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(1)

Annales Academire Scientiarum Fennictr Series A. I. Mathematica

Volumeo 6, 1981 , 197-2A3

ON THE ITERATION OF POLYNOMIALS

OF DEGREE 4 WITH REAL COEFFICIENTS

P. BHATTACHARYYA and Y. E. ARUMARAJ

1. Introduction

The theory of iteration of a rational or entire fanction f(z) of a complex variable z deals with the sequence of natural iteratesf,(z) defined by

fo(z): z, f,*r(z):f(f"(z)), n:0,1,2,...

.

In

the theory developed by Fatou [5,6] and Julia [8], the fundamental object of study is the set

F(f)

of those points of the complex plane in no neighbourhood of which

{f"(r)\

forms a normal family in the sense of Montel. Unless

f(z)

is a

rational function of order 0 or

l,

the set

,F(/)

has the following properties, proved for rational functions in [5] and for entire functions in [6].

I. f("f)

is a nonempty perfect set.

II. F(f"):P(/)

for anY integer a>'1.

lll. F(f)

is completely invariant under the mapping

z*f(z),

i.e.

if

a belongs

to F(f),

then so do

/(a)

and every solution B of

f($:q.

A

fixpont a of order n of

f(z) is a

solution

of f,(z):z; it

has exact order

n

if fi(u):q for k-n,

but not for any

k=n,

and in this case

fi@)

is called the

multiplier

of

a, provided

a#-.

The fixpoint is called attractive, indifferent or repulsive according as the modulus of its multiplier is less than, equal to or greater than

l,

respectively. With this nomenclature we have:

An indifferent fixpoint a is called rationally indffirent

if f'(u)-ez"',,

where p

is a rational number.

With this nomenclature we have:

fV

a) Attractive fixpoints belong to the complement of

F(f).

b)

Rationally indifferent fixpoints belong to

F(f\.

c)

The

set F(/) is

identical

with the

closure

of the

set

of

repulsive

fixpoints.

In

their original work Fatou [5, 6] and Julia [8] have derived

a

number of

general properties ofthe set

F(f).

The structure of

F(f)

depends on the

functionl

If

f

is a polynomial,

F(/)

depends on the coeffi.cients of

f

in a very complicated man-

ner.

Myrberg

[9-13],

Brolin [4], and Bhattacharyya and Arumaraj [3] have con- doi:10.5186/aasfm.1981.0605

(2)

198 P. BuarrACHARyyA and Y. E. Anuuanar

sidered the cases where

/

is a polynomial of degree

2

and 3 with real coeffi.cients.

In

this paper we investigate the structure

of r(/)

when

/

is a polynomial of

degroe 4 with real coeff.cients.

Baker

El

has shown that

for

suitably chosen

k,

F(kze") is the whole prane.

Bhattacharyya

[2]

has shown

that

F(e"

-l)

has inaccessible boundary points.

Other examples are studied by Van Haselen [7] and Topffer [14].

Two polynomials

f(z)

and g(z) are said

to

be similar

if

there exist constants

a and b such that

f(az+b):qg(z)+b.

clearly, if a is a fixpoint of order

r

of multi- plier k of g(z), then

aa*b

is a fixpoint of

f(z)

of the same order n and same multi- plier fr.

The most general polynomial

of

degree

4

with real coefficients can easily be shown to be similar to

f(r): z4*pzz*rz*s,

p, r, s reals.

By IV above it is enough if we consider this simpler polynomial to study the structure of

F(f)

where/is a polynomial of degree 4 with real coefficients.

We consider the following separate cases:

Case

A. -f(r): za*pz, p

real, Case

B. f(z\: ,n+r, r

real, Case

C. f(z): zn+p22, p

real,

Case

D. f@ : zaa-p2z1-l$, p

real, Case

E. f(r):

z4+pz2+rz+§, p, Q,

r,,t

reals.

In Section 2 we give a detailed analysis of Case

A.

In Section 3, we shall state

our results on the other cases omitting the proofs, as the proofs are similar.

We need the following definitions.

Definition l. If

the equation

f"(z)-c:0

has

a

multiple root, then

c

is called a critical point of the inverse function

f-"(z).

Definition

2. The immediate attractiue set A*(a)

of

a first order attractive fixpoint a is the maximal domain of normality

of {f"(z))

which contains

a.

The attractive set

l(a) ofa

is defined by

d).

A modern survey of the main results of iteration theory can be found in Brolin [4].

We shall refer the reader to this paper for the results from iteration theory needed

for our proofs.

A(u):

{rlJ,*T .f,(r)

-

(3)

On the iteration of polynomials of degree 4 with real coefficients t99

2. On the iteration of

f (z):

za a P7

Let

f(z):za*pz,

wherc p is real. Thenf(z) has four finite first order flxpoints, 4r, ez, qrand qn, and

f-r(r)

has three finite critical points, cr, cr, and cr. These are

er

: 0,

Qz

:

(1

-

p)t

l',

Qt

:

Qzw, QE

:

42w2,

,,: -1(+),', c2:

c1w5

cs:

c1w2e

where w and wz are the complex cube roots of unity.

Lemma

l.I. If lpl=\,

then

cr,cr,cr(A*(0\.

Proof.4r:0

is an attractive fixpoint

for lpl-l

and a rationally indifferent one

for lpl:l;

at least one critical point belongs to ,4+(0) [4, Theorem 3.1,

p'

109].

Suppose

c$A"(Q).

Since

c2:c1w and cs:crwz and

zCA*(Q) implies that wz€A*(0), the result follows.

Lemma

2.

The polynomial equation

g(t):t1(t3-4)s+27(4t-3):0

has only one negatiue real root.

Proof. Since

g(0)=0

and

g(--):-,

there is at least one negative real root.

Clearly if there are more than one negative roots, there must be at least three. This would imply

that g"(t):O will

have at least one negative real

root. But g"(t):

48t5(t3-l)(t}-4)(5ts-14)>0 for t=0.

Hence the lemma follows.

Lemma 3.

If

-4013=p=413.4tt2, ilten

cr,c2,cr§A(*).

Otherwise

Cr, Cr, Cg€A(*\,

(1) and (2)

We first prove

Lemma 4.

ff -4013=p=0,

then

lfk)l =

Qz

Proof. First we

if f(cr)=qr,

i.e.

if

lfr@)l

= Qz if

lf@)l

=

tz.

consider

(l).

Since

p=0,

we have

f(cr)=0,

and so (1) holds

kl+PtJa = t-*p,

i.e.

if g(t)s0,

where

t-3p14

and

s(t) -

tz (tt

-

4)' + 27 (4t

-

3).

By Lemma

2, g(t)=O if -4e\<p=0.

Thus (1) is proved.

Next we prove (2). We have

(3)

fr(x)-Qz: {f(x)-q}h[f(x)],

where h(y)

- y3*qry'+q|l+1.

(4)

200 P. BgarrAeHARyyA and Y. E. ARuu,rnar

We claim that

hlf(x))=O

for every

x.

For, since the discriminant of h'(y) is nega- tive,

h(y):Q

has only one real root and h(y) is an increasing function of

y.

Denote the real root

of h(y):O

by

a.

Hence the real zeros of

hlf(x)J

are the real roots

of the equation

f(x):a, i.e. xaapx-a:0.

The minimum value

of xa+px-a

is cr-u.

We assert

that cr-a>0. If

not,

let

cr=6s. Then

å(cr)-/z(a)

since

h(y\is

increasing,

i.e. å(c)=0.

But

h(c,)

: f(c)- ct-

4zq'

-

o,

since both

f(cr)-q,

and

cr-q,

are negative when

-4013=p=0,

which is a con- tradiction. Hence

cr-u,>O,

showing that no real root

of xa+px-a:0

exists,

i.e.

h[f(x\]

has no real zeros, and since

hlf0»:l, it

follows

that å[/(x)]>0

for all x.

Since

fi(x)-qr:(f(x)-ar)n11x11,

we have

-fr(x)=q, if lf@)l=qr.

We show next lhat f2(x)>

-q'r.

Now,

fn@)* q,

-

[1'@)]u+

Plf@))*

q,

>

0

lf

27pa-256(l

-p)=0,

which is obvious when

-4013=p<0,

i.e.

fr(x)> -q,

and

Lemma 4 is proved.

Remark. The approximate value

of

4013 is 1.401.

Proof of Lemma 3

We prove the lemma in four steps.

Step

I.

p=.-4013

For x=q,

we have

f(x)=x

and hence

x€A(*). If

p=4013, then

g(l)>0,

i.e.

f(cr)=-qr. It

follows

that cr€,4(-).

Since

lf"@r)l:lf,kr)l:lf,(cr)1, it

fol- lows

that

cr, cr, cs€A(-).

Step

II.

-4013<p<O

When

p:

-4013,

g(/):0,

i.e.

f(cr):q,

and hence

fr(cr):q2 for

eyery n.

If -4013<p<0,

by Lemma

4 lf(cr)l<q, and lf@)l=5,

implies l.fr(x)l=q,,

i.e. lf,(cr)l<q2

for every

n.

Thus cr, c2,

cs{A(*).

Step

III.

0<p<(413) - 4rt|

We claim

that lf"(cr)l=lcrl

for eyety

n.

We have

f(cr):(crl3)(4t-ta),

where

p:4t13.

Since

0=(4t-t,)13<l when o=t=4rt3, it

follows

that lf(cr)l=lcrl.

Also

f(cr):cy14,

where

0=p.=1. Now, /|(cr):(c113)(4prt-pltn)

and hence

lfrk)l=lcrl.

Similarly,

lf"@)l=lcrl for

eyery n,

i.e.

cr, c2,

cr{A(*).

(5)

On the iteration of polynomials of degree 4 with real coefficients 20t

Step

IV.

p>(413).4LIB

ff

p=@13).4us, then

f(x)=x for x>0

and hence

x<A(-).

Now

/(c)>0

if

p=(413).41t8, and

it

follows

that cr,c2,cs(A(*).

The lemma is completely

proved.

Theorem l.

Let

f(z):zaapz

with

p

real.

i) If

-4013=-p<(413)-4u2, then

F is

connected. Otherwise

F

is totally dis' connected and

mrF:O,

where m, denotes the planar measure of F.

ii)

,F rs a Jordan curue

if

and only

if lpl=I.

iii)

F c

{zl

lzl< (t + lpl)'/'}.

Proof.

i\ If

-4013=p<(413)'41t3, then

by

Lemma

3 cvc2,cr§A(*),

and

if

p ( l-4013, (4lZ7.4rt'1,

then

cy, c2,

csQA(*).

The result

then

follows from Theorems

ll.2

and 11.4 of Brolin 14, p. 1251.

ii)

By Lemma

l, Crcl*(0) for lpl=l,

and

it

follows from Theorem 11.3

that F is a Jordan curve.

Now

let lpl=1. lf p=1,

the tays

z:r, z:re'iol'

and

z:re-zinl\,

Q=.r=:*, all belong

to ,4*(-)

for in

all

cases

f(z):ru+pr=lzl

andhence

lf,,(z)l*-.

Simi'

larly, when

p=-1,

the rays

z:-r, z:reinls

and

z:re-i'l3,0<r<-,

belong

to ,4*(-).

In this case

l/(z)l:ra+lplr=lzl

and

lf"@)l**.

It

is clear that when

lpl=\,0€F,

and so by the symmetry of iteration F can- not be a Jordan curve.

iii) If lzl=(l+lpl)'/',

then

lf(z)l>lzl

and hence

zqA(-),

i'e.

This completes the proof of the theorem.

3. Other cases

In this section we state our results on the remaining cases mentioned

in

Sec-

tion

1.

We omit the proofs.

Theorem 2. Let

f(z):zaar, r

real.

i) t7

-2rtt=r=31@.4us), then

F(f)

is connected. Otherwise

F(f)

is totally

disconnected and

mrF(f):O,

where m, denotes the planar measure of

F(f)' ii) F(J)

is a Jordan curue

if

and onty

if

-51(4'41t\=r=31(4'4us).

Theorem 3.

If f(z):241r, r

real, then the iterates

f"(z)

haue only constant

limit functions in their domain of normality.

Theorem 4. Let

f(z):za1pzz, p

real, Then:

i) F(f)

is connected

if

and only

if -2(lt2+l)'t'=p=2(fi-l)'/'.

(6)

202 P. BH.qrrACHARyyA and Y. E. Anul{,q.nar

ii) F(f)

is a Jordan curue

if

and only

if

-314r/t=.p=2(y5-t7rtt.

iii) If -sll2rtz-p=-3l4ttt

and

pl-2,

then

F(f)

contains an infinite num- ber of lordan curues.

Theorem 5.

If f(z):znlpr', p

real, then the iterates

f,(z\

haue only constant

Iimit functions in their domains of normality.

Theorem 6. Let

f(z):zn-pz,+/p, p>0.

Then:

i) F(f)

is connected if and only

if 0=p<4. If

p=-4,

F(f)

is real and

F(f)c l-lp,/p-1. tf p-4, F(f)

is the interoal

l-2,21.

ii) F(f)

is a Jordan curue

if

and only

if O<p<ll4tt}.

Theorem 7. Let

.fQ):rn-prr+/p,p=-p. If p>4,

then

F(f)

containsinfinite number of single point components.

Theorem 8.

If f(z):zt-pszly'p, p=0,

then the iteratesf*(z) haue only con- stant limit functions in their domains of normality.

Theorem 9.

Let f(z):z'+pzzarzfs, p, e, r, s

reals.

If

rzlS+pBl27=O and

(r-l)2<4ps,

then

F(f)

is totally disconnected and

mrF:O,

where mrdenotes the planar measure. Further,f"(z) haue only constant limitfunctions in their domains

of normality.

The authors wish to thank

Dr. L

N. Baker of Imperial College, London, for his comments in the preparation of this paper.

References

[1] Berun,

l.

N.: Limit functions and sets of non-normality in iteration theory. - Ann. Acad.

Sci. Fenn. Ser. A I Math. 467, 1970,

l-ll.

[2] Bnerrn'cnenwl, P.: On the set of normality and iteration of e2-1. - Publ. Math. Debrecen 19,1972,29-34.

[3] Bna'rucnanw.l, P., and Y. E. Ax.uunna.r: On the stiucture of the Fatou set for the poly- nomial zB +pz, p< -3. - Math. Rep. Toyama Univ. 1980, l:3, 123_141.

[4] BxouN, H.: Invariant sets under iteration of rational functions - Ark. Mat. 6, 1965-67,

to4-tM.

[5] Ferou,M.P.:Surles6quationsfonctionnelles.-Bull.Soc.Math.France47,lglg,l61-J71and 48,1»0,33-94.

[6] Farou, M. P. : Sur l'it6ration des fonctions transcendant esentiöres. - Acta Math. 47 , 1926, 337*

370.

lZ V.c,N Hasrr"nN, A.: Sur l'itörarion de log(l*z). - Enseign. Math. 30, 1931, 267---271.

[8] Julu, G.: M6moire sur I'it6ration des fonctions rationnelles. - J. Math. pures Appl. (g) l,

1918,47---245.

[9] MvnsERc, P. J. Iteration der reellen Polynome zweiten Grades.

-

Ann, Acad. Sci. Fenn.

Scr. A

I

Math.256, 1958, t-10.

U0] Mvnsnnc, P, J.: Iteration der reellen Polynome zweiten Gradesll. - Ibid. 26g, 1959,

l-13.

(7)

On the iteration of polynomials of degree 4 with real coefficients 203

[lU Mynsrno,P.J.: Sur l'it6ration des polynomes rdels quadratiques. - J. Math. Pures Appl.

(9) 41,1962,339-3s1.

[12] Mynsrna, P. J.: Iteration der reellen Polynome zweiten Grades III. - Ann. Acad. Sci. Fenn.

Ser. A I Math. 336/3, 1963, 1-18.

[t3] Mvnnrnc.P.J.: IterationderBinomebeliebigenGrades.-Ibid.348, 1964,

l-14.

[4]

Toprnrn, H.: Uber die Iteration der ganzen transzendenten Funktionen insbesondere von sin und cos. - Math. Ann. 117, 1940, 65-84.

Indian Institute of Technology Department of Mathematics Madras 600036

India

Received 8 September 1980

Revision received 23 February l98l

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