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PROCEEDINGS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 104 Number October 198S

CONCERNING PERIODIC POINTSIN MAPPINGS OF CONTINUA

INGRAM

Communicated by Dennis Burke

ABSTRACT In this paper we present some conditions which are sufficient for

mapping to have periodic points

THEOREM 1ff is mapping of the space Xinto and there exist subcontinua

and of such that every subcontinuum of ha the fixed point property

and every subcontinuum of 1H are in class contains

contains HUK and if is positive integer such that fIHKintersects then then contains periodic points of of every period

greater than

Also included is fixed point lemma

LEMMA Suppose is mapping of the space into and is subcontinuum

of such that contains If every subcontinuunn of has the fixed point

property and every subcontinuum of is in class then there is point

xofKsuch that 1xFurther we show that If is mapping of into and has

periodic point which is not power of then lim contains an

indecomposable continuum Moreover for each positive integer there is

mapping of into with periodic point of period and having

hereditarily decomposable inverse limit

Introduction In his book An Introduction to Chaotic Dynamical Systems

Theorem 10.2 62 Robert Devaney includes proof of Sarkovskii Theorem Consider the following order on the natural numbers

is continuous If and has periodic point of prime period then has

periodic point of period In working through proof of this theorem for

the author discovered the main result of this paperTheorem For an alternate

proof of Sarkovskiis Theorem for see also For further look at this

theorem for ordered spaces see

By continuum we mean compact connected metric space and by mapping

we mean continuous function By periodic point of period for mapping of

continuum into is meant point such that fx The statement that

has prime period means that is the least integer such that

continuum is said to have the fixed point property provided if is mapping of

into there is point such that fx mapping of continuum onto

continuum is said to be weakly confluent provided for each subcontinuum of

Received by the editors May 22 1987 and in revised form September 14 1987

1980 Mathematics Subject lassiJication 1985 Revision Primary 54F20 54H20 Secondary

54F62 54H25 54F55

Key words and phrases Periodic point fixed point property class indecomposable contin

uum inverse limit

1988 American Mathematical Society

0002-9939/88 $100 $25 per page

643

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644 INGRAM

some component of is thrown by onto continuum is said to be in

Class provided every mapping of continuum onto it is weakly confluent Thecontinuum is triod provided there is subcontinuum of such that TK has

at least three components continuum is atriodic provided it does not contain

triod continuum is unicoherent provided if is the union of two subcontinua

and then the common part of and is connected continuum is

hereditarily unicoherent provided each of its subcontinua is unicoherent If is

mapping of space into the inverse limit of the inverse limit sequence

X5 f2 where for each is and f2 is will be denoted limX For

the inverse sequence X2 the inverse limit is the subset of the product of the

sequence of spaces X1 X2.. to which the point x1 x2.. belongs if and only

if fx11There has been considerable interest in periodic homeomorphisms of continua

where homeomorphism is called periodic provided there is an integer such that

h2 is the identity Wayne Lewis has shown that for each there is chainable

continuum with periodic homeomorphism of period theorem of Michel Smith

and Sam Young should be compared with Theorem of this paper Smith and

Young show that if chainable continuum has periodic homeomorphism of

period greater than then contains an indecomposable continuum In this

paper we consider the question of the existence of periodic points in mappings of

continua

fixed point theorem The problem of finding periodic point of period

for mapping is of course the same as the problem of finding fixed point

for ffl Not surprisingly we need fixed point theorem as lemma to the main

theorem of this paper The following theorem which the author finds interesting

in its own right should be compared with an example of Sam Nadler of

mapping with no fixed point of disk to containing disk corollary to Theoremis the well-known corresponding result for mappings of intervals

THEOREM Suppose is space is mapping of into and is

subcontinuum of such that contains If1 every subcontinuum of has

the fixed point property and every subcontinuum of is in Class then

there is point of such that fx

PROOF Since is in Class and is subset of there is subcon

tinuum K1 of such that Then IKi K1 is weakly confluent

since every subcontinuum of is in Class thus there is subcontinuum K2of K1 such that K1 Since K1 is in Class K2K2 K1 is weakly

confluent therefore there is subcontinuum K3 of K2 such that K2 Continuing this process there exists monotonic decreasing sequence K1 K2 K3..of subcontinua of such that for 123 Let denote

the common part of all the terms of this sequence and note that since

fl0 fld0 Kd Since fIH throws onto

and has the fixed point property there exists point of and therefore of

such that fxREMARK Note that and of the hypothesis of Theorem are met if

is chainable Theorem 236 and respectively while is met if is

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CONCERNING PERIODIC POINTS IN MAPPINGS OF CONTINUA 645

atriodic and acyclic and is met by planar tree-like continua such that each

two points of subcontinuum lie in weakly chainable subcontinuum of

Periodic points In this section we prove the main result of the paper

THEOREM If is mapping of the space into and there exist subcon

tinua and of such that every subcontinuum of has the fixed point

property and every subcontinuum of are in class contains contains and if is positive integer such that

fIHYK intersects then then contains periodic points of of

every period greater than

PROOF Suppose There is sequence H1H2. H-_1 of subcontinua

of such that note that fH is weakly confluent and

for 12. in case There is subcontinuum of so that

H_1 Thus and so contains so by Theoremthere is point of such that fmx We must show that if

then fJx is not If and f3x then and is in H2Since fmx and f2x f2x Since is in fIHY2K is in

H4 and in contrary to of the hypothesis Therefore is periodic

of prime period

REMARK If is mapping of the continuum into itself and has periodic

point of period then the mapping of limM induced by has periodic points

of period e.g xfc_lx fxx... Thus although Theorem does not

directly apply to homeomorphisms it may be used to conclude the existence of

homeomorphisms with periodic points

COROLLARY If is chainable continuum is mapping of into and

there are subcontinua and of such that contains HUK and

if IHK intersects then then has periodic points of every period

A/2

B/2

813

FIGURE


646 INGRAM

EXAMPLE Let be the mapping of the simple triod to itself given in

The mapping is represented in Figure above Letting A/2 and

B/21 it follows from Theorem that has periodic points of every period

EXAMPLE Let be the mapping of the simple triod to itself given in

The mapping is represented in Figure below Letting 3B/8 and

C/8 it follows from Theorem that has periodic points of every

period

___ /H___ //

C/32 C/8 C/2

B/2 C/2

FIGURE

EXAMPLE Let be the mapping of the unit circle S1 to itself given by fz z2

Letting ezOjO 3ir/4 and eIir 3ir/2 it follows from

Theorem that has periodic points of every period Similarly if is mapping

of onto itself which is homotopic to z7 for some ri then has periodic

points of every period

FIGURE

COROLLARY 1ff is mapping of an interval to itself with periodic point of

period then has periodic points of every period



PROOF To see this it is matter of noting that the hypothesis of Theorem is

met We indicate the proof for one of two cases and leave the second similar case

to the reader Suppose and are points of the interval with and

fa fb and fc other case is fa fb and fcJf f1 is nondegenerate then there exist mutually exclusive intervals and

lying in and respectively so that is and is and

Theorem applies

Suppose f1c Choose lying in and lying in so that

and For each denote by H5 the set IHK Note

that is not in for 23.. so is not in for 234.. and thus

is not in for Further is not in H1 since is not in Thus if

intersects then Consequently the hypothesis of Theorem is met

REMARK Condition of Theorem seems bit artificial more natural

condition the author experimented with in its place is requirement that and

be mutually exclusive In fact in each of the examples the and given

are mutually exclusive However replacing condition with this proved to be

undesirable in that the Sarkovskii Theorem for is not corollary to Theorem

if the alternate condition is used That condition may not be replaced by

the assumption that and are mutually exclusive can be seen by the following

For the function which is piecewise linear and contains the points

and 10 there do not exist mutually exclusive intervals and

such that contains HuK and contains To see this suppose and

are such mutually exclusive intervals By Theorem contains periodic point

of of period Note that f3 has only four fixed points and Since

is fixed point for must contain one of and We complete the proof

by showing that each of these possibilities leads to contradiction

Suppose is in Then is in since f10 and contains

But since fl is in both and

Suppose is in Since f11 is in Since f1and and do not intersect is in and is in But f10 so is

in

Suppose is in As before one of and is in Since f10if is in then is in both and Thus is in Then f1 contains

two points and one less than so P1 is in Since P1 contains two

points and one between and is in Thus fi is in Since

contains two points and one less than P2 is in Continuing

this process we get sequence P1 P2.. of points of which converges to

Thus is in

Periodic points and indecomposability In this section we show that

under certain conditions the existence of periodic point of period three in mapping of continuum to itself implies that limM contains an indecomposable

continuum Of course the result is not true in general since rotation of S1 by 120

degrees yields homeomorphism of and copy of S1 for the inverse limit

THEOREM Suppose is mapping of the continuum into itself and is

point of which is periodic point of of period three If is atriodic and

hereditarily unicoherent then limM contains an indecomposable continuum

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648 INGRAM

Moreover the inverse limit is indecomposable if clU0 f2 where M1is the subcontinuum of irreducible from to fx

PROOF Suppose is periodic point of of period three Denote by M1 M2and M3 subcontinua of irreducible from to fx fx to f2x and f2x to

respectively Note that since is hereditarily unicoherent M1 fl M2 M3M1 M2 M3 is continuum so there is point common to all three

continua

The three continua M1 fl M2 M2 fl M3 and M1 fl M3 all contain the point

so since is atriodic one of them is subset of the union of the other two

Suppose M1flM2 is subset of M2flM3UM1flM3 M3flM1UM2 M3 Thelast equality follows since M3 fl M1 M2 is subcontinuum of M3 containing

and f2x and M3 is irreducible between and f2x Then M1 M2 is subset

of M3 for if not there is point of M1 M2 such that is not in M3 Since

M1 fl M2 is subset of M3 is in M1 or in M2 but not in M1 fl M2 Suppose is

in M1 M1 fl M2 Since is not in M3 is in M1 M1 fl M3 and thus is in

M1 M1But M1 flM2 UM3 is subcontinuum of M1 containing and fx so it contains

M1 since M1 is irreducible between and 1x Thus M1 M1 fl M2 M3 and

so M1 M2 is subset of M3Note that is continuum containing fx and 2x so fl M2

is subcontinuum of M2 containing these two points Since M2 is irreducible

from fx to f2x fl M2 M2 Therefore M2 is subset of

Similarly contains M3 and contains M1 However since M3 contains

M1 uM2 M3 contains 1x and f2x so contains M1 UM2 UM3 Thus

fn2 contains ffll which contains fm which contains M1 UM2 UM3for 123 and so clJ0 clJ20 clU20 ftThen cltJno is continuum such that fIH Denote by

the inverse limit limH IH We show that is indecomposable by showing the

conditions of Theorem 267 are satisfied Suppose is positive integer

and is positive number There is positive integer such that if is in then

dt fk Suppose is subcontinuum of containing two of the three

points 1x and f2x Then contains one of M1 M2 and M3 In any case

f2 contains M3 and thus if rn dtfm for each tin By

Kuykendalls Theorem is indecomposable

THEOREM If is mapping of to and has periodic point

whose period is not power of then lim contains an indecomposable

continuum Moreover for each positive integer there exists mapping which has

periodic point of period and hereditarily decomposable inverse limit.1

PROOF Suppose has periodic point which has period and is not

power of Then 2k for some and f22 has periodic point of

period 2k By the Sarkovskii Theorem j2J has periodic point of period so

in proofi Theorem first appeared with slightly different proof as Theorem of

Chaos periodicity and snakelike continua by Marcy Barge and Joe Martin in publication MSRI014-84 of the Mathematical Sciences Research Institute Berkeley California in January 1984



f22 has periodic point of period Since lim is homeomorphic to

lim1J by Theorem lim contains an indecomposable continuum

In the family of maps fx for /L 3.5699456.. all the

inverse limits forjt

in this range are hereditarily decomposable and for each power

of there is map in this collection with periodic point of period that power of

In fact for the inverse limit is an arc for the inverse limit

becomes as increases first sinusoid then sinusoid to double sinusoid etc

For more details on this see

REFERENCES

James Davis Atriodic acyclic continua and class Proc Amer Math Soc 90 1984477482

James Davis and Ingram An atriodic tree-like continuum with positive span which

admits monotone mapping to chainable continuum Fund Math to appearRobert Devaney An introduction to chaotic dynamical systems Benjamin/Cummings Menlo

Park Calif 1986

Hamilton fixed point theorem for pseudo arcs and certain other metric continua Proc

Amer Math Soc 1951 173174

Ingram An atriodic tree-like continuum with positive span Fund Math 77 197299107Daniel Kuykendall Irreducibility and indecomposability in inverse limits Fund Math 84

1973 265270

Tien-Yien Li and James Yorke Period three implies chaos Amer Math Monthly 82 1975985992

Wayne Lewis Periodic homeomorphisms of chainable continua Fund Math 117 1983 8184Jack McBryde Inverse limits on arcs using certain logistic maps as bonding maps Masters

Thesis University of Houston 1987

10 Piotr Minc fixed point theorem for weakly chainable plane continua preprint11 Sam Nadler Examples of fixed point free maps from cells onto larger cells and spheres Rocky

Mountain Math 11 1981 31932512 David Read Confluent and related mappings Colloq Math 29 1974 233239

13 Helga Schirmer topologist view of Sharkovsky Theorem Houston Math 11 1985385395

14 Michel Smith and Sam Young Periodic homeomorphisms on T-like continua Fund Math 104

1979 221224

15 Sorgenfrey Concerning triodic continua Amer Math 66 1944 439460

DEPARTMENT OF MATHEMATICS UNIVERSITY OF HOUSTON HOUSTON TEXAS 77004


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