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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
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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
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CONCERNING PERIODIC POINTS IN MAPPINGS OF CONTINUA 647
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
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CONCERNING PERIODIC POINTS IN MAPPINGS OF CONTINUA 649
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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