Iterates of holomorphic self maps on pseudoconvex domains of finite and infinite type in ℂn

Iterates of holomorphic self maps on pseudoconvex domains of finite and infinite type in ℂn tài liệu, giáo án, bài giảng...

Article electronically published on May 23, 2016

ITERATES OF HOLOMORPHIC SELF-MAPS

ON PSEUDOCONVEX DOMAINS OF FINITE

AND INFINITE TYPE IN Cn

TRAN VU KHANH AND NINH VAN THU (Communicated by Franc Forstneric) Abstract Using the lower bounds on the Kobayashi metric established by the first author, we prove a Wolff-Denjoy-type theorem for a very large class of pseudoconvex domains in Cn This class includes many pseudoconvex domains

of finite type and infinite type.

1 Introduction

In 1926, Wolff [22] and Denjoy [9] established their famous theorem on the be-havior of iterates of holomorphic self-mappings of the unit disk Δ ofC that do not admit fixed points

Theorem (Wolff-Denjoy [9, 22], 1926) Let φ : Δ → Δ be a holomorphic self-map without fixed points Then there exists a point x in the unit circle ∂Δ such that the sequence {φ k } of iterates of φ converges, uniformly on any compact subsets of Δ,

to the constant map taking the value x.

The generalization of this theorem to domains inCn , n ≥ 2, is the focus of this

paper This has been done in several cases:

• the unit ball (see [13]);

• strongly convex domains (see [2, 4, 5]);

• strongly pseudoconvex domains (see [3, 14]);

• pseudoconvex domains of strictly finite type in the sense of Range [20] (see

[3]);

• pseudoconvex domains of finite type in C2

(see [15, 23])

The main goal of this paper is to prove a Wolff-Denjoy-type theorem for a general class of bounded pseudoconvex domains in Cn that includes many pseudoconvex domains of both finite and infinite type In particular, we shall prove the following (the definitions are given below)

Received by the editors July 16, 2015 and, in revised form, December 25, 2015, December 28,

2015, January 13, 2016 and February 4, 2016.

2010 Mathematics Subject Classification Primary 32H50; Secondary 37F99.

Key words and phrases Wolff-Denjoy-type theorem, finite type, infinite type, f -property,

Kobayashi metric, Kobayashi distance.

The research of the first author was supported by the Australian Research Council DE160100173.

The research of the second author was supported by the Vietnam National University, Hanoi (VNU) under project number QG.16.07 This work was completed when the second author was visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM) He would like to thank the VIASM for the financial support and hospitality.

c

2016 American Mathematical Society

1

Theorem 1 Let Ω ⊂ C n

be a bounded, pseudoconvex domain with C2-smooth boundary ∂Ω Assume that

(i) Ω has the f -property with f satisfying

 ∞ 1

ln α

αf (α) dα < ∞; and (ii) the Kobayashi distance of Ω is complete.

If φ : Ω → Ω is a holomorphic self-map such that the sequence of iterates {φ k } is compactly divergent, then the sequence {φ k } converges, uniformly on a compact set,

to a point of the boundary.

We say that a Wolff-Denjoy-type theorem for Ω holds if the conclusion of Theo-rem 1 holds We will prove TheoTheo-rem 1 in Section 3 using the (known) estimates of

the Kobayashi distance on domains of the f -property and the work by Abate [2–4].

We now recall some definitions of the f -property (see also [16, 17]) and the

Kobayashi distance

Definition 1 Let f :R+ → R+ be a smooth, monotonically increasing function

so that f (α)α −1/2 is decreasing We say that Ω⊂ C n has the f -property if there

exists a family of functions{ψ η } such that

(i) the functions ψ η are plurisubharmonic,|ψ η | ≤ 1, and C2 on Ω;

(ii) i∂ ¯ ∂ψ η ≥ c1f (η −1)2Id and |Dψ η | ≤ c2η −1 on {z ∈ Ω : 0 < δΩ (z) < δ } for some constants c1, c2> 0, where δΩ (z) is the Euclidean distance from z to the boundary ∂Ω.

This is an analytic condition where the function f reflects the geometric “type”

of the boundary For example, viewing Catlin’s results on pseudoconvex domains of

finite type through the lens of the f -property [6, 7], a domain is of finite type if and only if there exists an  > 0 such that the t -property holds If the domain is convex

and of finite type m, then the t 1/m-property holds [18] Furthermore, there is a

large class of infinite type pseudoconvex domains that satisfy an f -property [16, 17].

For example (see [17]), the ln1/α-property holds for both the complex ellipsoid of infinite type

Ω =

⎧

⎨

⎩z ∈ C n:

n



j=1

exp



− |z1

j | α j



− e −1 < 0

⎫

⎬

⎭ (1)

with α := max j {α j }, and the real ellipsoid of infinite type

(2) Ω =˜

⎧

⎨

⎩z = (x1 + iy1, , xn + iy n)∈ C n

:

n



j=1

exp



− |x1

j | α j



+ exp



− |y1

j | β j



− e −1 < 0

⎫

⎬

⎭

with α := max j {min{α j , β j }}, where α j , β j > 0 for all j = 1, 2, The influence

of the f -property on estimates of the Kobayashi metric and distance will be given

in Section 2

On hyperbolic manifolds, completeness of the Kobayashi distance (or

k-complete-ness for short) is a natural condition For a bounded domain Ω⊂ C n

,

k-complete-ness means

kΩ (z0, z)→ ∞ as z → ∂Ω

for any point z0 ∈ Ω where kΩ (z0, z) is the Kobayashi distance from z0 to z It

is well known that this condition holds for strongly pseudoconvex domains [11], convex domains [19], pseudoconvex domains of finite type inC2

[23], pseudoconvex Reinhardt domains [21], and domains enjoying a local holomorphic peak function

at any boundary point [12] We also remark that the domain defined by (1) (resp

(2)) is k-complete because it is a pseudoconvex Reinhardt domain (resp convex

domain) These remarks immediately lead to the following corollary

Corollary 2 Let Ω be a bounded domain in Cn

with smooth boundary ∂Ω The Wolff-Denjoy-type theorem for Ω holds if Ω satisfies at least one of the following settings:

(a) Ω is a strongly pseudoconvex domain;

(b) Ω is a pseudoconvex domain of finite type and n = 2;

(c) Ω is a convex domain of finite type;

(d) Ω is a pseudoconvex Reinhardt domain of finite type;

(e) Ω is a pseudoconvex domain of finite type (or of infinite type having the

f -property with f (t) ≥ ln 2+

(t) for any  > 0) such that Ω has a local, continuous, holomorphic peak function at each boundary point, i.e., for any

x ∈ ∂Ω there exist a neighborhood U of x and a holomorphic function p on

Ω∩ U, continuous up to ¯Ω ∩ U, satisfying

p(x) = 1, p(z) < 1, for all z ∈ ¯Ω ∩ U \ {x};

(f ) Ω is defined by (1) or (2) with α < 1

2.

Finally, throughout the paper we use  and  to denote inequalities up to a

positive multiplicative constant, and H(Ω1, Ω2) to denote the set of holomorphic maps from Ω1 to Ω2

2 The Kobayashi metric and distance

We start this section by defining the Kobayashi metric

Definition 2 Let Ω be a domain in Cn , and T 1,0Ω be its holomorphic tangent

bundle The Kobayashi (pseudo)metric KΩ: T 1,0Ω→ R is defined by

(3) KΩ (z, X) = inf {α > 0 | ∃ Ψ ∈ H(Δ, Ω) : Ψ(0) = z, Ψ  (0) = α −1 X}, for any z ∈ Ω and X ∈ T 1,0

Ω, where Δ is the unit open disk ofC

In the case that Ω is a smoothly pseudoconvex bounded domain of finite type, it

is known that there exists  > 0 such that the Kobayashi metric KΩhas the lower

bound δ −Ω (z) (see [8], [10]), in the sense that,

KΩ (z, X)

δ 

Ω(z) ,

where

lower bounds on the Kobayashi metric for a general class of pseudoconvex domains

inCn, that contains all domains of finite type and many domains of infinite type

Theorem 3 Let Ω be a pseudoconvex domain inCn with C2-smooth boundary ∂Ω Assume that Ω has the f -property with f satisfying

 ∞

s dα

αf (α) < ∞ for s ≥ 1, and

denote by (g(s)) −1 this finite integral Then,

K(z, X)  g(δ −1

Ω (z))

(4)

for any z ∈ Ω and X ∈ T 1,0

The Kobayashi (pseudo)distance kΩ: Ω× Ω → R+ on Ω is the integrated form

of KΩ kΩis given by

kΩ (z, w) = inf

 b a

γ(a) = z, γ(b) = w } for any z, w ∈ Ω An essential property of kΩ is that it is a contraction under

holomorphic maps, i.e.,

(5) if φ ∈ H(Ω, ˜Ω), then k˜Ω(φ(z), φ(w)) ≤ kΩ (z, w), for all z, w ∈ Ω.

We need the following lemma from [1, 11]

Lemma 4 Let Ω be a bounded C2-smooth domain inCn

and z0 ∈ Ω Then there exists a constant c0 > 0 depending on Ω and z0 such that

kΩ (z0, z)≤ c0 −1

2ln δΩ(z)

for any z ∈ Ω.

We recall that the curve γ : [a, b] → Ω is called a minimizing geodesic with respect to the Kobayashi metric between two points z = γ(a) and w = γ(b) if

kΩ (γ(s), γ(t)) = t − s =

 t s

KΩ (γ(τ ), ˙γ(τ ))dτ, for any s, t ∈ [a, b], s ≤ t.

This implies that

K(γ(t), ˙γ(t)) = 1, for any t ∈ [a, b].

The relation between the Kobayashi distance kΩ(z, w) and the Euclidean distance

δΩ (z, w) is contained in the following lemma, itself a generalization of [15, Lemma

36]

Lemma 5 Let Ω be a bounded, pseudoconvex, C2-smooth domain inCn

satisfying the f -property with

 ∞ 1

ln α

αf (α) dα < ∞ and z0 ∈ Ω Then there exists a constant c only depending on z0 and Ω such that

 ∞

e 2kΩ(z0,γ)

c0 + ln α

αf (α) dα, for all z, w ∈ Ω, where γ is a minimizing geodesic connecting z to w and c0 is the constant given in Lemma 4 Here, kΩ (z0, γ) is the Kobayashi distance from z0 to the curve γ.

Proof We may assume that z = w Let p be a point on γ of minimal distance

to z0 We can assume that p = z (if not, we interchange z and w) and denote by γ1 : [0, a] → Ω the reparametrized piece of γ going from p to z By the minimality

of kΩ(z0, γ) = kΩ(z0, p) and the triangle inequality we have

(7) kΩ (z0, γ1(t)) ≥ kΩ (z0, γ)

kΩ (z0, γ1(t)) ≥ kΩ (p, γ1(t)) − kΩ (z0, p) = t− kΩ (z0, γ)

for any t ∈ [0, a] Substituting z = γ1 (t) into the inequality in Lemma 4, it follows

1

δΩ (γ1(t)) ≥ e 2kΩ(z0,γ1(t)) −2c0

for all t ∈ [0, a] Since γ1 is a unit speed curve with respect to KΩwe have

δΩ (p, z) ≤

 a

0



1(t)



 a

0



g

 1

δΩ (γ1(t))

−1

KΩ (γ1(t), γ1 (t))dt



 a

0



g



e 2kΩ(z0,γ1(t)) −2c0

−1

dt.

(8)

We now compare a with 2kΩ(z0, γ) + c0 In the case a > 2kΩ(z0, γ) + c0, we split the

integral into two parts and use the inequalities (7) and the fact that g is increasing.

We then have

δΩ (p, z)

 2kΩ(z0,γ)+c0

0



g



e 2kΩ(z0,γ1(t)) −2c0

−1

dt

+

 a 2kΩ(z0,γ)+c0



g



e 2kΩ(z0,γ1(t)) −2c0

−1

dt



 2kΩ(z0,γ)+c0

0



g



e 2kΩ(z0,γ) −2c0

−1

dt

+

 ∞

2kΩ(z0,γ)+c0



g



e 2t −2kΩ(z0,γ) −2c0

−1

dt

 2kΩ(z0, γ) + c0

g

e 2kΩ(z0,γ) −2c0 + ∞

e 2kΩ(z0,γ)

dβ βg(β)





c0 + ln s g(se −2c0)+

 ∞

s

dβ βg(β) s=e 2kΩ(z0,γ)

(9)

By the definition of (g(s)) −1 in Theorem 3 and the fact that f (α)α −1/2is decreasing,

it follows

1

g(se −2c0) =

 ∞

se −2c0

dα

αf (α) =

 ∞

s

dα

αf (αe −2c0)

=

 ∞

s

e c0dα

α 3/2 (αe −2c0)−1/2 f (αe −2c0) ≤

 ∞

s

e c0dα

α 3/2 α −1/2 f (α) =

e c0

g(s) ,

(10)

thus obtaining

δΩ (p, z) ≤ c



c0 + ln s g(s) +

 ∞

s dβ βg(β) s=e 2kΩ(z0,γ) ,

where c is the multiplication of e c0 with a positive constant We also notice that

 ∞

s

dβ βg(β) =

 ∞

s

1

β

 ∞

β

dα

αf (α)



dβ =



{(α,β): β≤α<∞,s≤β<∞}

dαdβ βαf (α)

=



{(α,β): s≤α<∞,s≤β≤α}

dαdβ βαf (α) =

 ∞

s

1

αf (α)

 α s

dβ β



dα

=

 ∞

s

ln α − ln s

αf (α) dα =

 ∞

s

ln α

αf (α) dα − ln s

g(s) .

Therefore, in this case we obtain

δΩ (p, z) ≤ c



c0 g(s) +

 ∞

s

ln α

αf (α) dα s=e 2kΩ(z0,γ) = c

 ∞

e 2kΩ(z0,γ)

c0 + ln α

αf (α) dα.

In the case a < 2kΩ(z0, γ)+c0, we make the same estimate but without decomposing

the integral By a symmetric argument with w instead of z, we also have

δΩ (p, w) ≤ c

 ∞

e 2kΩ(z0,γ)

c0 + ln α

αf (α) dα.

The conclusion of this lemma now follows by the triangle inequality 

Corollary 6 Let Ω be a bounded, pseudoconvex domain in Cn

with C2-smooth boundary satisfying the f -property with

 ∞ 1

ln α

αf (α) dα < ∞ Furthermore, assume that Ω is k-complete Let {w n }, {z n } ⊂ Ω be two sequences such that w n → x ∈ ∂Ω and z n → y ∈ ¯Ω \ {x} Then kΩ (w n , z n)→ ∞.

Proof Fix a point z0 ∈ Ω and let γ n : [a n , b n] → Ω be a minimizing geodesic connecting z n = γ(a n ) and w n = γ(b n ) Since x = y, it follows δ(z n , w n) 1 By Lemma 5, it follows

1 c

 ∞

e 2kΩ(z0,γn)

c0 + ln α

αf (α) dα.

This inequality implies that kΩ(z0, γn)  1 because lim

s →∞

 ∞

s

c0 + ln α

αf (α) dα = 0. Consequently, there is a point p n ∈ γ n such that kΩ(z0, pn ) = kΩ(z0, γn)  1 Moreover,

kΩ (z0, wn)≤ kΩ (z0, pn ) + kΩ(p n , w n)

≤ kΩ (z0, pn ) + kΩ(w n , z n)

 kΩ(w n , z n ) + 1.

Since Ω is k-complete, it follows that kΩ(z0, wn) → ∞ as w n → x ∈ ∂Ω This

3 Proof of Theorem 1

In order to prove Theorem 1, we recall the definition of small and big horospheres

and F -convexity from [2, 3].

Definition 3 (see [2, p 228]) Let Ω be a domain inCn Fix z0∈ Ω, x ∈ ∂Ω and

R > 0 Then the small horosphere E z0(x, R) and the big horosphere F z0(x, R) of center x, pole z0 and radius R are defined by

E z0(x, R) = {z ∈ Ω: lim sup

Ωw→x [kΩ(z, w) − kΩ (z0, w)] < 1

2ln R }

F z0(x, R) = {z ∈ Ω: lim inf

Ωw→x [kΩ(z, w) − kΩ (z0, w)] < 1

2ln R }.

Definition 4 (see [3, p 185]) A domain Ω ⊂ C n is called F -convex if for every

x ∈ ∂Ω

F z0(x, R) ∩ ∂Ω ⊆ {x}

holds for every R > 0 and for every z0∈ Ω.

Remark 1 The bidisk Δ2 in C2

is not F -convex Indeed, since dΔ 2((1/2, 1 − 1/k), (0, 1 − 1/k)) − dΔ2((0, 0), (0, 1 − 1/k)) = dΔ (1/2, 0) − dΔ (0, 1 − 1/k) → −∞

(0,0) ((0, 1), R) ∩ ∂(Δ2

) for any R > 0.

Remark 2 If Ω is either a strongly pseudoconvex domain inCn, or a pseudoconvex domain of finite type inC2

, or a pseudoconvex domain of strict finite type inCn

,

then Ω is F -convex (see [2, 3, 23]).

Now, we prove that F -convexity holds on a larger class of pseudoconvex domains.

Proposition 7 Let Ω be a domain satisfying the hypotheses of Theorem 1 Then

Ω is F -convex.

Proof Let R > 0 and z0 ∈ Ω Assume by contradiction that there exists y ∈

F z0(x, R) ∩∂Ω with y = x Then we can find a sequence {z n } ⊂ Ω with z n → y ∈ ∂Ω

and a sequence{w n } ⊂ Ω with w n → x ∈ ∂Ω such that

(11) kΩ (z n , w n)− kΩ (z0, wn)≤1

2ln R.

Moreover, for each n ∈ N ∗ there exists a minimizing geodesic γ

n connecting z n to

w n Let p n be a point on γ n of minimal distance kΩ(z0, γn ) = kΩ(z0, pn ) to z0 We consider the following two cases for the sequence{p n }.

Case 1 There exists a subsequence {p n k } of the sequence {p n } such that p n k → p0 ∈ Ω as k → ∞,

kΩ (w n k , z n k)≥ kΩ (w n k , p n k ) + kΩ(p n k , z n k)

≥ kΩ (w n k , z0)− kΩ (z0, pn k ) + kΩ(p n k , z n k ).

(12)

From (11) and (12), we obtain

kΩ (p n k , z n k)≤ kΩ (w n k , z n k)− kΩ (w n k , z0 ) + kΩ(z0, pn k)≤ 1

2ln R + kΩ(z0, pn k) 1 This is a contradiction since Ω is k-complete.

Case 2 Otherwise, p n → ∂Ω as n → ∞ By Lemma 5, there are constants c and c0 only depending on z0 such that

 ∞

e 2kΩ(z0,γn)

c0 + ln α

αf (α) dα.

On the other hand, δΩ(w n , z n) 1 since x = y Thus, the inequality (13) implies

that

(14) kΩ (z0, γn ) = kΩ(z0, pn) 1.

Therefore,

kΩ (z n , w n)≥ kΩ (z n , p n ) + kΩ(p n , w n)

≥ kΩ (z0, zn ) + kΩ(z0, wn)− 2kΩ (z0, pn ).

(15)

Combining with (11) and (14), we get

kΩ (z0, zn)≤ kΩ (z n , w n)− kΩ (z0, wn ) + 2kΩ(z0, pn) ln R + 1.

This is a contradiction since z n → y ∈ ∂Ω and hence the proof is complete.  The following theorem is a generalization of Theorem 3.1 in [3]

Proposition 8 Let Ω be a domain satisfying the hypothesis in Theorem 1 and fix

z0 ∈ Ω Let φ ∈ H(Ω, Ω) such that {φ k } is compactly divergent Then there is a point x ∈ ∂Ω such that for all R > 0 and for all m ∈ N

φ m (E z0(x, R)) ⊂ F z0(x, R).

Proof Since {φ k } is compactly divergent and Ω is k-complete,

lim

k →+∞ kΩ (z0, φ

k (z0)) =∞.

For every ν ∈ N, let k ν be the largest integer k satisfying kΩ(z0, φk (z0))≤ ν; then

(16) kΩ (z0, φk ν (z0))≤ ν < kΩ (z0, φk ν +m

(z0))∀ν ∈ N, ∀m > 0.

Again, since{φ k } is compactly divergent, up to a subsequence, we can assume that

φ k ν (z0)→ x ∈ ∂Ω.

Fix an integer m ∈ N Without loss of generality we may assume that φ k ν (φ m (z0))→

y ∈ ∂Ω Using Corollary 6 and the fact that

kΩ (φ k ν (φ m (z0)), φ k ν (z0))≤ kΩ (φ m (z0), z0) (by (5))

it must hold that x = y.

Set w ν = φ k ν (z0) Then w ν → x and φ m

(w ν ) = φ k ν (φ m (z0))→ x From (16),

we also have for m ≥ 0

ν →+∞ [kΩ(z0, wν)− kΩ (z0, φm (w ν))]≤ 0.

Now, fix m > 0, R > 0 and take z ∈ E z0(x, R) Then

lim inf

Ωw→x [kΩ(φ

m (z), w) − kΩ (z0, w)]

≤ lim inf

ν →+∞ [kΩ(φ

m (z), φ m (w ν))− kΩ (z0, φm (w ν))]

≤ lim inf

ν →+∞ [kΩ(z, w ν)− kΩ (z0, φm (w ν))]

≤ lim inf

ν →+∞ [kΩ(z, w ν)− kΩ (z0, wν)]

+ lim sup

ν →+∞ [kΩ(z0, wν)− kΩ (z0, φm (w ν))]

≤ lim inf

ν →+∞ [kΩ(z, w ν)− kΩ (z0, wν)]

≤ lim sup

Ωw→x [kΩ(z, w) − kΩ (z0, w)]

< 1

2ln R, (18)

that is, φ m (z) ∈ F z0(x, R) Here, the first inequality follows by φ m (w ν)→ x, the

second follows by (5), the fourth follows by (17), and the last one follows from the

Lemma 9 Let Ω be an F -convex domain in Cn

Then for any x, y ∈ ∂Ω with

x = y and for any R > 0, we have lim

a →y E a (x, R) = Ω, i.e., for each z ∈ Ω, there exists a number  > 0 such that z ∈ E a (x, R) for all a ∈ Ω with |a − y| <  Proof Suppose that for some z ∈ Ω there exists a sequence {a n } ⊂ Ω with a n → y and z ∈ E a n (x, R) Then we have

lim sup

w →x [kΩ(z, w) − kΩ (a n , w)] ≥1

2ln R.

This implies that

lim inf

w →x [kΩ(a n , w) − kΩ (z, w)] ≤1

2ln

1

R . Thus, a n ∈ F z (x, 1/R), for all n = 1, 2, · · · Therefore, y ∈ F z (x, 1/R) ∩ ∂Ω = {x},

Now we are ready to prove our main result

Proof of Theorem 1 First we fix a point z0 ∈ Ω By Proposition 8 there is a point

x ∈ ∂Ω such that for all R > 0 and for all m ∈ N

φ m (E z0(x, R)) ⊂ F z0(x, R).

We need to show that for any z ∈ Ω

φ m (z) → x as m → +∞.

Indeed, let ψ(z) be a limit point of {φ m (z) } Since {φ m } is compactly divergent, ψ(z) ∈ ∂Ω By Lemma 9, for any R > 0 there is a ∈ Ω such that z ∈ E a (x, R) By Proposition 8, φ m (z) ∈ F a (x, R) for every m ∈ N ∗ Therefore,

ψ(z) ∈ ∂Ω ∩ F a (x, R) = {x}

Acknowledgment

We gratefully acknowledge the careful reading by the referees The exposition

of the paper was improved by the close reading

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School of Mathematics and Applied Statistics, University of Wollongong, NSW, Australia, 2522

E-mail address: tkhanh@uow.edu.au

Department of Mathematics, Vietnam National University at Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

E-mail address: thunv@vnu.edu.vn

, or a pseudoconvex domain of strict finite type inCn... Ω is F -convex (see [2, 3, 23]).

Now, we prove that F -convexity holds on a larger class of pseudoconvex domains.

Proposition Let Ω be a domain satisfying the hypotheses... contradiction since z n → y ∈ ∂Ω and hence the proof is complete.  The following theorem is a generalization of Theorem 3.1 in [3]

Proposition Let Ω be a domain