Chaos, Periodicity, and Snakelike Continua

Barge, Marcy; Martin, Joe

doi:10.2307/1999705

Cited by 38 publications

(72 citation statements)

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“…Namely, consider the family of functions fa: [0, 1] -► [0, 1] given by fa(x) = a(\ -\2x -l|)/2, where a is a parameter belonging to (1,2]. Using some estimations from [23] one can describe Ch(^) in an effective way.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Paradoxical functions on the interval

López¹

1994

Proc. Amer. Math. Soc.

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Abstract. In this paper it is shown that any expanding with Lipschitz derivative function / has a contradictory behaviour from the point of view of chaos in the sense of Li and Yorke. On the one hand it cannot generate scrambled sets of positive Lebesgue measure. On the other hand the two-dimensional set Ch(/) including the pairs (x, y) such that {x, y} is a scrambled set of / has positive measure. In fact, both the geometric structure (almost everywhere) and measure of Ch(/) can be explicitly obtained.

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Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Paradoxical functions on the interval

López¹

1994

Proc. Amer. Math. Soc.

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“…In [2], [3], Barge and Martin proved results which yield the equivalence of (l)-(4) in the following theorem.…”

Section: A For Continuous Maps Of the Circle With Periodic Points Tmentioning

confidence: 89%

“…Then/ 2 has no fixed points in (0, t) and f 2 • A standard compactness argument shows that the integer n = n(J) in the lemma depends only on the length of /. A piecewise monotone map / : / -» / is called strongly transitive [7] if for every interval / c /, there is an n such that {J" k=o f k {J) = I.…”

Section: Suppose Then That/has Non-zero Degree It Suffices To Show Tmentioning

confidence: 99%

Transitivity and the centre for maps of the circle

Coven

Mulvey

1986

Ergod. Th. Dynam. Sys.

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“…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].…”

Section: Introductionmentioning

confidence: 99%

Periodicity and Indecomposability

Ingram¹

1995

Proceedings of the American Mathematical Society

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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

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Chaos, Periodicity, and Snakelike Continua

Cited by 38 publications

References 0 publications

Paradoxical functions on the interval

Paradoxical functions on the interval

Transitivity and the centre for maps of the circle

Periodicity and Indecomposability

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