Periodicity and Indecomposability

Abstract

Abstract. In this paper we characterize the existence of periodic points of odd period greater than one for unimodal mappings of an interval onto itself. The interesting juxtaposition of this condition with the occurrence in inverse limits of the well-known Brouwer-Janiszewski-Knaster continuum is explored. Also obtained is a characterization of indecomposability of certain inverse limits using a single unimodal bonding map. IntroductionThis paper grew out of investigations of inverse limits on [0, 1] using logistic bonding maps and inverse limits on [0,1] using families of piecewise linear bonding maps. In each case, a "standard" indecomposable continuum, the socalled Brouwer-Janiszewski-Knaster continuum (throughout this paper we will refer to this continuum as the B-J-K continuum) [8, p. 204], kept occurring. It showed up often enough that the author isolated a theorem (Theorem 2) which produces the phenomenon. Then, the author observed that, by perturbing the condition which gives rise to the B-J-K continuum, for a certain class of mappings (which includes the logistic family, unimodal maps, and the tent family) a characterization of the existence of periodic points of odd period for this class of mappings is obtained (Theorem 6). By considering composites and looking at appropriate subintervals on which the restricted mapping belongs to the appropriate class, this theorem yields the existence of periodic points of periods which are odd multiples of powers of two. Dynamicists have long been interested in periodic points whose period is not a power of two (see [10], for example). The author's interest arises from the connection with the existence of indecomposable continua in the inverse limit [1], [6].By a continuum we mean a compact, connected subset of a metric space. By a mapping we mean a continuous function. If Xx, X2, X$, • • • is a sequence of metric spaces and fx, f2, h, •■■ is a sequence of mappings (called bonding maps) such that, for each positive integer z, / : Xi+X -> X¡ then by the inverse limit of the inverse limit sequence {X¿, fi} is meant the subset of the

The inverse limits approach to chaos

“…(Ingram (1995), Theorem 2). Suppose f is a type (1) unimodal mapping of an interval [a, b] onto itself with critical point c, and q is a point in (c, p] such that f 2 (q) = q and f (a) = q.…”

“…(Ingram (1995), Theorem 7). Suppose f is a type (1) Type (1) unimodal maps go up and then come down; maps from the CIA model that we will investigate in Section 3 go down and then come up.…”

“…The theorems below from Ingram (1995Ingram ( , 2002Ingram ( , 2003 deal with properties of the bonding map f and the associated inverse limit or periodic points of f. The properties of the bonding map include monotone, unimodal, and type (1). We define these properties next.…”

Implicit Equilibrium Dynamics

“…is the only existing forward admissible orbit. 20 See Ingram (1995), in particular Theorems 2 and 3. Moreover, remember that, for any n > 1, lim…”

“…Next, we make use of a result of Ingram (1995), which we restate using our own notation: THEOREM [Ingram (1995, Th. 6)].…”