds for N + 1. Let D ⊂ CN+1 be a convex
domain. Then, by Proposition 3.6, either D is complete hyperbolic or D contains an affine line,
say, up to a linear change of coordinates
lN+1 = {z1, . . . , zN = 0} ⊂ D.
Clearly, there exists c ∈ C such that D ∩ {zN+1 = c} 6= ∅. Up to translation we can assume
c = 0. Let us define
DN = D ∩ {zN+1 = 0}.
DN ⊂ C
N is convex. We claim that D = DN × C. Induction will then conclude the proof.
Let z0 ∈ DN . We want to show that (z0, ζ) ∈ D for all ζ ∈ C. Since lN+1 ⊂ D then
(0, ζ) ∈ D for all ζ ∈ C. Assume z0 6= 0. Fix ζ ∈ C. Since DN is open, there exists
ε0 > 0 such that z1 := (1 + ε0)z0 ∈ DN . Since D is convex, for any t ∈ [0, 1] it follows
t(z1, 0)+ (1− t)(0, ξ) ∈ D for all ξ ∈ C. Setting ξ0 :=
ζ ∈ C and t0 = (1+ ε0)
−1 ∈ (0, 1)
we obtain
(z0, ζ) = t0 (z1, 0) + (1− t0) (0, ξ0) ∈ D,
completing the proof. �
In order to finish the proof of Theorem 1.1 we need to show that the first eight conditions,
which are all and the same thanks to Proposition 3.6, are equivalent to (9), (10), (11).
Proof of Theorem 1.1. Conditions (1) to (8) are all equivalent by Proposition 3.6.
(10)⇒(9): obvious.
(4)⇒(10): By Lemma 3.1, the Caratheodory distance cD equals the Kobayashi distance kD,
thus, since D is (Kobayashi) complete hyperbolic, cD is a distance which induces the topology
on D and the cD-balls are compact. By Proposition 2.1 then D admits the Bergman metric and
it is complete with respect to it.
(9)⇒(4): Assume D is not complete hyperbolic. Then by Proposition 1.2, up to a linear
change of coordinates, D = D′ × Ck for some complete hyperbolic domain D′ and k ≥ 1. By
the product formula (see [9, Prop. 4.10.17]) lD = lD′ · lCk ≡ 0 and thus D does not admit the
Bergman metric.
HYPERBOLICITY IN UNBOUNDED CONVEX DOMAINS 9
(11) ⇒ (4): Assume D is not complete hyperbolic. We have to exhibit a holomorphic self-
map f : D → D such that {f ◦k} is not compactly divergent but there exists no z0 ∈ D such
that f(z0) = z0.
By Proposition 1.2, up to a linear change of coordinates, D = D′ × Ck for some complete
hyperbolic domain D′ and k ≥ 1. Let
f : D′ × Ck−1 × C ∋ (z, w′, w) 7→ (z, w′, ew + w) ∈ D′ × Ck−1 × C.
Then clearly f has no fixed points in D. However, if w0 = log(iπ), then f
◦2(z, w′, w0) =
(z, w′, w0), and therefore the sequence {f
◦k} is not compactly divergent.
(4) ⇒ (11): According to the theory developed so far, if D is complete hyperbolic, then it
is taut and its Kobayashi balls are convex and compact. With these ingredients, the proof for
bounded convex domains go through also in the unbounded case (see [1, Thm. 2.4.20]). �
4. APPLICATIONS
Corollary 4.1. Let D ⊂ CN be a convex domain. If there exists a point p ∈ ∂D such that ∂D
is strongly convex at p then D is complete hyperbolic.
Proof. By Proposition 1.2, if D were not complete hyperbolic, up to linear changes of coor-
dinates, D = D′ × Ck for some complete hyperbolic convex domain D′ and k ≥ 1. Then
∂D = ∂D′ × Ck could not be strongly convex anywhere. �
Note that the converse to the previous corollary is false: the half-plane {ζ ∈ C : Re ζ > 0} is
a complete hyperbolic convex domain in C with boundary which is nowhere strongly convex.
Proposition 4.2. Let D ⊂ CN be an (unbounded) convex domain. Then D has canonical
complete hyperbolic decomposition (up to a linear change of coordinates) D = D′ × Ck with
D′ complete hyperbolic, if and only if for every p ∈ ∂D and every separating hyperplane Hp,
(Hp ∩ ∂D) ∩ i(Hp ∩ ∂D) contains a copy of C
k but contains no copies of Ck+1.
Proof. (⇒) Since D = D′ × Ck, for every p ∈ ∂D and every separating hyperplane Hp,
(Hp ∩ ∂D) ∩ i(Hp ∩ ∂D) = [(Hp ∩ ∂D
′) ∩ i(Hp ∩ ∂D
′)]× Ck.
Since D′ is complete hyperbolic, its boundary does not contain complex lines.
(⇐) Since D is convex, D = D′ × Ck
, by Proposition 1.2. By the first part of the present
proof, for every p ∈ ∂D and every separating hyperplane Hp,
(Hp ∩ ∂D) ∩ i(Hp ∩ ∂D) = C
Hence k′ = k. �
Corollary 4.3. Let D ⊂ CN be an (unbounded) convex domain. If there exist p ∈ ∂D and a
separating hyperplane Hp such that (Hp ∩ ∂D) ∩ i(Hp ∩ ∂D) does not contain any complex
affine line then D is complete hyperbolic. Conversely, if D is complete hyperbolic, then for any
point p ∈ ∂D and any separating hyperplane Hp, it follows that (Hp ∩ ∂D)∩ i(Hp ∩ ∂D) does
not contain any complex affine line.
10 F. BRACCI AND A. SARACCO
As a final remark, we notice that, as Abate’s theorem 2.5 is the cornerstone to the study of
iteration theory in bounded convex domains, our Theorem 1.1 and Proposition 1.2 can be used
effectively well to the same aim for unbounded convex domains. In fact, if D = D′ ×Ck is the
canonical complete hyperbolic decomposition of D, then a holomorphic self map f : D → D
can be written in the coordinates (z, w) ∈ D′ × Ck as f(z, w) = (ϕ(z, w), ψ(z, w)), where
ϕ : D′ ×Ck → D′ and ψ : D′ ×Ck → Ck. In particular, since D′ is complete hyperbolic, then
ϕ depends only on z, namely, f(z, w) = (ϕ(z), ψ(z, w)). The map ψ(z, w) can be as worse as
entire functions in Ck are, but the map ϕ is a holomorphic self-map of a complete hyperbolic
convex domains and its dynamics goes similarly to that of holomorphic self-maps of bounded
convex domains. For instance, if the sequence {f ◦k} is non-compactly divergent, then f might
have no fixed points, but the sequence {ϕ◦k} must have at least one.
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HYPERBOLICITY IN UNBOUNDED CONVEX DOMAINS 11
F. BRACCI: DIPARTIMENTO DI MATEMATICA, UNIVERSITÀ DI ROMA “TOR VERGATA”, VIA DELLA
RICERCA SCIENTIFICA 1, 00133, ROMA, ITALY
E-mail address: fbracci@mat.uniroma2.it
A. SARACCO: SCUOLA NORMALE SUPERIORE, PIAZZA DEI CAVALIERI 7, 56126, PISA, ITALY
E-mail address: a.saracco@sns.it
	1. Introduction
	2. Preliminary
	3. The proof of Theorem ??
	4. Applications
	References
