PROCEEDINGS OF THE
`SOCIETY
`AMERICAN MATHEMATICAL
`Volume 104 Number
`October 198S
`
`CONCERNING PERIODIC POINTS
`IN MAPPINGS OF CONTINUA
`INGRAM
`
`Communicated by Dennis Burke
`
`ABSTRACT In this paper we present some conditions which are sufficient
`mapping to have periodic points
`THEOREM 1ff is mapping of the space Xinto
`and there exist subcontinua
`ha the fixed point property
`and
`every subcontinuum of
`such that
`of
`and every subcontinuum of 1H are in class
`contains
`positive integer such that fIHK
`contains HUK and
`
`if
`
`is
`
`for
`
`intersects
`
`then
`
`then
`
`contains periodic points of
`
`of every period
`
`than
`
`is
`
`fixed point
`
`lemma
`
`greater
`Also included
`LEMMA Suppose
`
`such that
`
`into
`
`and
`
`is
`
`subcontinuum
`
`has the fixed point
`
`is in class
`
`then there is
`
`point
`
`is mapping of the space
`contains
`every subcontinuunn of
`of
`If
`property and
`every subcontinuum of
`xofKsuch that 1x
`Further we show that
`mapping of
`then lim contains
`periodic point which
`power of
`is not
`for each positive integer
`continuum Moreover
`with
`into
`periodic point of period
`
`indecomposable
`
`mapping of
`
`If
`
`is
`
`into
`
`and
`
`has
`
`an
`
`there is
`and having
`
`hereditarily decomposable
`
`inverse limit
`
`Introduction In his book An Introduction to Chaotic Dynamical Systems
`The
`62 Robert
`Theorem 10.2
`includes
`proof of Sarkovskii
`Devaney
`orem Consider the following order on the natural numbers
`
`is continuous
`
`If
`
`and
`
`has
`
`has
`
`this
`
`then
`periodic point of prime period
`proof of this theorem for
`In working through
`periodic point of period
`the main result of this paperTheorem For an alternate
`the author discovered
`proof of Sarkovskiis Theorem for
`For
`further look at
`see also
`theorem for ordered spaces see
`continuum we mean
`By
`compact connected metric space and by mapping
`we mean
`continuous function By
`periodic point of period
`for mapping
`of
`continuum
`is meant
`into
`
`such that fx The statement
`
`that
`
`point
`such that
`means that
`is the least integer
`has prime period
`continuum
`is said to have the fixed point property provided if
`is mapping of
`such that fx
`continuum
`onto
`there is
`of
`mapping
`point
`is said to be weakly confluent provided for each subcontinuum
`
`into
`
`continuum
`
`of
`
`Secondary
`
`Received by the editors May 22 1987 and in revised form September 14 1987
`lassiJication 1985 Revision Primary 54F20 54H20
`1980 Mathematics Subject
`54F62 54H25 54F55
`Key words and phrases Periodic point
`uum inverse limit
`
`fixed point property class
`
`indecomposable
`
`contin
`
`1988 American Mathematical Society
`$100
`$25 per page
`0002-9939/88
`
`643
`
`License or copyright
`
`restrictions may apply to redistribution
`
`see http//vxvw.ams.org/journal-terms-of-use
`
`MYLAN - EXHIBIT 1032
`
`

`
`644
`
`INGRAM
`
`onto
`
`continuum is said to be in
`The
`
`is
`
`at
`
`least
`
`triod
`
`space
`
`is
`
`If
`
`is
`
`is thrown by
`some component of
`continuum onto it
`provided every mapping of
`Class
`is weakly confluent
`such that TK has
`continuum
`subcontinuum
`triod provided there is
`of
`continuum is atriodic provided it does not contain
`three components
`is the union of two subcontinua
`continuum
`is unicoherent provided if
`then the common part of
`and
`continuum is
`is connected
`and
`hereditarily unicoherent provided each of its subcontinua is unicoherent
`mapping of
`into
`the inverse limit of the inverse limit sequence
`X5 f2 where for each
`limX
`will be denoted
`For
`and f2 is
`the inverse sequence X2
`is the subset of the product of the
`the inverse limit
`sequence of spaces X1 X2.. to which the point x1 x2.. belongs
`if and only
`if fx11
`There has been considerable interest
`where
`homeomorphism
`is called periodic provided there is an integer
`h2 is the identity Wayne Lewis has shown
`for each
`there is
`chainable
`that
`theorem of Michel Smith
`continuum with
`periodic homeomorphism of period
`and Sam Young
`should be compared with Theorem of this paper Smith and
`Young show that
`periodic homeomorphism of
`chainable continuum
`has
`than
`then
`contains an indecomposable continuum In this
`period greater
`paper we consider the question of the existence of periodic points in mappings of
`continua
`
`in periodic homeomorphisms of continua
`such that
`
`if
`
`fixed point theorem The problem of finding
`periodic point of period
`is of course the same as the problem of finding
`for mapping
`fixed point
`for ffl Not surprisingly we need
`lemma to the main
`theorem as
`fixed point
`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
`corollary to Theorem
`mapping with no fixed point of
`containing disk
`disk to
`is the well-known corresponding result
`for mappings of intervals
`
`Suppose
`
`is
`
`THEOREM
`and
`into
`is mapping of
`If1 every subcontinuum of
`subcontinuum of
`such that
`the fixed point property and
`every subcontinuum of
`is in Class
`such that fx
`there is
`point
`of
`
`space
`contains
`
`is
`
`has
`
`then
`
`is in Class
`
`and
`
`is
`
`Then
`
`there is
`
`subcon
`
`PROOF Since
`subset of
`IKi K1
`tinuum K1 of
`such that
`is weakly confluent
`since every subcontinuum of
`subcontinuum K2
`is in Class
`thus there is
`K1 Since K1 is in Class
`K2K2
`K1 is weakly
`of K1 such that
`K2 Con
`subcontinuum K3 of K2 such that
`therefore there is
`confluent
`tinuing this process there exists monotonic decreasing sequence K1 K2 K3..
`123
`of subcontinua of
`such that
`Let
`denote
`for
`the common part of all
`the terms of this sequence and note that
`fl0
`onto
`Since fIH throws
`fld0 Kd
`and therefore of
`has the fixed point property there exists
`of
`point
`such that fx
`REMARK Note that
`Theorem
`is chainable
`
`and
`
`since
`
`and
`
`of the hypothesis of Theorem are met if
`236 and
`is met if
`respectively while
`
`is
`
`License or copyright
`
`restrictions may apply to redistribution
`
`see http//onNw.ams.org/journal-terms-of-use
`
`

`
`CONCERNING PERIODIC POINTS IN MAPPINGS OF CONTINUA
`
`645
`
`and
`atriodic and acyclic
`subcontinuum
`two points of
`
`is met by planar tree-like continua such that each
`lie in weakly chainable subcontinuum of
`
`Periodic points In this section we prove the main result of the paper
`
`THEOREM
`and
`tinua
`
`property
`
`tains
`
`If
`
`of
`
`into
`is mapping of the space
`every subcontinuum of
`has the fixed point
`such that
`con
`and every subcontinuum of
`are in class
`and
`positive integer such that
`contains
`fIHYK intersects
`contains periodic points of
`every period greater than
`
`and there exist subcon
`
`then
`
`then
`
`if
`
`is
`
`of
`
`of
`
`for
`
`PROOF Suppose
`such that
`
`12.
`H_1 Thus
`
`there is
`point
`then fJx is not
`Since fmx
`H4
`
`of
`
`If
`
`contrary to
`
`sequence H1H2. H-_1 of subcontinua
`There is
`note that fH is weakly confluent
`and
`in case
`There is
`subcontinuum
`and so
`such that fmx
`and f3x
`and f2x
`f2x
`of the hypothesis
`
`and in
`of prime period
`REMARK If
`into itself and
`is mapping of the continuum
`periodic
`then the mapping of limM induced by
`has periodic points
`point of period
`e.g xfc_lx fxx... Thus although Theorem does not
`of period
`directly apply to homeomorphisms
`it may be used to conclude
`homeomorphisms with periodic points
`
`of
`so that
`so by Theorem
`contains
`We must show that
`is in H2
`then
`and
`is in fIHY2K is in
`Therefore
`
`if
`
`Since
`
`is periodic
`
`has
`
`the existence
`
`of
`
`COROLLARY If
`there are subcontinua
`
`is
`
`and
`
`IHK intersects
`
`if
`
`chainable continuum
`such that
`of
`
`is mapping of
`
`into
`
`and
`
`contains HUK and
`has periodic points of every period
`
`then
`
`then
`
`A/2
`
`B/2
`
`813
`
`FIGURE
`
`License or copyright
`
`restrictions may apply to redistribution
`
`see http//onNw.ams.org/journal-terms-of-use
`
`

`
`646
`
`INGRAM
`
`EXAMPLE Let
`be the mapping of the simple triod
`to itself given in
`A/2 and
`The mapping
`above Letting
`is represented in Figure
`follows from Theorem that
`has periodic points of every period
`B/21 it
`EXAMPLE Let
`be the mapping of the simple triod
`to itself given in
`The mapping
`below
`3B/8 and
`is represented in Figure
`Letting
`C/8 it
`follows from Theorem
`has periodic points of every
`
`that
`
`period
`
`___
`___
`
`C/32
`
`/H
`
`//
`
`C/8
`
`C/2
`
`B/2
`
`C/2
`
`FIGURE
`
`be the mapping of the unit circle S1 to itself given by fz z2
`EXAMPLE Let
`eIir
`follows from
`3ir/4 and
`3ir/2 it
`ezOjO
`Letting
`Theorem that
`has periodic points of every period Similarly if
`onto itself which is homotopic to z7 for some ri
`then
`points of every period
`
`is mapping
`has periodic
`
`of
`
`FIGURE
`
`COROLLARY 1ff is mapping of an interval
`then
`has periodic points of every period
`period
`
`to itself with
`
`periodic point of
`
`License or copyright
`
`restrictions may apply to redistribution
`
`see http//onNw.ams.org/journal-terms-of-use
`
`

`
`CONCERNING PERIODIC POINTS IN MAPPINGS OF CONTINUA
`
`647
`
`and fc
`
`intervals
`
`is
`
`is
`
`and
`
`and
`
`PROOF To see this it
`the hypothesis of Theorem is
`is matter of noting that
`met We indicate the proof
`for one of two cases and leave the second similar case
`to the reader
`and
`are points of the interval with
`and
`Suppose
`fa
`other case is fa
`fb
`and fc
`fb
`Jf f1 is nondegenerate then there exist mutually exclusive
`and
`and
`respectively so that
`lying in
`Theorem applies
`Suppose f1c
`and
`
`that
`
`is not
`
`in
`
`for
`
`is not
`
`in
`
`for
`
`if
`
`Choose
`
`so that
`
`IHK Note
`
`that
`
`and
`
`and
`lying in
`lying in
`denote by H5 the set
`For each
`234.. and thus
`23.. so
`is not
`in
`for
`Thus if
`in H1 since
`Further
`is not
`is not
`in
`Consequently the hypothesis of Theorem is met
`then
`intersects
`REMARK Condition
`of Theorem
`seems
`more natural
`bit artificial
`condition the author experimented with in its place is
`and
`requirement
`and
`be mutually exclusive
`in each of the examples the
`In fact
`given
`are mutually exclusive However
`with this proved to be
`replacing condition
`corollary to Theorem
`undesirable in that the Sarkovskii Theorem for
`is not
`may not be replaced by
`the alternate condition is used That condition
`and
`are mutually exclusive can be seen by the following
`the assumption that
`For the function
`which is piecewise linear and contains the points
`and 10 there do not exist mutually exclusive
`and
`intervals
`contains HuK and
`such that
`To see this suppose
`contains
`intervals By Theorem
`are such mutually exclusive
`contains
`periodic point
`Note that
`and
`Since
`f3 has only four fixed points
`of
`of period
`We complete the proof
`contradiction
`
`is
`
`for
`
`and
`
`must contain one of
`fixed point
`by showing that each of these possibilities leads to
`since f10
`Then
`Suppose
`is in
`is in
`is in both
`
`is in
`
`Since f11
`
`and
`
`is in
`
`do not
`
`intersect
`
`is in
`
`and
`
`is in
`
`and
`
`contains
`
`Since f1
`But f10
`
`so
`
`is
`
`But since fl
`
`Suppose
`and
`
`and
`
`in
`
`if
`
`Suppose
`then
`
`is in
`
`is in
`
`and
`As before one of
`Thus
`is in both
`and
`
`is in
`
`is in
`
`Since f10
`Then f1 contains
`P1 contains two
`
`is in
`
`Since
`
`is in
`
`Continuing
`which converges
`
`to
`
`and one less than
`two points
`and
`and one between
`points
`P2
`and one less than
`contains two points
`sequence P1 P2.. of points of
`this process we get
`Thus
`
`so P1
`
`is in
`
`is in
`
`Since
`
`Thus fi
`
`is in
`
`In this section we show that
`Periodic points and indecomposability
`periodic point of period three in map
`under certain conditions the existence of
`implies that limM contains an indecomposable
`continuum
`ping of
`to itself
`continuum Of course the result is not true in general since
`rotation of S1 by 120
`homeomorphism of
`and
`copy of S1 for the inverse limit
`
`degrees yields
`
`THEOREM
`
`point of
`hereditarily unicoherent
`
`Suppose
`which is
`
`into itself and
`is mapping of the continuum
`is atriodic and
`of period three If
`periodic point of
`then limM contains an indecomposable
`continuum
`
`is
`
`License or copyright
`
`restrictions may apply to redistribution
`
`see httpthmnw.ams.org/journal-terms-of-use
`
`

`
`648
`
`INGRAM
`
`the inverse limit
`Moreover
`is the subcontinuum of
`
`is indecomposable
`
`is
`
`is not
`
`is
`
`is
`
`is
`
`is
`
`Similarly
`
`is
`
`points
`
`if clU0 f2 where M1
`irreducible from to fx
`of period three Denote by M1 M2
`PROOF Suppose
`periodic point of
`irreducible from to fx fx to f2x and f2x to
`and M3 subcontinua of
`is hereditarily unicoherent M1 fl M2 M3
`respectively Note that since
`M1 M2
`M3 is
`common to all
`continuum so there is
`continua
`The three continua M1 fl M2 M2 fl M3 and M1 fl M3 all contain the point
`subset of the union of the other
`two
`is atriodic one of them is
`so since
`M3 The
`M3flM1UM2
`subset of M2flM3UM1flM3
`Suppose M1flM2 is
`last equality follows since M3 fl M1 M2 is
`subcontinuum of M3 containing
`and f2x Then M1
`and f2x and M3 is irreducible between
`M2 is
`subset
`in M3 Since
`of M3 for if not
`M2 such that
`of M1
`there is
`point
`subset of M3 is in M1 or in M2 but not in M1 fl M2 Suppose
`M1 fl M2 is
`M1 fl M2 Since
`in M3
`is in M1 M1 fl M3 and thus
`in M1
`M1
`M1
`and fx so it contains
`But M1 flM2 UM3 is
`subcontinuum of M1 containing
`and 1x Thus M1
`M1 fl M2 M3 and
`M1 since M1 is irreducible between
`subset of M3
`so M1
`M2 is
`continuum containing fx and 2x so
`fl M2
`Note that
`subcontinuum of M2 containing these two points
`Since M2 is irreducible
`from fx to f2x
`M2
`fl M2
`Therefore M2 is
`subset of
`contains M1 However since M3 contains
`contains M3 and
`M1 uM2 M3 contains 1x and f2x so
`contains M1 UM2 UM3 Thus
`which contains fm which contains M1 UM2 UM3
`fn2 contains ffll
`clU20 ft
`123
`and so clJ0
`clJ20
`for
`continuum such that fIH
`Then
`Denote by
`cltJno
`limH IH We show that
`is indecomposable by showing the
`the inverse limit
`Theorem
`267 are satisfied Suppose
`conditions of
`positive integer
`dt fk Suppose
`positive number There is
`such that
`and
`then
`positive integer
`subcontinuum of
`containing two of the three
`1x and f2x Then
`contains one of M1 M2 and M3 In any case
`dtfm for each tin
`f2 contains M3 and thus if
`By
`Kuykendalls Theorem
`
`point
`
`three
`
`is not
`
`is
`
`is in
`
`is
`
`if
`
`is in
`
`rn
`
`is indecomposable
`
`THEOREM
`If
`whose period is not
`continuum Moreover
`periodic point of period
`
`is
`
`mapping of
`
`to
`
`and
`
`has
`
`periodic point
`
`then lim contains an indecomposable
`power of
`there exists mapping which has
`for each positive integer
`and hereditarily decomposable
`inverse limit.1
`
`PROOF Suppose
`has
`2k
`Then
`power of
`By the Sarkovskii Theorem j2J has
`period 2k
`
`for some
`
`periodic point which has period
`
`and
`
`is not
`
`and f22 has
`
`periodic point of
`
`periodic point of period
`
`so
`
`as Theorem of
`in proofi Theorem
`first appeared with
`slightly different proof
`publication MSRI
`continua by Marcy Barge and Joe Martin in
`Chaos periodicity and snakelike
`014-84 of the Mathematical Sciences Research Institute Berkeley California in January 1984
`
`License or copyright
`
`restrictions may apply to redistribution
`
`see http//onNw.ams.org/journal-terms-of-use
`
`

`
`CONCERNING PERIODIC POINTS IN MAPPINGS OF CONTINUA
`
`649
`
`f22
`
`has
`
`periodic point of period
`
`for
`
`Since lim is homeomorphic to
`by Theorem lim contains an indecomposable continuum
`lim1J
`In the family of maps fx
`/L
`3.5699456..
`the
`all
`and for each power
`inverse limits for jt
`in this range are hereditarily decomposable
`there is map in this collection with
`periodic point of period that power of
`of
`is an arc for
`In fact
`the inverse limit
`the inverse limit
`sinusoid then
`sinusoid to
`
`for
`
`becomes as
`increases first
`For more details on this see
`
`double sinusoid etc
`
`REFERENCES
`
`Davis Atriodic acyclic continua and class
`
`Proc Amer Math Soc 90 1984
`
`James
`477482
`James
`
`admits
`
`Ingram An atriodic
`Davis and
`tree-like continuum with positive span which
`chainable continuum Fund Math to appear
`monotone mapping to
`Devaney An introduction
`to chaotic dynamical systems Benjamin/Cummings Menlo
`Robert
`Park Calif 1986
`Hamilton
`theorem for pseudo
`fixed point
`Amer Math Soc
`1951 173174
`tree-like continuum with positive span Fund Math 77 1972
`Ingram An atriodic
`
`arcs and certain other metric continua Proc
`
`Irreducibility and indecomposability
`
`in inverse limits Fund Math 84
`
`Yorke Period three implies chaos Amer Math Monthly 82 1975
`
`99107
`Kuykendall
`Daniel
`1973 265270
`Tien-Yien Li and James
`985992
`Wayne Lewis Periodic homeomorphisms of chainable continua Fund Math 117 1983 8184
`Jack McBryde Inverse limits on arcs using certain logistic maps as bonding maps Masters
`Thesis University of Houston 1987
`10 Piotr Minc
`theorem for weakly chainable plane continua preprint
`fixed point
`11 Sam
`Nadler Examples of fixed point free maps from cells onto larger cells and spheres Rocky
`Math 11 1981 319325
`Mountain
`Read Confluent and related mappings Colloq Math 29 1974 233239
`12 David
`Math 11 1985
`13 Helga Schirmer
`Theorem Houston
`view of Sharkovsky
`topologist
`385395
`14 Michel Smith and Sam Young Periodic homeomorphisms
`1979 221224
`Sorgenfrey Concerning triodic continua Amer Math 66 1944 439460
`
`on T-like continua Fund Math 104
`
`15
`
`DEPARTMENT OF MATHEMATICS UNIVERSITY
`
`OF HOUSTON HOUSTON TEXAS 77004
`
`License or copyright
`
`restrictions may apply to redistribution
`
`see http//onNw.ams.org/journal-terms-of-use