PROCEEDINGS OF TH!E
`AMERICAN MATHEMATICAL SOCIETY
`Volume 104, Number 2, October 1988
`
`CONCERNING P ERIODIC POINTS
`IN MAPPINGS OF CONTINUA
`W. T. INGRAM
`
`(Communicated by Dennis Burke)
`
`ABSTRACT . In this paper we present some conditions which are sufficient for
`a mapping to have periodic points.
`THEORE:M. If/ is a mapping of the space X into X and there exist subcontinua
`Hand K of X such that (1) every subcontinuum of K has the fixed point property,
`(2) / [K) and every .mbcontinuum of /[HI are in class W , (3) / [K l contains H,
`(4) /[HI contains HUK, and (5) ifn is a positive integer such that (llH)-n(K)
`intersects K ,, then n = 2, then K contains periodic points off of every period
`greater than 1.
`Also incltuded is a fixed point lemma:
`LEMMA. Suppose f is a mapping of the space X int.o X and K is a subcontinuum
`of X such that /[Kl contains K. If ( 1) every subcontinuum of K has the fixed point
`property, and (2) every subcontinuum of / [Kl is in clCUiS W, then there is a point
`x of K such t:hat f(x ) = x .
`Further we show that: If f is a mapping of [O, II into [O, II and f has
`a periodic point which is not a power of 2, then lim{ [O, lj, /} contains an
`indecomposable continuum. Moreover, for each positive integer i, there is a
`mapping of {O, ll into [O, 11 with a periodic point of period 2; and having a
`hereditarily decomposable inverse limit.
`
`1. Introduction. In his book, An Introduction to Chaotic Dynamical Systems
`13, T heorem 10.2, p . 62], Robert L. Devaney includes a proof of Sarkovskii s' The(cid:173)
`orem. Consider the following order on the natural numbers: 3 t> 5 t> 7 t> · · · t> 2 · 3 t>
`2·51> · · · 1> 22 · 3 1> ~?2 · 5 t> · · · t> 23 · 31> 23 · 51> · · · 1> 23 1>22 1> 2 1> 1. Suppose f: R --+ R
`is continuous. If k 1> m and f has a periodic point of prime period k, then f has a
`periodic point of period m. In working through a proof of this theorem for k = 3,
`the author discove1red the main result of this paper- Theorem 2. For an al!ternate
`proof of Sarkovskii's Theorem for k = 3, see also [7]. For a further look at this
`theorem for ordered spaces see [13].
`By a continuum we mean a compact connected metric space and by a mapping
`we mean a continuous function. By a periodic point of period n for a mapping f of a
`continuum Minto Mis meant a point. x such that r(x) = x. The statement that
`x has prime period n means that n is the least integer k such that f k ( x) = x . A
`continuum M is saiid to have the fixed point property provided if f is a mapping of M
`into M there is a p1oint x such that f(x) = x. A mapping f of a continuum X onto
`a continuum Mis said to be weakly confluent provided for each subcontinuum K of
`
`Received by the ediitors May 22, 1987 and, in revised form, September 14, 1987.
`1980 Mathematics Subject Classification (1985 Revi.~ion). Primary 54F20, 54H20; S•~onda.ry
`54F62, 54H25, 54F55.
`Key words and phmses. Periodic point, fixed point property, class W, indecomposaMe contin(cid:173)
`uum, inverse limit.
`
`©1988 American Mathematic.al Society
`0002-9939/88 $1.00 + S.25· per page
`
`643
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`644
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`W. T. INGRAM
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`M some component of 1- 1 ( K) is thrown by f onto K. A continuum is said to be in
`Class W provided every mapping of a continuum onto it is weakly confluent. T he
`continuum T is a tri.od provided there is a su bcontinuum K of T such that T - K has
`at least three components. A continuum is atriodic provided it does not contain a
`triod. A continuum M is unicoherent provided if M is the union of two subcontinua
`H and K , then the common part of H and K is connected. A continuum is
`hereditarily unicoherent provided each of its subcontinua is unicoherent. If f is
`a mapping of a space X into X, t he inverse limit of the inverse limit sequence
`{Xi. /i} where, for each i, Xi is X and /; is f will be denoted lim{X, !}. For
`the inverse sequence {X;, /i}, the inverse limit is the subset of t he product of the
`sequence of spaces Xi, X2 , ... to which the point (xi, x2 ,. .. ) belongs if and only
`if fi(Xi+ i) = X;.
`There has been considerable interest in periodic homeomorphisms of continua
`where a homeomorphism h is called periodic provided there is an integer n such that
`hn is the identity. Wayne Lewis has shown (8] that for each n there is a chainable
`continuum with a periodic homeomorphism of period n. A theorem of Michel Smith
`and Sam Young [14] should be compared with Theorem 3 of this paper. Smith and
`Young show that if a chainable continuum M has a periodic homeomorphism of
`period greater than 2, t hen M contains an indecomposable continuum. In this
`paper we consider the question of the existence of periodic points in mappings of
`continua.
`
`2. A fixed point theorem. The problem of finding a periodic point of period
`n for a mapping f is, of course, the same as the problem of finding a fixed point
`for r . Not surprisingly, we need a fixed point theorem as a 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 (11] of a
`mapping with no fixed point of a disk to a containing disk. A corollary to Theorem
`1 is the well-known corresponding result for mappings of intervals.
`
`THEOREM I. Suppose X is a space, f is a mapping of X into X, and K is a
`subcontinuum of X such that f [K] contains K. If (1) every subcontinuum of K has
`the fixed point property, and (2) every subcontinuum of f[K J is in Class W, then
`there is a point x of K such that f(x) = x.
`
`PROOF. Since /[K] is in Class Wand K is a subset of /[K], there is a subcon.(cid:173)
`t inuum K1 of K such that /[Ki] = K. Then /IK1:K1 -> K is weakly confluent
`since every subcont inuum of J[K] is in Class W; thus there is a subcontinuum K2
`of K1 such that /[K2] = K1 . Since K 1 is in Class W, / IK2: K2 -> K 1 is weakly
`confluent; therefore there is a subcontinuum K3 of K2 such that /[K3J = K2· Con(cid:173)
`t inuing t his process there exists a monotonic decreasing sequence Kli K 2 , K3, .. .
`of subcontinua of K such that J[Ki+i] = K i for i = 1, 2, 3, .. . . Let H denote
`the common part of all the terms of this sequence and note that ![H J = H , since
`![HJ = !lni>O Ki] = ni>O / [Ki] = ni>O Ki = H . Since JIH throws H onto H
`and H has the fixed point property, there exists a point x of H (and therefore of
`K) such that /(x) = x.
`REMARK. Note t hat (1) and (2) of the hypothesis of Theorem I are met if /[K]
`is chainable ([12, Theorem 4, p. 236 and 4], respectively), while (2) is met if /[KJ is
`
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`CONCERNING P ERIODIC POINTS IN MAPPINGS OF C ONTINUA
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`645
`
`atriodic and acyclic [l ] and (1) is met by planar, tree-like continua such that each
`two points of a subcontinuum L lie in a weakly chainable subcontinuum of L [10].
`
`3. Periodic points. In this section we prove the main result of the paper.
`THEOREM 2. If f is a mapping of the space X into X and there exist subcon(cid:173)
`tinua H and K of X such that ( 1) every subcontinuum of K has the fixed point
`property, (2) f[K] and every subcontinuum of f[H] are in class W, (3) /[K] con(cid:173)
`tains H , (4) ![HJ contains H U K, and (5) if n is a positive integer such that
`(JIH) -n(K) intersects K , then n = 2, then K contains periodic points off of
`every period greater than I .
`
`PROOF . Suppose n ~ 2. There is a sequence H 1 , H2, ... , Hn- l of subcontinua
`of H such that / [H 1] = K (note that JIH is weakly confluent) and /[Hi+il = Hi
`for i = 1, 2, .. . , n - 2 (in case n > 2). There is a subcontinuum Kn of K so that
`! [Kn] = Hn-1 · Thus, r[Kn] = Kand so r [Kn] contains Kn, so, by Theorem
`1, there is a point x of Kn such that r(x) = x. We must show that if j < n
`then Ji(x ) is not x. If j < n and Ji(x) = x, then j = n - 2 and x is in H2 •
`Since r(x) = x and r -2 (x ) = x, f 2 (x) = x. Since x is in (J IH)- 2 (K ), x is in
`(J IH )- 4 (K) and in K contrary to (5) of the hypothesis. Therefore, x is periodic
`of prime period n.
`REMARK. If f is a mapping of the continuum M into itself and f has a periodic
`point of period k, then the mapping of lim{M, J} induced by f has periodic points
`of period k, e.g. (x,Jk- I(x) , ... ,f(x), x, ... ). Thus, although Theorem 2 does not
`directly apply to homeomorphisms, it may be used to conclude the existence of
`homeomorphisms with periodic points.
`
`COROLLARY. If M is a chainable continuum, f is a mapping of Minto M, and
`there are subcontinua Hand K of M such that f[K] = H , /[HJ contains HuK, and
`if (!IH)-n(K) intersects K then n = 2 then f has periodic points of every period.
`
`A
`
`A/2
`
`~ B/2 V1
`!-.
`v
`
`~
`
`., /
`
`8
`
`ol
`I
`
`B/3
`
`FIGURE 1
`
`A
`B
`c
`
`c
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`646
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`W . T . I NGRAM
`
`EXAMPLE. Let f be the mapping of the simple triod T to itself given in [5].
`The mapping f is represented in Figure 1 above. Letting H = [O, A/2] and K =
`[B/3, B/2] it follows from Theorem 2 that f has periodic points of every period.
`EXAMPLE. Let f be the mapping of the simple t riod T to itself given in [2].
`T he mapping f is represented in Figure 2 below. Letting H = [O, 3B/8] and
`K = [C /32, C /8], it follows from Theorem 2 that f has periodic points of every
`period.
`
`K
`
`B
`
`B
`
`0 C/32
`
`C/8
`
`6 /2
`
`F IGURE 2
`
`H
`
`C/2
`
`38/8
`
`B/ 2
`
`C/2
`
`A
`c
`
`c
`
`EXAMPLE. Let f be the mapping of the unit circle S1 to itself given by f (z) = z2 .
`Letting H = { ei0 IO S () S 311" / 4} and K = { eio 111" S () S 311" /2}, it follows from
`Theorem 2 that f has periodic points of every period. Similarly, if f is a mapping
`of S 1 onto itself which is homotopic to zn for some n > 1, then f has periodic
`points of every period.
`
`H
`
`z2
`
`K
`
`FIGURE 3
`
`COROLLARY. If f is a mapping of an interval to itself with a periodic point of
`period 3, then f has periodic points of every period.
`
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`CONCERNING PERIODIC POINTS IN MAPPINGS OF CONTINUA
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`647
`
`PROOF. To see this it is a matter of noting that the hypothesis of Theorem 2 is
`met. We indicate the proof for one of two cases and leave the second similar case
`to the reader. Suppose a, b and c are points of the interval with a < b < c and
`J(a) = b, J(b) = c and f(c) =a [the other case is f(a) = c, f(b) =a and /(c) = b].
`If J-I ( c) is nondegenerate, then there exist mutually exclusive intervals H and
`K lying in [b, c] and [a, b], respectively, so that f [H J is [a, c] and f[ K] is (b, c) and
`Theorem 2 applies.
`Suppose 1-1(c) = {b}. Choose K lying in [a,b] and H lying in (b,c] so that
`/(K] = [b, c] and / (H J = [a, c] . For each i , denote by H; the set (!IH)- 1(K). Note
`that a is not in H; for i = 1, 2, 3, ... soc is not in H; for i = 2, 3, 4, ... and thus b
`is not in H; for i = 3, 4, .... Further, bis not in H 1 since c is not in K. Thus, if
`H ; intersects K, then i = 2. Consequently, the hypothesis of Theorem 2 is met.
`REMARK. Condition (5) of Theorem 2 seems a bit artificial. A more natural
`condition the author experimented with in its place is a requirement that H and
`K be mutually exclusive. In fact, in each of the examples, the H and K given
`are mutually exclusive. However, replacing condition (5) with this proved to be
`undesirable in that the Sarkovskii Theorem for k = 3 is not a corollary to Theorem
`2 if the alternate condition is used. That condition (5) may not be replaced by
`t he assumption that H and K are mutually exclusive can be seen by the following.
`For the function f: (0, I] --+ [O, lJ, which is piecewise linear and conta ins the points
`(0, t), ( ! , 1) and ( 1, 0), there do not exist mutually exclusive intervals H and K
`such that / [H J contains HuK and /[K] contains H . To see this suppose Hand K
`are such mutually exclusive intervals. By Theorem 2, K contains a periodic point
`of f of period 3. Note t hat / 3 has only four fixed points: 0, t, ~, and 1. Since ~
`is a fixed point for f, K must contain one of 0, !, and 1. We complete the proof
`by showing that each of these possibilities leads to a contradiction.
`(1) Suppose 0 is in K. Then 1 is in H since 1- 1(0) = {l } and ![HJ contains K.
`But since 1- 1(1) = { t }, t is in both Hand K.
`(2) Suppose 1 is in K. Since 1- 1(1) = {! } , ! is in H. Since 1-1(!) = {O, V
`and Hand K do not intersect 0 is in Hand ~ is in K. But, 1- 1 (0) = {l} so 1 is
`inH.
`(3) Suppose ! is in K. As before, one of 0 and i is in H . Since J- 1(0) = {1},
`if 0 is in H t hen 1 is in both H and K . Thus ~ is in H. Then 1- 1 ( i) contains
`two points, i and one less than ! , so P 1 = i is in H. Since 1-1 (Pi ) contains two
`( k) = !~ is in H . Since
`points, ~ and one between ~ and ~, k is in K. Thus, 1- 1
`r 1 (!~) contains two points, ~~ and one less than ! , P2 = ~~ is in H. Continuing
`this process, we get a sequence Pi, P2, . . . of points of H which converges to ! .
`Thus t is in H.
`4. Periodic points and indecomposablllty. In this section we show that
`under certain conditions the existence of a periodic point of period three in a map(cid:173)
`ping of a continuum M to itself implies that lim{M, !} contains an indecomposable
`continuum. Of course the result is not true in general since a rotation of 8 1 by 120
`degrees yields a homeomorphism of 8 1 and a copy of 8 1 for the inverse limit.
`THEOREM 3. Suppose J is a mapping of the continuum M into itself and x is
`a point of M which is a periodic point of f of period three. If M is atriodic and
`hereditarily tmicoherent, then lirn{M, !} contains an indecomposable continuum.
`
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`
`648
`
`W. T. I NGRAM
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`Moreover, the inverse limit is indecomposable if cl(Ui>O Ji[Mi]) = M, where Mi
`is the subcontinuum of M irreducible from x to f(x).
`
`PROOF. Suppose xis a periodic point off of period three. Denote by M 1 , M2
`and M3 subcontinua of M irreducible from x to f(x), J(x) to f 2 (x) and / 2 (x) to
`x, respectively. Note that since M is hereditarily unicoherent, M1 n (M2 u M3 ) =
`(Mi n M2) u (M1 n M3) is a continuum, so there is a point p common to all three
`continua.
`The three continua M1 n M2, M2 n M 3 and M1 n M3 all contain the point p
`so, since M is atriodic, one of them is a subset of the union of the other two (15].
`Suppose M1nM2 is a subset of (M2n M3)U(M1nM3) = M3n(M1UM2) = M3. (The
`last equality follows since M3 n (M1 U M2 ) is a subcontinuum of M3 containing x
`and / 2 (x) and M3 is irreducible between x and / 2(x)). Then, M1 UM2 is a subset
`of M3 for if not there is a point t of M 1 U M2 such that t is not in M3 . Since
`M1 n M2 is a subset of M3, t is in M1 or in M2 but not in M1 n M2· Suppose t is
`in M1 - (Mi n M2)· Since t is not in M3, t is in M1 - (M1 n M3) and thus t is in
`Mi - [(M1 n M2) u (Mi n M3)] = M1 - [Mi n (M2 u M3)].
`
`But, Mi n (M2UM3 ) is a subcontinuum of M 1 containing x and f(x), so it contains
`Mi since Mi is irreducible between x and f(x). T hus, Mi = Min (M2 u M3) and
`so Mi u M2 is a subset of M3.
`Note that /[Mi] is a continuum containing f(x) and / 2 (x) , so f[Mi] n M 2
`is a subcontinuum of M2 containing t hese two points. Since M2 is irreducible
`from f(x) to / 2(x), /[Mi ] n M2 = M2. Therefore, M2 is a subset of /[Mi].
`Similarly, /[M2] contains M3 and /[M3] contains Mi. However, since M3 contains
`Mi U M2, M3 contains x, f(x) and / 2(x), so JIM3] contains M 1 U M2 U Ms. Thus,
`r +2[Mi] contains r+i[M2] which contains r(M3] which contains Mi UM2UMs
`for n = l , 2, 3, ... and so cl(Ui>o Ji[M1]) = cl(Ui>o Ji[M2]) = cl(U,>o /i[M3]).
`Then, H = cl(LJn>o r[Mi]) is a continuum such that /IH: H --+ H. Denote by K
`the inverse limit, lim{H,/IH}. We show that K is indecomposable by showing the
`conditions of [6, Theorem 2, p. 267] are satisfied. Suppose n is a positive integer
`and e is a positive number. There is a positive integer k such that if t is in H then
`d(t , Jk[M3 ]) < e. Suppose C is a subcont inuum of H containing two of the three
`points, x, f(x) and f 2(x). Then C contains one of M 1 , M2 and M3 . In any case
`/ 2(C] contains M3, and thus, if m = k + 2, d(t,fm[C]) < e for each tin H. By
`Kuykendall's Theorem, K is indecomposable.
`THEOREM 4. If f is a mapping of [O, l] to [O, l ] and f has a periodic point
`whose period is not a power of 2, then lim{ ([O, I],/)} contains an indecomposable
`continuum. Moreover, for each positive integer i, there exists a mapping which has
`a periodic point of period 2' and hereditarily decomposable inverse limit.1
`PROOF. Suppose f has a periodic point which has period n and n is not a
`power of 2. Then, n = 2i (2k+ 1) for some j , k ~ 0, and / 2; has a periodic point of
`period 2k + 1. By the Sarkovskii Theorem, / 2; has a periodic point of period 6, so
`1 Added in proof: Theorem 4 first appeared, with a slightly different proof, as Theorem 1 of
`Chaos, periodicity, and sna~like continua by Marcy Barge and J oe Martin in a publication (MSRI
`014-84} of the Mathematical Sciences Research Institute, Berkeley, California in January, 1984.
`
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`CONCERNING PERIODIC POINTS IN MAPPINGS OF CONTINUA
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`649
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`g = (!2;)2 has a periodic point of period 3. Since lirn{(O, l ], !} is homeomorphic to
`lim{(O, !J, g}, by Theorem 3 lim{[O, lJ, !} contains an indecomposable continuum.
`In the family of maps fµ.(x) = µx( l - x), for 2 < µ < µc ~ 3.5699456 . . . all the
`inverse limits for µ in this range are hereditarily decomposable and for each power
`of 2, there is a map in this collection with a periodic point of period that power of
`2. In fact for 2 < µ < 3 the inverse limit is an arc, for 3 < µ < µc the inverse limit
`becomes, as µ increases, first a sinusoid, then a sinusoid to a double sinusoid, etc.
`For more details on this, see (9J.
`
`R EFERENCES
`
`l. James F . Davis, Atriodic acyclic continua and class W, Proc. Amer. Math. Soc. 90 ( 1984),
`477-482.
`2. James F. Davis and W. T . Ingram, An atriodic tree-like continuum with positive span which
`admits a monotone mapping to a chainable continuum, Fund. Math. (to appear).
`3. Robert L. Devaney, An introduction to chaotic dynamical systems, Benjamin/Cummings, Menlo
`Park, Calif., 1986.
`4. 0 . H. Hamilton, A fixed point theorem for pseudo arcs and certain other metric continua, P roc.
`Amer. Math. Soc. 2 (1951), 173- 174.
`5. W. T . Ingram, An atriodic tree-like continuum with positive span, Fund. Math. 77 (1972),
`99-107.
`6. Daniel P. Kuykendall, Irreducibility and indecomposability in inverse limits, Fund. Math. 84
`(1973), 265- 270.
`7. Tien-Yien Li and James A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975) ,
`985-992.
`8. Wayne Lewis, Periodic homeomorphisms of chainable continua, Fund. Math. 117 (1983), 81- 84.
`9. Jack McBryde, Inverse limits on arcs using certain log'i$tic maps as bonding maps, Master's
`Thesis, University of Houston, 1987.
`10. Piotr Mine, A fixed point theorem for weakly chainable plane continua, preprint.
`11 . Sam B. Nadler, Examples of fixed point free maps from cells onto larger cells and spheres, Rocky
`Mountain J. Math. 11 (1981), 319- 325.
`12. David M. Read, Confluent and related mapping$, Colloq. Math. 29 (1974), 233- 239.
`13. Helga Schirmer, A topologi3t 's view of Sharkovsky's Theorem, Houston J. Math. 11 (1985),
`385- 395.
`14. Michel Smith and Sam Young, Periodic homeomorphisms on T -like continua, Fund. Math. 104
`(1979), 221-224.
`15. R. H. Sorgenfrey, Concerning triodic continua, Amer. J. Math. 66 (1944), 439-460.
`
`DEPARTMENT OF MATHEMATICS, UNIVERSITY OF H OUSTON, HO USTON, T EXAS 77004
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