Periodic homeomorphisms of chainable continua

Lewis, Wayne

doi:10.4064/fm-117-1-81-84

Cited by 18 publications

(8 citation statements)

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“…Chainable continua provide a context that is suitably linear for portions of Sarkovskii's proof to be extended. However, Sarkovskii's Theorem does not hold for maps of indecomposable chainable continua (see [5] and [6]). It is for hereditarily decomposable chainable continua that the folding patterns of maps with prescribed periodic orbits mimic those of maps of the real line.…”

Section: Kuratowski Mapsmentioning

confidence: 96%

The Sarkovskii order for periodic continua II

Ryden¹

2007

Topology and its Applications

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In this paper it is shown that the existence of three maximal proper periodic continua for a map of a hereditarily decomposable chainable continuum onto itself implies the existence of a maximal proper periodic continuum with odd period greater than one. Hence, while the periods of such continua do follow the Sarkovskii order apart from the case in which the ambient space is the union of two maximal proper periodic continua with period two, for any nondegenerate terminal segment of the Sarkovskii order that fails to contain an odd integer greater than one, there does not exist a map of a hereditarily decomposable chainable continuum onto itself for which the set of all periods of such continua is the prescribed terminal segment. It is also shown that, for any terminal segment of the Sarkovskii order that does contain an odd integer greater than one, there is a map of [0, 1] onto itself for which the set of all periods of such continua is the prescribed terminal segment.

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Section: Kuratowski Mapsmentioning

confidence: 96%

The Sarkovskii order for periodic continua II

Ryden¹

2007

Topology and its Applications

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“…Lewis [8] proved that for an n > 1 there exists a period-^ homeomorphism on the pseudo-arc. Since a periodic homeomorphism generates a finite and therefore compact group, we have the following.…”

Section: = {D} = {«-'Imumentioning

confidence: 99%

Small Compact Actions on Chainable Continua

Toledo

1986

Can. j. math.

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1. Introduction. In 1931, Newman [9] showed that a connected manifold cannot accept arbitrarily small period-n homeomorphisms, for any n > 1. In this paper we are concerned with the existence of chainable continua with arbitrarily small homeomorphisms.For a long time the only known periodic homeomorphisms of chainable continua had periods 1, 2 or 4 [4]. Recently, Wayne Lewis [8] showed that the pseudo-arc admits periodic homeomorphisms of every order, as well as p-adic cantor group actions. We will see that such homeomorphisms can be made arbitrarily small.In Section 4, a different chainable indecomposable continuum accepting arbitrarily small period-2 homeomorphisms is constructed.

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“…We describe briefly here an example of a map of a chainable (indecomposable) continuum into itself having a point of period 3 but none of period 2, and then proceed with the proof of the theorem. A better example was given earlier by Wayne Lewis [6], who showed that for every « there is a chainable continuum with a periodic homeomorphism of period «. The general outline of the proof of Theorem 1.1 is patterned on that of Block, Guckenheimer, Misiurewicz and Young [1] as described in Devaney's book [2].…”

Section: Introductionmentioning

confidence: 96%