On asymptotic values of functions in a polydisk domain and Bagemihl Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 4(22), No. 2, 2015, pp. 23-31

DOI: 10.15393/j3.art.2015.2951

23

UDC 517.57

E. G. Ganenkova

ON ASYMPTOTIC VALUES OF FUNCTIONS IN A POLYDISK DOMAIN AND BAGEMIHL'S THEOREM

Abstract. Asymptotic sets of functions in a polydisk domain of arbitrary connectivity are studied. We construct an example of such function, having preassigned asymptotic set. This result generalizes well-known examples, obtained by M. Heins and W. Gross for entire functions. Moreover, it is found out that not all results on asymptotic sets of functions in C can be extended to functions in Cn. In particular, this fact is connected with the failure of Bagemihl's theorem on ambiguous points for functions in , n > 3.

Key words: asymptotic value, analytic set, ambiguous point 2010 Mathematical Subject Classification: 32A40, 26B99

Let Di,..., Dn be domains in C, D = Di x- ■ -xDn, z0 = ..., z^) G G dD be an accessible boundary point i.e., there exists an open arc r C D with endpoint z0. Let f be a function defined in D.

Definition 1. [1, Section 1.6, p.8], [2] We say that a G C is an asymptotic value of f at the point z0 if there exists an arc ja C D with endpoint z0 such that

lim f (z) = a.

Ya Bz^zo

The arc Ya is called an asymptotic curve corresponding to the asymptotic value a. The set of all asymptotic values (or, briefly, the asymptotic set) of f at the point z0 is denoted by As(f, z0).

Asymptotic sets were actively studied for entire functions and mero-morphic functions in c.

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It is well known that for a nonconstant entire function this set is analytic (in the sense of Suslin) [3]. This means that As(f, to) can be represented in the form

As(f\ to) = n Anin2 n Anin2n3 n ... }

(n 1 ,n2,... )

where nk, k = 1, 2,..., are integer numbers, Ani...nk are closed sets and the union extends over all sequences (n±,n2,...) (see [4, p. 105], [5, p. 136] for details). From the Iversen theorem [6] (see also [1, ch. 1], [7, section 5.1, p. 171]) it follows that for such functions this set contains the infinity. Many articles are devoted to constructing the examples of functions, having prescribed asymptotic sets. W. Gross [8] constructed an entire function whose set of asymptotic values at the infinity is equal to c. M. Heins [9] proved that every analytic set containing the infinity is an asymptotic set of some entire function.

In [10] and [11] functions analytic in planar domains of arbitrary connectivity were considered. For such functions theorems of W. Gross and M. Heins were generalized. Here the case when z0 belongs to an isolated boundary fragment was considered.

Definition 2. [12] A domain D C c has an isolated boundary fragment if one of the following conditions holds:

(I) There exist a continuum K C dD and an open set U such that K C U and (dD \ K) n U = 0.

(II) There exist a Jordan arc r C dD with distinct ends n and an open disc B such that £,n G dB, r \ {£, n} C B and (dD \ r) n B = 0.

(III) There exist a point a G dD and an open disc B(a) centered at a such that (B(a) \ {a}) n dD = 0, i.e., a is an isolated point of the set dD.

The continuum K from (I), the arc r from (II), and the point a from (III) are called isolated boundary fragments of D.

Theorem A. [11] Let D C C be a domain with isolated boundary fragment T. Let point (o belong to this fragment. If this fragment has type (I), then we assume, in addition, that Z0 is an accessible and it is an impression of some prime end of D. Let A be an analytic set, tog A. Then there exists an analytic function f such that As(f, Z0) = A.

Remark. The fact that Z0 is an impression of some prime end of D (D may not be simply connected) means that Z0 is an impression of some prime end of the simply connected domain G D D, dG = T.

In this note Theorem A is extended to functions analytic in a poly-disk domain D. We describe simple construction of such function with a preassigned asymptotic set.

Theorem 1. Let k be a fixed natural number, 1 < k < n. Suppose D = Di x ■ ■ ■ x Dn, where Di, 1 < i < n, i = k, are arbitrary domains in c, Dk C c is a domain with an isolated boundary fragment T. Suppose z0 = (z0,..., z^) G dD, moreover z0, i = k, is either points of the domains Di or accessible boundary points of Di, z° G T. If T is a fragment of type (I), then we assume in addition that z° is an accessible from Dk and it is an impression of some prime end of Dk. Let A be an analytic set, containing the infinity. Then there exists an analytic in D function f such that As(f, z0) = A.

Proof. Consider the domain Dk C c and the point z^ G T. By Theorem A there exists an analytic in Dk function F(z), possessing the property As(F, z°) = A. Let a G A and ra be an asymptotic curve, corresponding to the value a. This means that

lim F (zk) = a.

ra 3zk

We construct the analytic in D function

f (z) = f (zu . . . , zn) d=f F(zk).

Denote by ra, 1 < i < n, i = k, a curve in Di with endpoint z0. Then for any curve Ya with endpoint z0, Ya C ra x ■ ■ ■ x , we have

lim = lim F (zk) = a

Ya Bz ^zo ra Bzk

for all a G A. Therefore, As(f, z0) = A. □

Remark. If we put n = 1, we obtain Theorem A.

In the case n = 1 the following fact takes place (see [11]): if card A > 1, then the set of all points z0 such that a function f defined in a simply connected domain D, z0 G dD, possesses the property

As(f,z0) = A

is at most countable. This fact follows easily from Bagemihl's theorem on ambiguous points.

If a function f is defined in a domain D C rn, r C D, z0 G dD n r, then the cluster set C(f,z0, r) of f at the point z0 along r [1, ch. 1] is the set of all numbers w G c such that there exists a sequence zn G r, zn —> zo and f (zn) —> w.

n—^^o n—^^o

Definition 3. [13], [1, ch 4.7] A point z0 G D is an ambiguous point of f if there exist two arcs r and r2 in A with endpoint z0 such that

C(f, zo, ri) n C(f, zo, r2)= 0.

Bagemihl proved in [13] (see also [1, ch 4.7]) that an arbitrary function in the open unit disk can have at most countable set of ambiguous points.

Unlike the case n = 1, for n > 2 there exist functions f analytic in a simply connected polydisk domain D such that

As(f, z0) = A, (card A> 1)

at an uncountable set of points z0 G dD.

Suppose all domains Dk in Theorem 1 be simply connected. Let f0(z) be the function constructed in the proof of Theorem 1. Then for every point z0 = (z°,..., z<n) from the conditions of Theorem 1 we have

As(fo ,zo) = A.

The set of all such points zo is uncountable.

This difference of the case n > 2 from the case n = 1 is connected with the fact that Bagemihl's theorem is not true in rn for n > 3. There are functions in Euclidean ball bn = {x G rn : ||x|| < 1}, n > 3, with an uncountable set of ambiguous points. Examples are given in [14]-[16]. One more example can be obtained using Theorem 1. Take Dk = A and A such that card A > 2. Let f be the function constructed in Theorem 1, g be a homeomorphism of b2n onto the polydisk An = A x • • • x A. Then the set of ambiguous points of the composition f o p is uncountable.

In [17] P. J. Rippon introduced a new definition of ambiguous point of function in rn, n > 3. He replaced one arc by the boundary of a subdomain of bn. More precisely, P. J. Rippon says that a point z0 G dbn is an ambiguous point of a function f defined in bn if there exist

1) a subdomain S of bn, dS n dbn = {zo},

2) an arc r C S with endpoint zo

such that

C (f,zo, r) n C (f,zo, dS \{zo})= 0.

This definition allows to obtain the analog of Bagemihl's theorem in Rn, n > 3 [17]: for any function in bn the set of all ambiguous points (in the sense of Rippon) is at most countable.

One can pose the following problem: how to define the ambiguous point of function in bn, n > 3, using object of the same nature (like two arcs in Bagemihl's definition), saving the statement about countability of the set of such points. In view of Rippon's definition it is natural to consider points z0 G dbn for which there exist subdomains Si and S2 of bn, dSi n dbn = dS2 n dbn = {z0}, and

C(f, z0, dSi \ {z0}) n C(f, z0, dS2 \ {z0}) = 0.

The following example shows that this assumption does not save Ba-gemihl's theorem true even for continuous functions.

Example. Take the function

g(t) =

|1 - t| 1- Itl

, t G A = {z G c : |z| < 1}.

Construct our example, using the function g(t). For x = (xi,..., xn) G bn put

f (x) =

\

1 -

Xl

a/1 X1 ••• xn

+

_ rp2 _ . . . _ rp2

3 n

1

Ж2 I /-v»2

1 i ju 2

_ rf2 _ . . . _ rf>2

tXy Q tXy n

where

1

xi

+ i-

X2

V/l"x3-----xn V/l"x2-----

1

xi

V/l"xi-----x

+ i

X2

V/l"xi-----a

= g(t),

t =

x1

+ i-

X2

a/1 — x3 — ■ ■ ■ — V1 — x2 — ' ' ' —

A.

(1)

Consider the set l = dbn n {(xi, 0, x3,..., xn), xi > 0, x3,..., xn G r}. Let

z = (Zi,0,Z3,...,Zn) G I.

2

x

Fix s G (0; n/4). By n denote the plane

x3 = Ca

Cn.

Let Yo be the segment [(0, 0, (3,..., Zn), (Zi, 0, (a,..., Zn)] C n. By j4-e denote the segment [(Zi, 0, (3,..., Zn), (0, Zitg(4s), (3,..., Zn)]. Consider the open triangle Tn-s C n bounded by the segments j0, Yf -s and the ray {(0, X2, Z3,..., Zn), X2 > 0}.

Let Si be a subdomain of Bn such that

1.1) dSi n dBn = {Z},

1.2) for each x = (xi,..., xn) G Si the projection X = (xi,x2, Z3,..., Zn) of x to n belongs to Tn-e,

1.3) |xk| < IZk|, k = 3,... ,n for each x = (xi,...,xn) G Si. Suppose p G C(f, Z, dSi \{Z}). This means that there exists a sequence

w

N

= (w

w

N

G OSi, wN — Z as N —y to, such that f (wN) — p

as N — to. Using (1), for each point wN G rn let us construct the point tN G A, substituting xk by wJN in (1). Then we get tN — 1 as N — to. By 3(tN) denote the angle between the segments [tN; 1] and [0; 1] in A. By condition 1.3) we have

1 - (wN)2-----(wN)2 >A/1 - Z3-----zn.

Therefore, 0 < Re tN =

0 < Im tN =

w

N i

1-(

w

3N )2-----

wN

(wN )2

<

w

N

<

v/1-ZÏ-T7T-Z2 :

wN

V1 - (wN)2-----(wN)2 " v/1-Z3r-TTT3Z2 '

(2)

(3)

From condition 1.2) it follows that the point

NN

wN + iwN

N

NN

wN + iwN

ZT

V1 - Z3-----Zn

N

def i

belongs to the triangle An-s = z1 ■ Tf -s C A. Consequently, taking into account (21) and (31), we obtain that tN belongs to An-s too. Hence,

n

0 < ¡(tN ) < 4 - s

n

i

for all wN G dSl.

Denoting a = 11 — tN |, we have

|1 - tN | a

<

1 — |tN 1 1 — va2 + 1 — 2a cos 0(tN) 1 ^a2 + 1 — 2a cos (n — e)' Then

|1 — tN |

p=Nlimof (wN)=Nlimog(tN)=Nlimo r-^ <

a1 < lim--= -(-). (4)

1 — yja2 + 1 — 2acos (f — e) cos (f — £)

Now consider a domain S2 C bn, possessing the following properties:

2.1) dS2 n dbn = {Z},

2.2) for every x = (xl,... ,xn) G S2 the projection x = (xl,x2,Z3,... ,Zn) of x to n belongs to the set Tn +£, bounded by the circle dbn n n and the straight line, passing through the points (Zl, 0, Z3,..., Zn) and (0, Zltg(f +

+e),Z3,..., Zn); (0,0,Z3,..., Zn) GTf+.

2.3) |xk| > IZk|, k = 3,..., n, for each x = (xl,... ,xn) G S2.

If p' e C(f, Z, dS2\{Z}), then there exists a sequence iN = ,..., ) e e dS2, iN ^ Z as N ^m, such that f (un) ^ p'. For un G rn calculate tN G A by formula (1). By 2.3),

1 - ("3N)2-----("N)2 < V 1 - Z3-----zn.

Consequently,

Re tn = " > =

V1 - ("3N)2-----("N)2 " V1 - Z2-----zn

"N "N

Im tn = "2 > "2

V1 - ("N)2-----("N)2 " v/1-ZÏ-T7T-Z2 '

Since, by condition 2.2),

NN

U1 + ^ A def 1

—. G An+ = — T n +=■,

V1 — Z32-----Zn 4 + Zi 4

a

we have rn e An+e. This yields that p(rN) > f + e for all N e n. Hence, denoting b = 11 — rN |, we get

|1 — rN| b b

>

1 — |rN1 1 — Vb2 + 1 — 2b cos p(tN) 1 — ^b2 + 1 — 2b cos (f + e) This implies that

b1

p = Jim f (^N) > lim-/ =-(n , ).

N^ 1 — Jb2 + 1 — 2bcos (f + e) cos (f + e)

We have proved that for all p e C(f, Z, dSi \ {Z}) and p e C(f, Z, dS \ \{Z}) the following inequality holds

11

p < -jn-v < -(n , ) < p.

cos (f — ej cos (f + ej

This yields

c (f, Z, dSi \ {Z}) n c (f, Z, dS2 \ {Z}) = 0 (5)

for each Z e Z. The set of all points, possessing property (5), contains Z. Therefore this set is uncountable.

Remark. The above example shows that the requirement for r to be a subset of D in Rippon's definition is essential. If we take D = S1 from the example and r be a curve, contained in the domain S2, then the function f has the property

C(f,Z, r) n C(f,Z,dD \{Z}) = 0

at uncountable set of points Z.

Acknowledgment. This work was partially supported by a grant from the Simons Foundation and RFBR (project N 14-01-00510a).

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Received December 10, 2015.

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