Periodic Points of small Periods of Continuous Mappings of Trees

Annals of Discrete Mathematics 27 (1985) 443-446 OElsevier Science Publishers B.V. (North-Holland)

443

PERIODIC POINTS OF SMALL PERIODS OF CONTINUOUS MAPPINGS OF TREES Wilf r i e d I m r ich

1

M o n t a n u n i v e r s i t a e t Leoben AUSTRIA

L e t f b e a c o n t i n u o u s s e l f - m a p o f a tree T w i t h e e n d p o i n t s . We show t h a t f h a s a p o i n t o f p e r i o d m , 1 < m 5 e , i f f has a periodic point of period > e .

91.

INTRODUCTION

I n [ l ] S a r k o v s k i i ' s t h e o r e m [21 on p e r i o d i c p o i n t s o f c o n t i n u o u s s e l f mappings o f t h e i n t e r v a l w a s g e n e r a l i z e d t o trees and c o n d i t i o n s were g i v e n under which t h e e x i s t e n c e o f a p e r i o d i c p o i n t o f p e r i o d s e l f - m a p o f a tree

T

of a continuous

i m p l i e s t h e e x i s t e n c e of p e r i o d i c p o i n t s o f o t h e r ,

Here w e show t h a t a c o n t i n u o u s s e l f - m a p o f a tree

l a r g e r periods.

e n d p o i n t s a l w a y s h a s a p e r i o d i c p o i n t of p e r i o d

,1<

m

with

T

, if

m C e

e

i t has

> 1.

p e r i o d i c p o i n t s of p e r i o d

52.

n

DEFINITIONS We c o n s i d e r trees a s m e t r i c s p a c e s , e v e r y e d g e b e i n g i s o m e t r i c t o t h e u n i t

interval.

If

x,y

a r e p o i n t s ( n o t n e c e s s a r i l y v e r t i c e s ) o f a tree

e x i s t s a unique s h o r t e s t path

[ x , y l from

x

to

y

in

.

T

i s o m e t r i c t o a n i n t e r v a l of t h e r e a l l i n e w e c a l l i t a n i n t e r u u l o f [x,yI\Ix,yl

we w r i t e

i n an i n t e r v a l

A s u s u a l we s a y 1 5 i < n. set

{fl(x)

2

v'

'Supported

in

is

For

x 6 T

has

f-period

Orb(x)

n

f 7

if

of a t r e e

[x,yl

fn(x)

f o r t h e o r b i t of

x

=

x

,

has a fixed

T

and i f

y 6 [x,f(y)]. but fi(x)

i.e.,

4x

for

for the

01.

For e v e r y s u b t r e e vertex

[x,yl i f [ f ( x ) , f ( y ) l

We f u r t h e r w r i t e

Ii

.

b u t f o r o n e r e f e r e n c e t o 11, Lemma 1 1 , w h e r e w e

u s e t h e s p e c i a l c a s e t h a t a c o n tin u o u s self-map z

T

(x,y).

The p a p e r i s s e l f - c o n t a i n e d point

there

T

[x,yl

As

S

of

T

t h e r e is a unique

which i s c o n t a i n e d i n e v e r y p a t h from

v

to

by NSERC

S

of

T

and every p o i n t

v

S

.

We c a l l

v1

W.Imrich

444 t h e projection

ps(v)

o r a n x-branch of

53.

of

to

S

.

If

f ,

f

Let

a

U

2 endpoints l e t

t

T

be t h e x-branch of

y

f(x)

b e t h e e n d p o i n t of

Thus t h e r e e x i s t p o i n t s

on

t

d

If

z

T

f(x).

d i f f e r e n t from 5 3)

U

9

has only

, otherwise

x

W

of

x such t h a t

let

y

which i s c l o s e s t t o U 0 f(U) =

0.

[x,yl w i t h for all

[x,yl w i t h d i s t a n c e

w e i n f e r by c o n t i n u i t y t h a t

y

W

If

of

s € [x,tl.

b e t h e supremum of t h e d i s t a n c e s of s u c h p o i n t s

b e t h e p o i n t on

z

i s i n the t-branch of

f(t)

containing

W

s C [x,f(x)]

Let

. z

t h e r e i s a neighbourhood

x

4x

f(x)

and

€ [x,zl.

be t h e r a m i f i c a t i o n p o i n t ( v e r t e x o f d e g r e e Since

,

x

a l s o contains a f i x e d point

can be chosen such t h a t

W

Let

f(x)

which contains

for a l l

z

Proof.

let

and

with respect to

T

be a continuous self-map o f a t r e e T

Moreover, z

containing

.

T

T

RESULTS LEMMA 1.

x

i s a p o i n t of

x

U {XI i s c a l l e d a b r m c h of

U

T.

Then t h e x-braneh of of

v

T {XI then

component of

d

from

from

t

.

x

x

and

s a t i s f i e s t h e a s s e r t i o n s of t h e

z

Lemma. If

i t is still possible t h a t

z = y

implies

f(z)

=

z

However, i f

if

w

f(y)

f

W

, which

LEMMA 2. [f

(x), f ( y ) l

a c o n t i n u i t y argument shows t h a t

y

W

containing

f(y).

f

[x,yl.

be a continuous self-rap of (x,y)

Then

v € [x,f(v)l

and

If

a € [x,bl

f o l l o w s from [l, Lemma 1 1 . a € [b,yl. a-branch o f Suppose

a

, whereas

n

[y,f(v)l

By c o n t i n u i t y t h e r e are p o i n t s

f ( b ) = y.

z = y

y € [x,f(y)l.

But

I t h a s fewer e n d p o i n t s

(and

T

and suppose

contains e i t h e r a f i x e d p o i n t or a point

such that

Proof.

also

a l l o w s t o conclude t h e proof by i n d u c t i o n .

Let

3

z

h a s o n l y two e n d p o i n t s .

t h e n w e c o n s i d e r t h e y-branch of than

f ( z ) = z , and t h e n

I n p a r t i c u l a r , we note t h a t

s a t i s f i e s t h e a s s e r t i o n s of t h e Lemma.

a

.

and

b

in

[x,yl

with

b C [x,al

By Lemma 1 t h i s i m p l i e s t h a t t h e r e e x i s t s a f i x e d p o i n t x

b € [a,zl.

Then

f(b)

(and

b).

f(b) = y

Let

f(a) = x

the existence of a fixed point

b € [a,yl)

We can t h e r e f o r e assume t h a t

T containing

v

z

and i n the

z

b e such a f i x e d p o i n t .

is i n t h e

b-branch of

would have t o be i n t h e b-branch of

T

T

containing

containing z

by

Periodic Points of Small Periods in Trees Lemma 1.

1

b

Thus

[a,zl.

T h i s means t h a t

t a i n i n g a and t h e r e f o r e t h e p r o j e c t i o n z

iu

is not already

f(z')

(a,b),

i s i n t h e 2'-branch

THEOREM 1.

T

containing

z

2'

c [a,f(z')l

n

5 e

If

By Lemma 1,

[b,f(z')l.

T

be a continuous self-map of a f i n i t e t r e e

o f period

con-

T

[a,bl is i n (a,b).

and t h e r e f o r e

have a periodic point of period

f

y

periodic point

f

Let

endpoints and l e t

onto

z

must b e a r a m i f i c a t i o n p o i n t .

2'

of

i s i n t h e b-branch o f

z

of

z'

445

n > e

.

.

Then

T

and

Lth f

e

has a

I t s u f f i c e s t o prove

Let

THEOREM 2 .

be a continuous self-map o f a t r e e

f

periodic point of period

.

n

1 < m < n , i f the subtree o f

T

Then T

x

a

has a periodic point of period

m

Orb(x)

spanned b y

has fewer than n

,

end-

points. Proof. period

m

It o b v i o u s l y s u f f i c e s t o show t h e e x i s t e n c e of a p e r i o d i c p o i n t o f

,1<

of t h e s u b t r e e

m < n.

is a fixed p o i n t

w

fixed point or

w

if

v f S

L e t t h e n o t a t i o n be chosen s u c h t h a t

spanned by

S

v of

i n the

fs

of

ft

Orb(x).

i n the

i s n o t f i x e d by

Suppose

the

v'

v'-branch

of

T

containing

We wish t o show t h a t

If

fS-l(v')

{

B

x

v

f(v')

v

.

If either

t'

a r e f i x e d by

to

S

B

.

is also i n

then there e x i s t s a

w

and v

.

and a

x

we have found a p e r i o d i c p o i n t o f p e r i o d

f

b e t h e p r o j e c t i o n of

B

By Lemma 1 t h e r e

containing

containing

T

i s a cut-point

x

C [xs,xtl.

T

x -branch o f

x -branch o f

W e can t h e r e f o r e assume t h a t b o t h let

x

f

By Lemma 1,

.

v

.

< n

Moreover,

fS(v')

is i n

Suppose t h i s i s n o t t h e c a s e .

v" € [ v ' , v l

with

fs-l(v")

=

v'

and

then f ( v ' ) = fs(v") by Lema 1,

Thus

fs-2(v'),fs-3(v'),

fS-l ( v ' )

...,f ( v ' )

S i m i l a r l y we d e f i n e e l e m e n t s of Let

P

Orb(x)

w'

C B.

R e p e a t i n g t h i s argument w e see t h a t

must a l l b e i n if

w

f

S.

T

B

.

Clearly neither

and b o t h are b r a n c h p o i n t s of

be t h e set of a l l branchpoints ps(f(b))

Since

f B

b

of

F

T

in

v'

nor

S

with

w'

can b e

= b.

h a s o n l y f i n i t e l y many b r a n c h p o i n t s i n

Further, l e t

.

T

d e n o t e t h e set of f i x e d p o i n t s o f

t h e set

S

f

in

S

is f i n i t e .

P

.

F

is closed.

W. Imrich

446 Setting x -branch of

containing

S

0

x -branch o f

for the

Ss

x

and a p o i n t

Further, l e t

x

%

and

j

d

b e p o i n t s of

Orb(x)

and

J

x

C [c,%]

t h e r e is a

f(k-')([c'yx.l) I Since

f j [ x ,dl

[x

3

dl

jy

c'-component of z

2

i s g r e a t e r t h a n t h a t of

If

z

which i s mapped i n t o t h e can show t h a t because most

k

xJ

f

lXj,C'1

3

*

in

> d(c,xo).

with

[xoydl

fk-j(c')

.

.

= x

Hence

.

To t e r m i n a t e t h e proof i t s u f f i c e s t o

from

f

z'

. f .

S i n c e t h e d i s t a n c e of

i t is c l e a r t h a t

x T

containing

i n t o t h i s component.

Hence t h e p e r i o d o f

is s m a l l e r than

n

.

z

z

by

z

from

cannot b e i n

z

i s a b r a n c h p o i n t of

psz

2'-branch of

a l s o maps

d(z',xo)

, which

c

we note t h a t

S

3

we have

i s a f i x e d p o i n t of

x

is not i n

x

F) fl S t .

0 < j < k 5 n-1.

J

cannot b e a f i x e d p o i n t of

Suppose t h a t

u

s a t i s f y i n g t h e a s s e r t i o n s of Lemma 1 i n t h e

z

containing

t

(P

with

C [c,x.l

[xo,\l

3

[xjyc'l

3

has a fixed point

show t h a t

in

' [c,"kI. c'

fk([c',x.l) J

Thus, f k

xo

[ C , X ~ + ~ and ] a fortiori

3

fk-j( [c,x.l) J Since

for the

d C [xo,\l.

Without l o s s of g e n e r a l i t y w e can assume t h a t f([c,x,l)

St

of maximal d i s t a n c e from

c

of maximal d i s t a n c e from

c C [x , x , l 0 3

Clearly

and

(P U F) fl St

and

Thus t h e r e e x i s t s a p o i n t

(P U F) fl S s

xs

w e o b s e r v e t h a t we have j u s t shown t h a t b o t h

t

(P U P) fl S s a r e nonempty.

containing

S

in

T

fk

.

S

But t h i s i s n o t p o s s i b l e

i s l a r g e r t h a n 1 and a t

REFERENCES

[l] W.

[21

A.N.

I m r i c h and R. Kalinowski, trees, see t h i s volume. Y

P e r i o d i c p o i n t s o f c o n t i n u o u s mappings of

v

S a r k o v s k i i , Coexistence o f c y c l e s of a continuous map o f a l i n e i n t o i t s e l f ( R u s s i a n ) , Ukr. Mat. 1 6 (1964) 61-74.

i.

.

[c',x.] J A s above one