Complex Variables and Elliptic Equations

map f the Wolff Lemma states the existence of a unique point T E dA, called the WoH point of f, such that each horocycle (i.e., each...

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Commuting holomorphic maps and linear fractional models Cinzia Bisi a; Graziano Gentili a a Dipartimento di Matematica "U. Dini", Firenze

Online Publication Date: 01 July 2001 To cite this Article: Bisi, Cinzia and Gentili, Graziano (2001) 'Commuting holomorphic maps and linear fractional models', Complex Variables and Elliptic Equations, 45:1, 47 - 71 To link to this article: DOI: 10.1080/17476930108815368 URL: http://dx.doi.org/10.1080/17476930108815368

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Commuting Holomorphic Maps and Linear Fractional Models ClNZlA BISI* and GRAZIANO GENTILI~ Dipartimento di Matematica "U. Dini: Viale Morgagni 67/A, 50134 Firenze Communicated by R. P. Gilbert (Received22 December 1999)

Let f be a holomorphic map of the open unit disc A of C into itself, having no fixed points in A and WolB point T E aA. In the open case in which f (T)= 1 we study the centralizer o f f , i.e., the family Gf of all holomorphic maps of A into itself which commute with f under composition. We prove that if the sequence of iterates {f} converges to T non tangentially, then Gf coincides with the set of all elements of the pseudo-iteration semigroup off (in the sense of Cowen, see [5,6]) whose WolfFpoint is T . In the same hypotheses we give a representation of the centralizer Gf in Aut(A) or Aut(@), study its main features and generalize a result due to Pranger ([IS]). Keywordr: Iteration and composition of holomorphic maps; Linear fractional models; Centralizer of families of holomorphic maps; W O Epoint and fundamental set AMS Subject CZass$cation Numbers: 30D05, 30C99, 32H50

1. INTRODUCTION

In this paper we give a contribution to the investigation of the connection between iteration theory and the study of sets of commuting holomorphic maps, in the open unit disc A of C. We will be concerned with the case of a holomorphic map of the open unit

* e-mail: [email protected] +corresponding author. Partially supported by GNSAGA of the I.N.D.A.M. and by M.U.R.S.T. e-mail: [email protected]

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disc A into itself,f E Hol(A,A), without fixed points in A. For such a map f the Wolff Lemma states the existence of a unique point T E dA, called the WoH point of f, such that each horocycle (i.e., each Euclidean disc contained in A and tangent to aA at r ) is sent into itself. The WoH Lemma and the Julia-WoH-CarathCodory Theorem assert that the non-tangential derivative of such an f at r is a strictly positive real number smaller or equal than 1 ([I]). The WoH point is fundamental in the study of the iterates of f: the WOE-Denjoy theorem asserts that the iterates off converge, uniformly on compact sets, to the Wolff point off ([1,20]). Classical results point out that, in the case of a holomorphic map of the unit disc A into itself having a fixed point in A, the behaviour of the iterates of the map depends on the value of the derivative at the fixed point itself ([I 1,161). In a very similar way the value of the nontangential derivative off at its WoH point T E dA is strictly connected with the behaviour of the iterates of$ if the value of ft(r) is strictly smaller than 1, then for any z E A the sequence of iterates {f "(z)),, converges to T non-tangentially and the behaviour off has been widely investigated in recent years, also in connection with the study of the family of maps which commute with f with respect to composition ([2,5,9]). The case in which the value of ft(r) equals 1 is still open for many questions, in particular for what concerns the study of the family of maps which commute with f under composition. In this framework Cowen proves in his papers [5,6] a fundamental theorem for the study of the geometry, and iteration theory, of a holomorphic map f E Hol(A,A) with no fixed points in A, and gives the nice and fertile definition of pseudo-iteration semigroup of such a map. Essentially, Cowen studies a boundary version of the Schroeder equation; he proves, among the other things, that (see Theorem [4]), given f E Hol(A,A) with Wolff point r E dA, then there exist a holomorphic ~ (or a f Hol(A,@)) ~ and a holomorphic automormap u f Hol(A,A) phism af of A (or, respectively, of @), unique up to conjugation by automorphisms, such that

,

The automorphism af contains interesting informations on f, and allows the definition of pseudo-iteration semigroup o f f : a map g

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belongs to the pseudo-iteration semigroup of f if there exists an automorphism XP, of A (or, respectively, of @) such that

and that

The pseudo-iteration semigroup Pf of a map f turned out to be an interesting tool for the study of the geometric theory of holomorphic mappings and for iteration theory (see also [3,5,6,13]) and the results obtained by Cowen will be widely used here. In this paper we prove two main results. The fist one concerns the study of the family Gf of all holomorphic maps from the unit disc A into itself which commute under composition with a given f E Hol(A,A) having no fixed points in A and W O E point T E dB. We prove that, in the open case in which f1(7-)= 1, if for same zo E A (and hence for all z E A) the sequence of iterates {fn(z0))n N converges to T non-tangentially, then the family Gf coincides with the set of all elements of the pseudo-iteration semigroup Pf of f whose Wolff point is T (see theorem 6). This result is a nice improvement of a result obtained by Vlacci [8,19]. The second main result of this paper is related to an interesting statement due to Pranger [15]. In the case in which f~Hol(A,A) has 0 E A as a h e d point, and if Gf denotes the family of all g E Hol(A,A) such that f o g =g oS, let X : Gf -+ h denote the map g Hg'(o). Pranger [15], proves the following result:

1 Let 'I THEOREM

c 1be a set with the following properties:

z;

(a) I' is a closed subset of (b) 0 and 1 belong to I' and r n A # (0); (c) 'I is closed under multiplication; (6) I?*, the complement of I', is connected. Then, there is a map f E Hol(A,A) such that f (0) = 0,0 < 1fl(0)J < 1, f is locally univalent and I? = X(Gf). Conversely, given a map f E Hol(A,A) such that f(0) =O,O < If'(O)l< 1,f is locally injective, then X(Gf) satisjiesproperties (a). . .(d).

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In the same spirit in which the Julia-WOE-CarathCodory Theorem and the Wolff Lemma (see Section 2) can be viewed as boundary versions of the Schwarz Lemma, we prove a (partial) boundary version of the above theorem of Pranger. Let Gf still denote the set of all holomorphic maps which commute with a given f E Hol(A,A) with no fixed points in A, and let X :Gf+A be the map which associates to g€Gf the automorphism Q, of A (or, respectively, of @) which appears in the definition of pseudo-iteration semigroup off. We prove the following (see Section 4):

THEOREM 2 Let f E Hol(A,A) be such that its Wolflpoint

r is equal to 1. Suppose that there exists zo E A such that {f "(z~)),, N converges to 1 non-tangentially. Then the set X(Gf) satisfies the following properties:

(a) X(Gf) is a subset of Aut(A) (or, respectively, of Aut(@)) topologically closed; (b) the identity map I E X(Gf); (c) X(Gf) is closed with respect to composition; (4 the complement of X(Gf) in Aut(A) (or, respectively, in Aut(@)) is connected. Section 2 contains some fundamental preliminary results and definitions, which are essential to understand the setting in which the paper is located, and which are used extensively. Section 3 contains the new result which completely describes the family of all holomorphic maps which commute with a given f E Hol(A,A) having no fixed points in A, and whose iterates converge non-tangentially to the WoH point T E I ~ AAs . we have already mentioned this description is made in terms of the pseudo-iteration semigroup Pf off. Some consequences are also presented. In Section 4 the boundary version of the above mentioned theorem due to Pranger (see Theorem 8) is presented, together with some related results. The authors are happy to thank Francesco Borgatti for his technical support which has permitted the typesetting of this paper. 2. PRELIMINARIES AND A FEW BASIC RESULTS

In this section we will present some preliminary results which will be extensively used in the rest of the paper. Some of them are quite

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classical results and some other are recently discovered statements, all concerned with the role played by fixed points of holomorphic maps of the unit disc into itself (including the case of boundary fixed points) in the theory of iteration and in connection with the study of families of commuting holomorphic functions. It is in the theory of iteration that the horocycles play the role of "Poincarb discs at the boundary of A": let T E 6A; then for all R > 0 the open disc of A tangent to a A at T defined as

is called the horocycle of center T and radius R. The following, well known, WoH Lemma (see e.g. [I, 16]), guarantees the existence of a fixed point at the boundary for a function f E Hol(A,A) having no fixed points in A. It also plays the role of a boundary Schwarz Lemma. LEMMA 1 (Wolff) Let f E Hol(A,A) be withoutfixedpoints. Then there is a unique T E 6A such that for all z E A

that is

where E(T, R) is the horocycle of center T and radius R > 0. Moreover, the equality (1) holds at one point (and hence at allpoints) ifand only iff is a (parabolic) automorphism of A leaving r j x e d . Iff E Hol(A,A) has a h e d point in A (and f# id) then we denote this fixed point by TV). Otherwise, T C ~denotes ) the point constructed in Lemma 1. In both cases TU) is called the Wolff point off.

DEFINITION 1 Take a E 6A and M > 1. The set

is called Stolz region K (u, M ) of vertex u and amplitude M.

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The Stolz region K(a,M) is an "angular region" of A with vertex at give the following

u and opening a! < n. Stolz regions are necessary to

DEFINI~ON 2 Let f : A 4 be a function. We say that c is the nontangential limit (or angular limit) off in u E dA iff (z)+c as z tends to u within K(u,M), for all M > 1. We shall write

K - limf (z)= c. 2-0

The definition of non-tangential limit is used in (see [I]).

THEOREM 3 (Julia-Wolff-Carathkodory) Let f E Hol(A,A) and let T, a be any two points in dA, then

K - lim-72-0

-f (4 u -z

exists. If the above non-tangential limit is finite, then

K

- lirnf (z)= T 2-m

and there exists r E R, r 2 0, such that 7 -f (4 K - lirnf'(z) = K - lim = err 2'0

2-0

(7 - Z

In particular, ifr = a then the non-tangential limit r off' at a is a strictly positive real number.

We say that r E dA is a fixedpoint o f f on the boundary of A,

K

if

- lirnf ( z ) = T ; 2'7

at the same time we call derivative o f f at afixedpoint T on the boundary of A the value of

As a consequence of the Julia-Wolff-CarathCodory theorem, it follows that iff E Hol(A,A) has no fixed points then the value of the derivative ft(r) at the Wolff point T of f (TE dA) is real and such that 0
DEFINITION 3 Let f be a holomorphic function mapping the unit disc A into itself. The functions f n =f of" - defined inductively for the

'

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natural numbers (where f 1 =f) will be called the "natural iterates" off. The two following lemmas give some description of the behaviour of the iterates off (see, e.g. [5]). LEMMA 2 Suppose that a mapf E Hol(A,A) has its Wolffpoint T C ~on ) the boundary of A. If

,

then for any z E A the sequence of iterates {fn(z)), N converges to non-tangentially, that is, converges within K (T, M), for some M > 1.

T

LEMMA 3 Suppose that a mapf E Hol(A,A) has its Wolfpoint T C ~on ) the boundary of A. If for some zo in A the sequence of iterates Cf"(zo)},, N converges to T non-tangentially, then for any compact set K in A, the sequence of iterates {fn(K)),,, converges to T non-tangentially. Inparticularfor any z E A, the sequence of iterates {f"(z)), N converges to T non-tangentially.

,

If f,g E Hol(A,A) commute under composition, then it is well known thatf; g have the same fixed point in A or the same Wolff point in aA, unless both of them are hyperbolic automorphisms of A (see, e.g. [I, 41). Iff, g E Hol(A,A) are commuting holomorphic maps, then a connection between the value of their derivatives at a common k e d point T E is presented in the following:

a

1 Zff and g commute and T is their common Wolfpoin t, PROPOSITION then (1) W ( T ) =0, then gt(r) = 0;

(2) $0 < 1ft(r)) < 1, then 0 < lgt(r)(< 1; )(3) r( ' ffi = 1, then g'(r) = 1. Now we will state the "main theorem" of Cowen and give the definition of pseudo-iteration semigroup of a map f E Hol(A,A). We need the following: DEFINITION 4 An open, connected, simply connected subset Vof A is called a fundamental set for f E Hol(A,A) iff (V) c V and if for any compact set K i n A, there is a positive integer n so that fn(K) c V.

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The "fundamental set" of a map f is a set of points "near" the Wolff point T (f) "small enough" that f is "well behaved" on it, and "large enough" that fn(z) belongs eventually to this set. THEOREM 4 (Cowen) Let f E Hol(A,A) be neither a constant map nor an automorphism of A. Let T be the Wolffpoint off and suppose that fl(r) #O. Then there exists a fundamental set Vf for f in A on which f is injective. Furthermore there also exist: (1) a domain R which is either the complex plane C or the unit disc A, (2) a linear fractional transformation cp mapping R onto R, (3) an analytic map uf mapping A into 0 , such that (i) uf is univalent on Vf, (ii) q ( V ) is a fundamental set for cp in R, (iii) af of = cp o q-. Finally, cp is unique up to conjugation under linear fractional transformations mapping R onto 0 , and the maps cp and a- depend only on f and not on the choice of the fundamental set Vf. It has been proved, [5], that the map cp and the set R in the statement of the above theorem, fall into four essentially different cases: (1) R=C, af(r)=0, cp(z)=sz, 0 < Is1 < 1; cp(z) = (((1 s)z 1-s)/((l -s)z 1 s)), (2) R = A, 0-(7.) = 1, O
+ +

++

Following Cowen, [5,6], we will now define the pseudo-iteration semigroup of a map f c Hol(A,A). DEFINITION 5 Let f, g be holomorphic maps of A into A. Let T be the W O E point off and suppose that fl(r) #O. Let VfiR, uf and cp be as in theorem (4), relative to f. We say that g "is in the pseudo~ if there exists a linear fractional iteration semigroup off" ( g Pf), transformation $ which commutes with cp, such that o--0 g = $ o uf The following results relate the study of the pseudo-iteration semigroup of a map f~Hol(A,A) with the study of the family of all maps g E Hol(A,A) which commute with f under composition.

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THEOREM 5 Let f, g e H o l ( A , A ) be neither constants nor automorphisms of A, and let f o g =g of. Let r E aA be the (common) W o l f point o f f and g. I f there exist zo and wo in A so that g"(zo)-r T and f n(wo)-+ r non-tangentially, then g is in the pseudo-iteration semigroup off. PROPOSITION 2 Let f be a holomorphic map of A into A neither constant nor an automorphism of A and let r E aA be its W o l f point. Let g be in the pseudo-iteration semigroup o f f . Then f andg commute if and only if there is an open set U in A such that g ( U ) and g ( f ( U ) ) are contained in the fundamental set Vf o f f .

If the sequence of all iterates of a point zo E A under5 {fn(z0)),, N, has a non-tangential behaviour, then the fundamental set Vf o f f , which is constructed in Theorem 4, has nice geometric properties, which help much to understand the geometric structure of the pseudoiteration semigroup. Before stating the geometric results contained in Propositions 3 and 4, we need a few definitions: DEFTNITION 6 Let a E aA and let T be the line containing the diameter of & passing through a. An angular sector of vertex a and opening 0 in A, is the intersection of A with the open angle having vertex in a, bisectrix line T and opening 8. A small angular sector of vertex a and opening 8 in A, is the intersection of an angular sector of vertex a and opening 0 in A with any open, Euclidean disc of positive radius centered at a. LEMMA 4 Let f~ Hol(A,A) be without fixed points and with W o l f point T E aA. Thenf sends Stolz regions of vertex r into Stolz regions of vertex T .

Proof Using the inequality (1) stated in the W O E Lemma, we have:

Away from the Wolff point r, the fraction

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56

is bounded from above; on the other hand, if we suppose that z belongs to some Stolz region K (T,M ) and z is "near" T, then

because K - lim 2-'T

IT -I. 17

-f ( 4 I

= K - limp=1 z-T

Ift(z)I

-> 1 Ifl(T)l -

(8)

(by the Julia-Wolff-Carathtodory theorem, 0
Hence, the right-hand member of inequality (7) is bounded from above, if we suppose that z belongs to some Stolz region K(T, kt);SO we have proved that f sends Stolz regions of vertex T into Stolz regions of vertex T. DEFINITION 7 A set V G A is said to be angular in 7 E a A if it contains small angular sectors of vertex T and opening 8, for all 8 < T . DEFINITION 8 Let f E Hol(A,A). A set U is said to be f -absorbing if for any compact set Kin A there exists no E N such that y ( K ) c U, for all n E N, n > no. The proof of the following proposition, which gives a nice geometric property of the fundamental set of a map f E Hol(A,A), can be found in [5].

3 Let f~Hol(A,A) be neither a constant map nor an PROPOSITION automorphism of A and let the Wolfpoint rCf) belong to the boundary of A. If,for same point zo of A, the sequence ( f " ( ~ ~ ) ) ~converges to rCf) non-tangentially, then anyfundamental set V of f is angular at TC~). At this point, we will state the Noshiro Lemma, an elementary result (see e.g. [5]) which, in particular, will be used later as a powerful tool to guarantee injectivity of a map f E Hol(A,A) on convex subsets of the fundamental set Vf.

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LEMMA 5 (Noshiro) If U is a convex open subset of the plane @, f is holomorphic on U and Refl(z) > 0 for all z E U, then f is injective on U. To end this section, we will present the following result (whose proof is sketched in [19]) which states a very nice geometric property of holomorphic maps at their Wolff point. This property, which is interesting for itself, will be important in the sequel of the paper. PROPOSITION 4 Let f E Hol(A,A) be such that T E dA is its Wolffpoint. Let W be a subset of A which is angular at 7. Thenf (W) is also angular at 7.

Proof Indeed, fixed a < n, consider in A the angular sector S, of vertex T and opening a. Take E > 0 and define r?+€ and G+Ethe two sides of the angular sector S,,, of opening a+& at T. The angle at T between f(<+€) and f(G+€) has amplitude a + &(f is isogonal at T E aA, i.e., the angles between curves starting in T are preserved, see, e.g. [14]). Since W is angular, it is possible to find a horocycle O1of center T such that O1n S,,, c W and f is injective on 01 n S,+, (by the Noshiro Lemma). Now, the boundary of the region OlnS,+, consists of two portions of the two sides of S,+,-let us call the two portions 1;1+& and );+'-and of a portion of the boundary of O1denoted by 13.The map f (being injective on it) transform the region 0,n S,+. onto the region whose boundary is f (I?+") u f (I;+€) u f (I3). Since, as we have noticed, the angle between f (lr+') and f(l;+,) at T is a+&, then f(OlnS,+,) contains a small angular sector of vertex T and opening a. Since a < .rr can be chosen arbitrarily, then the assertion H follows.

3. PSEUDO-ITERATION SEMIGROUPS AND COMMUTING

HOLOMORPHIC MAPS

This section contains a new result which describes completely the family of all holomorphic maps which commute with a given f E Hol(A,A) having no fixed points in A and whose iterates converge non-tangentially to the Wolff point T E aA. The proof of this result is based on the following Lemmas.

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LEMMA6 Let f, g E Hol(A,A) be such that ~ ( f =) r(g)= 1. Suppose that 3z0 E A such that f n(zo)+ 1 non-tangentially. Zf Vf is a fundamental set for f (see Definition 4), then there exists a set with the following properties:

vf

(1) Pf is open and simply connected; (2) c vf; (3) _is angular at 1; (4) d V f ) c 5; (5) Regf( z )> 0, Vz 'z

vf vf

vf.

vf

Proof In order to construct , we choose an arbitrary a < 7r and let S, be an angular sector of vertex 1 and opening a . Thanks to Proposition 3, the set Vf is angular at 1. Then there exists a horocycle 0,of center 1 and positive radius such that ( S , n 0,) c Vf Moreover, since 0 < g'(1) = K- lim,,lgf(z) 5 1 (i.e., since g f is K-continuous) the horocycle 0, can be chosen in such a way that Reg'(z) > 0 for all z in S, n 0,. By Lemma 4, we can find /I < 7r, /I =p(a) depending on a and g, such that g(Sa)c SA,). Now, again by Proposition 4, in correspondence of the angular sector Sp(,) there exists a horocycle OA,) c 0, such that SP(,) n Op(,) c Vf and that Reg'(z) > 0 for all z in S,n Op(,,. Let us now define

We obtain: (1)

fjis open and simply connected because it is the union of open

angular sectors with not empty intersection; (2) c 5 because Va, S, n Op(,) G S, n 0, c Vj (3) is angular at 1; (4) Since for any horocycle Od,), centered at the Wolff point 1, we have g(Op(,)) c Op(,) then

vf vf

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(5) Reg'(z) > 0, for all z E

vf because

and Reg'(z) > 0 on S, n OH,), for all a < T. The following Lemma will permit us to use the result of Noshiro (Lemma 5) to obtain a set U, contained in the above encountered set U open and angular at 1 and on which the function g will be injective. Such a set U will play a key role in the sequel.

vf,

LEMMA7 Let W be an open set, angular in 1 E aA. Then there exists U c W, U open, angular at 1 and convex. Proof Let a, = T- (2/n),n E N, and let S, be the interior of an open angle contained in A with vertex in 1, of opening a, and symmetric with respect to the ray Since W is angular in 1, there exists a small open disc C,, with center 1 and radius smaller than 1, such that S,nC,c W, Vne N. Without loss of generality, one can take C,+ 1 c C,, that means that the sequence (r,), N of radii of (C,), N is a decreasing sequence. Finally let r = U , N(S,n C,) c W. Consider now the sequence of points:

a.

.

po

p,

=acln ( - 1 , l ) = (as, n ac,,~nH+)

.

(n > 0)

where (- 1,l) is the open interval bounded by - 1 and 1 on the real axis and where H+ is the open upper half-plane of 62. We can also suppose that the sequence of the open discs {C,), N is choosen in a such a way that, for all n E N, n > 0, p,+2 lies on the same

.

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side of the point 1 with respect to the straight line connecting pn and P,+~. Notice that the straight line containing pn and pn+l never contains the point 1 since W is angular at 1. We pass now to consider U bj?)nENU ( 1 ) consisting of the the set of points P , = {p,),, sequence {pnjnEN, of the sequence of the conjugates @ j l n E of (p,}, N and of the point 1. Let U be the interior of the convex hull of P,. The set U is such that: (1) the interior of U is not empty, since the segment p,p,+l is never radial; (2) U c I? and hence UC W; (3) U is angular at 1. Indeed, let S c A be an open angular sector with vertex 1 and amplitude strictly smaller than T . Then there exists n E N such that S E Sn and, by construction, the open disc C of center 1 and radius dCpn,l) is so that Snn C c U . In conclusion, the small angular sector Sn C is contained in U. (4) U is convex and in particular it is simply connected.

We will point out a nice connection between the property of being angular and that of beingf-absorbing, for a subset of the unit disc A. LEMMA 8 Let f E Hol(A,A) be without fixed points and suppose that there exists zo E A such that f ( z o ) 4 T(f ) (the Wolflpoint o f f ) nontangentially. Then any subset W of A which is angular at T(f ) is also f-absorbing. Proof We want to prove that: for any K compact set in A, there exists m E N such that f "(K)c W. From Lemma 3, f n(K)4 TOnon-tangentially, i.e., f ( K )-+ ~ ( f ) inside a Stolz angle S. Therefore for all E > 0 there exists ii, such that for all n > Ti,, we have f n ( K ) c O , n S , where 0, is the horocycle with ) radius E . Since W is angular at ~ ( f )then , for E center ~ ( f and sufficiently small, f n ( K ) c S n 0,C W for all n 2 ii,.

We are now ready to prove the main result of this section. THEOREM 6 Let f E Hol(A,A) be such that its Wolflpoint ~ ( f=) 1 and suppose there exists zo -EA such that f"(zo)-t 1 non-tangentially. Let g E Hol(A,A) with ~ ( g=) 1. Then f o g =g of if and only if g belongs to the pseudo-iteration semigroup Pf o f f .

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Proof By Proposition 2, and since f1(l)#O, if gePf, then g commutes with f if and only if there exists U open in A such that g(U) and g(f (U)) are contained in Vfi fundamental set off. Let S, A be an arbitrary open angular sector of opening a < .rr and vertex 1; by Lemma 4, there exist p, 7, S such that

Since for a zoE A, the iteration sequence {fn(z~,)}~ converges to 1 non-tangentially, the fundamental set o f f , Vfi is angular at 1 (see Proposition 3). Therefore there exists Op, horocycle of center 1, such that S p n O p c Vf and there exists 06, horocycle of center 1, such that

ssno6cv-

In general, let r(0) be the radius of the horocycle 0 , and let r(0,) = min(r(Op), r(O6)). Then define U = 0, n S,. The WoliT lemma yields g(0,) c 0,, f (0,) c 0, and therefore:

and also

Therefore g commutes with f under composition. On the other hand, suppose that f o g =g of. Let V'' a, uf and cp be the same as in Theorem 4 (Cowen). By using Proposition 3, we can find a fundamental set Vf of f that is angular at 1. Let U be obtained from V' like in Lemma 7. We want to prove now that uf(U) is cp-absorbing. Since uf(Vf) is fundamental for cp, given any compact set K C 0 there exists rno E N such that

for all rn >mo. Being now uf injective on Vf we obtain, if

q1:= (Uf,"J1:

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for all m > mo, and since U is $absorbing by Lemma 8, there exists ko E N such that

for all k > ko. Hence

for all k, m such that m+ k > mo+ ko. Taking into account that uf (U) is cp-absorbing, we are now ready to define:

where n is sufficiently large such that cpn(w)E of (U). The map $ is well-defined, because if cpn(w)E q ( U ) and if p E N, then:

Moreover, since uf is injective on Vf and g is injective on U then $ is injective on 0. It remains to prove that $ is also surjective on a. To this aim it is now necessary to prove that the set uf(g(U)) is also cpabsorbing. The procedure is similar to the one used above for uf(U). Notice at first that, by Proposition 4, since Uis angular at 1, so is g(U). Lemma 8 implies now that g(U) isf-absorbing. Consider that for any compact set K C a , there exists nl E N such that

for all n > nl (since uf(Vf) is fundamental for cp on a ) As usual this is to say that for all n > nl

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Since g(U) is $absorbing, there exists k2E IW such that for all k > kz

and then

for all k+n such that k > k2, n > nl; hence uf(g(U)) is cp-absorbing. The map $ is surjective on R if for all z E 52 there exists w E R such that $(w) =z, i-e., (cp-')"(nf (g(up(cp"(w)))))= z. Let z E R be given. Then, for n sufficiently large, since of (g(U))is cp-absorbing, set z, = cpn(z)OE of MU))and consider (uf o g o u ~ ' ) - ' on f?f(g(U)).

Then, since g is injective on U, we can d e k e g-' =(glv)-l and set z2 = (of o g o u;l)-' (zl) E q(LI). Finally, take w = cp-"(z2) E R. It is straightforward to prove now that $(w)=z, i.e., that $ is surjective. Hence $ is a linear transformation from R onto R. By following the construction, we have Il,o cp o $-' = cp and

And so g E Pf As an immediate consequence, we obtain a result proved directly by Cowen in [5,6]. COROLLARY 1 Let f E Hol(A, A) such that T(f ) = 1 and f '(1) < 1. Let g E Hol(A, A) with ~(g) = 1. Then g of =f o g if and only if g belongs to the pseudo-iteration semigroup Pf o n

4. A BOUNDARY GENERALIZATION OF A THEOREM OF PRANGER

The assertion of Theorem 6 can be restated as follows:

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THEOREM 7 Let f E Hol(A, A) be such that its Wolfpoint ~ ( f ) is equal to 1, not an hyperbolic automorphism, and let Pf denote as usual the pseudo-iteration semigroup off. Define:

GI= { g ~ H o l ( A , a:)f o g = g o f ) and

Gz= {g€Hol(A,A) : ~ ( g=) l,g€Pf). Ifthere exists zo E A such thatf 'l(zo)-+ 1 non-tangentially, then GI = Gz. In the hypotheses in which f €Hol(A, A) is such that {f"(zo)} converges to the Wolff point TO= 1 non-tangentially, by applying Theorem 4 (Cowen) one proves that, up to conjugation under linear fractional transformations, 0 , cp, af are such that (a) 0 = @cp(z)=z-t, 1, fl(rCf>)=l; 4 ((K1+s)z+(l -s)l)l([(l-4z+(l (b) 0 = A, ( ~ (= O
+s)l>),f'(.rCf)>< 1,

Case (a) or (b) occurs according as f'(1) = 1 or f '(1) < 1. In the hypotheses of Theorem 6 , we will now consider R, cp, af relative tof, fixed. Since GI = G2= Gf, given g E Gf there exists a linear fractional transformation a, belonging to Aut(@) or to Aut(A) such that:

and that

If A is set to be

A = { Aut(@) in case (a) Aut(A) in case (b) then the map

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is well defined both in case (a) and in case (b); in fact, given g, it is easy to see that Qg in unique as defined. The construction given above for X recalls the construction given by Pranger in his paper [15], in the case in which the Wolff point ~ ( f o)f f belongs to the interior of the unit disc. And in fact in the same spirit in which the Julia-WolffCarathkodory Theorem and the Wolff Lemma are boundary versions of the Schwarz's Lemma, we are now ready to prove a boundary version of a theorem of Pranger [15].

THEOREM 8 Let f E Hol(A, A) be such that ~ ( f=)1 and not an hyperbolic automorphism. Suppose there exists zo E A such that f "(zo) -t 1 non-tangentially. Let A be deJned as in (9) and let X :Gf -,A be the map defined in (10) which associates to each g€Gf the automorphism a,. Set X(Gf) =I?cA. Then the following properties hold: (1) (2) (3) (4)

J? is a subset of A topologically closed; the identity map I E I?; I? is closed with respect to composition; the complement of I?, I?*, is connected.

Proof We start by proving (3) (see also [7]): taken two maps h, k E Gf, we have, by definition of Gf:

It is straightforward that also (h o k) commutes withf, i.e.:

Hence (h o k) E Gf and Gf is closed with respect to composition. Therefore, it makes sense to consider ahok€X(Gf). NOW,by definition of pseudo-iteration semigroup, it follows that:

Since h E Gf, then

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and since also k E Gf, then ( @ h ~ a f ) ~ k = @ h O ( ~ f O k ) = @ h o @ = ( [email protected]~h ~0 @ f k)0gf

We conclude that

Hence I' is closed with respect to composition. It is immediate to prove (2). Let g = l; then g ~ Gf because I commutes with f, yielding @, = I and I E X(Gf). We continue by proving (I), (see also [7]). The set I? = X(Gf) is closed if given ( Q j } j , E I'such that it converges uniformly on compact sets to @, then @ E X(Gf) =I?. Given aje X(Gf), there exists g j Gf~ such that X(gj)=cpi, for all j E N. By definition of pseudo-iteration semigroup, we have

Take the limit for j 4 oo on each side of the equality and find

We have that { g j I j , is a normal family of holomorphic maps from A to A, and then, by the Montel's theorem, we can extract a convergent sub-sequence of maps: { g j k j k -t jj. Hence { g j k ) -+ j for k

-t

oo

It follows that lim (uf 0 gj) = lim (af0 gjk) = uf O 2

J-'m

Furthermore

k+m

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Therefore lim (uf o gjk)= lirn (ajko u f )

k+w

k-tw

Moreover, since

take the limit for j - r m, and obtain lim cp o aj = lim cpi o cp,

j-w

J+m

Hence

(i..

jEGf g E Pf and

~ ( g=) 1)

Now we prove (4). In our hypotheses on f, and if f'(1) < 1, then (see [5]) the automorphism cp of A associated to f is, up to conjugation, of the type:

and its matrix representation in SU(1,l) is:

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Now consider the set of all matrices associated to the automorphisms of X(Gf) and denote it by the same symbol X(Gf). By definition of X(GJ.), its elements commute with B,. In general, X(Gf) is a subset of the group C(B,) of all matrices which commute with B, and it turns out (see also [7]):

More precisely, X(Gf) is a subset of C+(B,), the semigroup of all the matrices of C(B,) which represent automorphisms of A commuting with B, and with Wow point equal to 1. After simple computations:

Now, C+(B,) c SU(1,l) is a real manifold of dimension 1, and SU(1,l) is a real connected manifold of dimension 3. Hence the complement of X (Gf) has, at least, dimension two and consequently it is connected. Instead, if f'(1) = 1, the automorphism of C associated to f is:

In this case recall that Aut(C) = { z c t a z + p , a € @ * , ~ € ~ } ~ u t ( @ ) r ~ u (C)) t ( ~c' SL(2, C) Hence

and, up to conjugation, the matrix which represent p is in this case:

Again, all the matrices associated to the automorphisms of @ which are in X(Gf), commute with A,. In general, X(Gf) is a subset of the

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group C(A,) of all matrices which commute with A,, and it turns out that:

Observe that C(A,) c Aut(@) is a real manifold of dimension 2, but Aut(@)is a real connected manifold of dimension 4; the complement of X(Gf) has at least dimension 2 and so it is connected. It is worthwhile noticing that, while proving Theorem 8, we have also proved that A : Gf+ A is a homomorphism of sernigroups. COROLLARY 2 Let f E Hol(A, A) be such that T(f)= 1. Suppose that f(1) < 1. Then, with the same notations as in Theorem 8, X(Gf) = F is a subset of the group of the Moebius transformations of A and

(1) J? is topologically closed and conjugated to a semigroup contained in C+(A,) (see W1); (2) the identity map ZE F; (3) I? is closed with respect to the composition; (4) the complement of r, r*is connected, If A denotes the group Aut(A) or the group Aut(@), (according to the hypotheses on f), then we can prove the following result on A: PROPOS~ION 5 Let f E Hol(A, A) be such that T(f ) = 1. Suppose there exists zo E A such that f "(zo) -t 1 non-tangentially. Let A be dejined as in (9) and let X :Gf -+ A be deJined as in (10). Then the map X : Gf + X (Gf) c A is injective. Proof Let a E X(Gf) and suppose that there exist h, g E Gf such that X(h) = X(g) = a. Let S, be an arbitrary open angular sector of opening a: < .rr and vertex 1. Then, by Lemma 4, the functions g, h send Stolz angles of vertex r in Stolz angles of vertex T (at least locally near 7). Hence, there exist p < n, 7 < n such that:

g(S,) h(S,)

c Sp locally near c S., locally near

T T

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Since by hypothesis, there exists zo E A such that f(zo) + 7,then Vf is angular at 1 and hence there exist horocycles Op and 0, such that:

Let r(O6) = min{r(Op), r(0,)); then:

Now, by the Wolff lemma:

and hence the open set U = S,

n O6c A is such that:

The map 9 is invertible on Vf and so is on g(U) and h(U). Since

we have

In particular

Therefore of is invertible on @(uf(U)) c uf (Vf) and by setting, as usual

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we obtain:

Then g = h on the open set U and therefore, by the analytic rn continuation principle, g = h on A and X is injective.

References [I] Abate, M., Iteration of Theory of Holomorphic Maps on Ta ut Manifolds, Mediterranean Press, 1989. [2] Baker, I. N. (1958). Zusamensetzungenganzer Funktionen, Math. Z., 69,121 - 163. [3] Baker, I. N. and Pommerenke, Ch. (1979). On the iteration of analytic functions in a half-plane, 11, J. Lond. Math. Soc., 20, 255-258. [4] Behan, D. F. (1973). Commuting analytic functions without fixed points, Proc. Amer. Math. Soc., 37, 114- 120. [5] Cowen, C. C. (1981). Iteration and the solution of functional equations for functions analytic in the unit disk, Trans. Amer. Math. Soc., 265, 69-95. [6] Cowen, C. C. (1984). Commuting analytic functions, Trans. Amer. Math. Soc., 283, 685-695. [I Frosini, C., Sui semigruppi di pseudo-iterazione nel disco unitb del piano complesso, Laurea Dissertation, University of Firenze, 1998. [8] Gentili, G. and Macci, F. (1994). Pseudo-iteration semigroups and commuting holomorphic maps, Rendiconti Lincei, V, IX,33-42. [9] Hadamard, J. (1944). Two works on iteration and related questions, Bull. Amer. Math. Soc., 50, 67-75. 1101 Hilgert, J. and Neeb, K. H., Lie Semigroups and Their Applications, Lecture Notes, Springer-Verlag, Berlin 1993. 1111 Koenigs, G. (1883). Recherches sur les substitutions uniformes, Bull. Sci. Math., 7, 340-357. [12] Pommerenke, Ch., Univalent Functions, Vandenhoeck and Ruprecht, Gottingen, 1975. [13] Pommerenke, Ch. (1979). On the iteration of analytic functions in a half-plane, I, J. Lond. Math. Soc., 19, 439-447. [14] Pommerenke, Ch., Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin-Heidelberg, 1992. [15] Pranger, W. A. (1970). Iteration of Functions Analytic on a Disk, Aeq. Math., 4, 201 - 204. [ l q Schroeder, E. (1870). h r unendlich viele Algorithmen zur Adosung der Gleichungen, Math. Ann., 2, 317-363. [17] Valiron, G., Functions Analytiques, Presses Univesitaires de France, Paris, 1954. [18] Vesentini, E., Capitoli Scelti della teoria delle funzioni olomorfe, Unione Matematica Italians, Oderisi, Gubbio, 1984. [19] Vlacci, F. (1996). On Commuting Holomorphic Maps in the Unit Disc of C, Complex Variables, 30, 301 - 313. [20] Wolff, J. (1926). Sur une gknkralisation d' un thkorhe de Schwarz, C. R. Acad. Sci. Paris, 18,918-920. [21] Worn, J. (1926). Sur une gknkralisation d' un thkorhe de Schwarz, C. R. Acad. Sci. Paris, 183, 500-502.