Chainable Continuum Research Articles

On Weakly Chainable Inverse Limits with Simplicial Bonding Maps

In this paper we prove that if an atriodic and weakly chainable continuum is the inverse limit of trees with simplicial bonding maps, then it is chainable.

On weakly chainable inverse limits with simplicial bonding maps

In this paper we prove that if an atriodic and weakly chainable continuum is the inverse limit of trees with simplicial bonding maps, then it is chainable.

Accessible Points of Hereditarily Decomposable Chainable Continua

In this paper it is proven that a chainable continuum $X$ can be embedded in the plane in such a way that every point is accessible from its complement if and only if it is Suslinean. An example is shown of an hereditarily decomposable chainable continuum which cannot be embedded in the plane in such a way that each endpoint is accessible.

Accessible points of hereditarily decomposable chainable continua

In this paper it is proven that a chainable continuum X X can be embedded in the plane in such a way that every point is accessible from its complement if and only if it is Suslinean. An example is shown of an hereditarily decomposable chainable continuum which cannot be embedded in the plane in such a way that each endpoint is accessible.

Periodic points for homeomorphisms of hereditarily decomposable chainable continua

In this paper it is shown that homeomorphisms of hereditarily decomposable chainable continua cannot have periodic points whose periods are not powers of two. Examples show that for each power of two there is a hereditarily decomposable chainable continuum and a homeomorphism of it which has a periodic point of period that power of two.

Absolute endpoints of chainable continua

An endpoint of chainable continuum is a point at which it is always possible to start chaining that continuum. Some endpoints appear to have the property that one is almost "forced" to start (or finish) the chaining at these points. This paper characterizes these "absolute endpoints", and this characterization is used to show that in a chainable continuum locally connected at p p is equivalent to connected im kleinen at p p .

An atriodic tree-like continuum with positive span which admits a monotone mapping to a chainable continuum

An atriodic tree-like continuum with positive span which admits a monotone mapping to a chainable continuum

The Preservation of Atriodicity by Semiconfluent Mappings

Grace and Vought have proven that the image of a chainable continuum under a semiconfluent mapping is atriodic. This motivates the question of whether the semiconfluent image of an atriodic continuum is atriodic. In this paper this question is answered affirmatively.

The preservation of atriodicity by semiconfluent mappings

Grace and Vought have proven that the image of a chainable continuum under a semiconfluent mapping is atriodic. This motivates the question of whether the semiconfluent image of an atriodic continuum is atriodic. In this paper this question is answered affirmatively.

Small Compact Actions on Chainable Continua

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.

The depth of tranches in 𝜆-dendroids

According to the well-known theory of Kuratowski, any hereditarily decomposable chainable continuum admits a decomposition into tranches. These tranches are themselves chainable and thus admit decompositions into their own tranches. We may thus define nested sequences { T α } \{ {T_\alpha }\} of tranches-within-tranches, indexed by countable ordinals α \alpha , and finally terminating in a singleton set. E. S. Thomas, Jr. has asked whether, for a given continuum C C , there is a countable ordinal bound on the length of all such nests { T α } \{ {T_\alpha }\} in C C . We answer Thomas’s question in the affirmative. By generalizing the definitions, we obtain the same result for λ \lambda -dendroids. We also answer, for chainable continua, a related question of Illiadis.

A chainable continuum not homeomorphic to an inverse limit on [0, 1] with only one bonding map

A chainable continuum not homeomorphic to an inverse limit on [0, 1] with only one bonding map

Open maps of chainable continua

It is apparently “well known” that the image of the closed unit interval under an open map is homeomorphic to the closed unit interval (see [13], [11], and [15]). In this paper, we generalize this result to chainable continua. In particular, the fact that the open continuous image of a chainable continuum is also chainable is proved, answering a question of A. Lelek (see [10]). This fact, as well as its proof, implies that the open continuous image of the pseudo-arc is also a pseudo-arc. An additional corollary (of the proof) is that a local homeomorphism of a chainable continuum is actually a homeomorphism. The proofs are all very elementary.

Inverse limits on graphs and monotone mappings

In 1935, Knaster gave an example of an irreducible continuum (i.e. compact connected metric space) K which can be mapped onto an arc so that each point-preimage is an arc. The continuum K is chainable (or arc-like). In this paper it is shown that every one-dimensional continuum M is a continuous image, with arcs as point-preimages, of some one-dimensional continuum $M’$. Moreover, if M is G-like, for some collection G of graphs, then $M’$ can be chosen to be G-like. A corollary is that every chainable continuum is a continuous image, with arcs as point-inverses, of a chainable (and hence, by a theorem of Bing, planar) continuum. These investigations give rise to the study of certain special types of inverse limit sequences on graphs.

A characterization of hereditarily indecomposable continua

Hereditarily indecomposable continua are characterized by giving a condition which some defining sequence for the continua must satisfy.

Arc Components of Certain Chainable Continua

AbstractIt is shown that if a chainable continuum has exactly two arc components, then one of them is an arc and the other is a half-ray.

The representation of chainable continua with only two bonding maps.

DEFINITION. A set S of continuous functions of [0, 1 ] into [0, 1 ] is called a set of bonding maps if every chainable continuum can be obtained as the inverse limit of an inverse mapping system each of whose coordinate spaces is [0, 1] and each of whose bonding maps is in S. It is shown in [1 ] that a dense set is complete. Jolly and Rogers in [3] demonstrate a complete set with only four elements. It follows from the theorem of Jarnik and Knichal [2 ] that there is a complete set with only two elements. Mahavier proves in [4] that every complete set must have at least two elements and therefore two is the minimum number. In this note, we prove the following:

On embedding cones over circularly chainable continua

It is known that every chainable continuum (i.e., every 1-cell-like continuum) can be embedded in the plane [1]. Every 2-cell-like continuum can be embedded in E4 [5]. It is natural to ask whether every 2-cell-like continuum can be embedded in El. Fearnley [3] has stated that the cone over a dyadic solenoid cannot be embedded in Et, but no proof has appeared. In this note we show that the cone over a nonplanar circularly chainable continuum cannot be embedded in E3. This provides a large class of 2-cell-like continua not embeddable in E3. A definition of what it means to say that a continuum is P-like may be found in [6]. For any continuum M, the cone over M is the decomposition space C(M) of the upper semicontinuous decomposition of MX [0, 11 whose only nondegenerate element is MX { 1). We call the point MX { 1 1 of C(M) the vertex of C(M) and the set of points { (m, 0), m in M}, we call the base of C(M). It is easy to see that if P is a continuum and M is P-like then C(M) is C(P)-like. Therefore every cone over a circularly chainable continuum is 2-cell-like.

A Chainable Continuum not Homeomorphic to an Inverse Limit on [ 0, 1 ] with only One Bonding Map

A Chainable Continuum not Homeomorphic to an Inverse Limit on [ 0, 1 ] with only One Bonding Map

A chainable continuum not homeomorphic to an inverse limit on [0,1] with only one bonding map

Introduction. In this note by a continuum we mean a nondegenerate, compact, connected metric space. It is known (see [4] or [6]) that each chainable continuum is homeomorphic to the inverse limit of a sequence of maps from [0, 1] onto [0, 1], and it is not difficult to show, conversely, that if each of fi, f2, * * * maps [0, 1 ] on [0, 1], then the inverse limit of this sequence is a chainable continuum. G. W. Henderson has recently shown [5] that there is a map of [0, 1 ] on [0, 1 ] such that the inverse limit with it as the only bonding map is a pseudo arc. We observe that not every chainable continuum can be so represented (using only one bonding map) but that each chainable continuum can be embedded in such an inverse limit. If each term of the sequence ax= {fi, f2, * } maps [0, 1] on [0, 1 ] then the inverse limit of the sequence ax, denoted by lim a, is the subspace of the infinite cartesian product [0, 1] consisting of all number sequences x1, x2, * * -such that for each positive integer i, fi(xi+1) =xi. Iff maps [0, 1] on [0, 1], then limf denotes lim ax where a = {f, f, * * } . For a discussion of properties of inverse limit spaces see [3 ].