Proper Subcontinuum Research Articles

Non-cut ordered arcs of the hyperspace of subcontinua

Let C(X) be the hyperspace of all subcontinua of a continuum X topologized by Hausdorff metric. For a non-empty closed subset A of a continuum X, consider the following properties: A is a strong non-cut subset, non-block subset, weak non-block subset, shore subset, not a strong center, and non-cut subset of X. The aim of this paper is to study the conditions under which an ordered arc from a singleton to a proper subcontinuum of a continuum X has one of these properties in C(X).

Some characterizations of the internal structure of Whitney levels

Let X be a continuum, and let C(X) denote the hyperspace of all subcontinua of X. It is known that there exist monotone maps μ from C(X) into [0, ∞) such that μ({x}) = 0 for each x ∈ X, and if A is a proper subcontinuum of B, then μ(A) < μ(B). The subcontinua μ−1(t) of C(X) are called Whitney levels of C(X). In this paper, a class of closed subsets of X is employed to characterize the Whitney levels of C(X) possessing one of the following properties: irreducibility, decomposability, being a Wilder continuum, aposyndesis, semiaposyndesis, n-aposyndesis, finite aposyndesis, and connectedness colocal.

Nonblockers in homogeneous continua

Given a continuum X, an element A∈2X∖{X} is said to be a set that does not block singletons of X provided that for every point x∈X∖A, there exists an order arc α:[0,1]→C(X) such that α(0)={x}, α(1)=X and α(t)∩A=∅ for all t∈[0,1). If A does not block singletons of X, we say that A is a nonblocker of X. In this paper, we present some properties about the sets that do not block singletons and, for a homogeneous curve X, we use the set NB(F1(X)) to construct a continuous decomposition of X that is a homogeneous curve and a Whitney level of C(X), where its elements are homogeneous, acyclic, hereditarily unicoherent and indecomposable continuum. Also, we prove that if X is a continuum such that NB(F1(X))=F1(X), then the next three properties are equivalent: 1) X is a homogeneous continuum, 2) X has the property that for all x∈X, there exists a proper subcontinuum A of X such that x∈Int(A), and 3) X≈S1.

Boundaries in hyperspaces II

Let C(X) be the hyperspace of subcontinua of a continuum X, and let A be a proper subcontinuum of X. We continue the study of the boundary Bd(C(A)) of C(A) in C(X). We show properties of continua for which these boundaries are 2-cells. Moreover, we define a natural boundary function of hyperspaces, and we characterize the simple closed curve as the unique arcwise connected continuum for which this function is continuous. Results in this paper, as those in [3], extend the study of semi-boundaries in hyperspaces done by A. Illanes in [4].

On non-metric continua that support Whitney maps

We discuss the construction of non-metric continua that support Whitney maps and their properties. We indicate a technique for producing non-metric indecomposable continua that support Whitney maps from certain compact totally disconnected spaces each of which allows a self-homeomorphism all of whose orbits are dense. These are non-metric examples that have the property that each proper subcontinuum is metric. Both perfectly normal and non-perfectly normal examples are constructed. We describe techniques for producing large collections of non-homeomorphic continua that support Whitney maps. An example of a continuum every non-degenerate subcontinuum of which is non-metric that supports a Whitney map is constructed; an example of a continuum that does not support a Whitney map which is the union of two subcontinua each of which supports a Whitney map is constructed.

Boundaries in hyperspaces

Let C(X) be the hyperspace of subcontinua of a continuum X. Given a proper subcontinuum A of X, we study the boundary Bd(C(A)) of C(A) in C(X). We show that Bd(C(A)) is always arcwise connected, and then a subcontinuum of C(X). We give conditions under which Bd(C(A)) coincides with the family of all subcontinua of A intersecting the boundary of A in X. We also study conditions for the local connectedness and the unicoherence of sets of the form Bd(C(A)).

The Sarkovskii order for periodic continua

Suppose f is a map of a continuum X onto itself. A periodic continuum of f is a subcontinuum K of X such that f n [ K ] = K for some positive integer n. A proper periodic continuum of f is a periodic continuum of f that is a proper subcontinuum of X. A proper periodic continuum of f is maximal if and only if X is the only periodic continuum that properly contains it. In this paper it is shown that the maximal proper periodic continua of a map of a hereditarily decomposable chainable continuum onto itself follow the Sarkovskii order, provided the maximal proper periodic continua are disjoint. The case in which the Sarkovskii order does not hold reduces to the scenario in which the map's domain is the union of two overlapping period-two continua, each of which is maximal.

Monotone images of W-sets and hereditarily weakly confluent images of continua

A proper subcontinuum H of a continuum X is said to be a W-set provided for each continuous surjective function f from a continuum Y onto X, there exists a subcontinuum C of Y that maps entirely onto H. Hereditarily weakly confluent (HWC) mappings are those with the property that each restriction to a subcontinuum of the domain is weakly confluent. In this paper, we show that the monotone image of a W-set is a W-set and that there exists a continuum which is not in class W but which is the HWC image of a class W continuum.

Buried points and lakes of Wada Continua

Let $f:\hat \mathbf C\rightarrow \hat \mathbf C$ be a rational map of degree $n\geq 3$ and with exactly two critical points. Assume that the Julia set $J(f)$ is a proper subcontinuum of $\hat \mathbf C$ and there is no completely invariant Fatou component under the iterates $f^{2}$. It is shown that if there is no buried points in $J(f)$, then the Julia set $J(f)$ is a Lakes of Wada continuum, and hence is either an indecomposable continuum or the union of two indecomposable continua.

Reduced [formula omitted] -to-1 maps and graphs with no [formula omitted] -to-1 cut set

A reduced map is a continuous function such that the preimage of every proper subcontinuum of the range is disconnected. A k -to-1 cut set is a finite subset B of a continuum X such that X\\B has at least k|B| components. If a continuum is the image of a reduced at most k -to-1 map from a continuum, then it does not contain a k -to-1 cut set. A connected graph is the image of a reduced at most k -to-1 map from a continuum if and only if the graph does not contain a k -to-1 cut set. A connected graph is the image of a reduced k -to-1 map from a continuum if and only if the graph does not contain a k -to-1 cut set or an endpoint.

Irreducibility of inverse limits on intervals

A procedure for obtaining points of irreducibility for an inverse limit on intervals is developed. In connection with this, the following are included. A se- miatriodic continuum is defined to be a continuum that contains no triod with interior. Characterizations of semiatriodic and unicoherent continua are given, as well as necessary and sufficient conditions for a subcontinuum of a semiatriodic and unicoherent continuum M to lie within the interior of a proper subcontinuum ofM.

Diophantine Conditions Imply Critical Points on the Boundaries of Siegel Disks of Polynomials

Let f be a polynomial map of the Riemann sphere of degree at least two. We prove that if f has a Siegel disk G on which the rotation number satisfies a diophantine condition, then either the boundary B of G contains a critical point or B is a Lakes of Wada indecomposable continuum with one of the lakes containing a critical point. Consequently, if the boundary B of G has only 2 complementary domains, then B contains a critical point. We also show, without any assumption on the rotation number, that each proper nondegenerate subcontinuum of the boundary B of G is tree-like, and any other bounded complementary domain of B is a preperiodic component of the grand orbit of G. Finally, we establish some conditions under which B contains no periodic point.

An atriodic simple-4-od-like continuum which is not simple-triod-like

The paper contains an example of a continuum K K such that K K is the inverse limit of simple 4 4 -ods, K K cannot be represented as the inverse limit of simple triods and each proper subcontinuum of K K is an arc.

$2$-to-$1$ maps on hereditarily indecomposable continua

Suppose / is a 2-to-l continuous map from the hereditarily in- decomposable continuum X onto a continuum Y. In order for it to be the case that each proper subcontinuum C in Y has as its preimage two disjoint continua each of which / maps homeomorphically onto C , it is obviously necessary that / satisfy the condition that each nondense connected subset of Y has disconnected preimage. We show that this condition is also sufficient, and thus any 2-to-l continuous map from a hereditarily indecomposable con- tinuum satisfying this condition must be confluent and have an image that is hereditarily indecomposable.

Stable extensions of homeomorphisms on the pseudo-arc

We prove the following: Theorem. If P ′ P’ is a proper subcontinuum of the pseudoarc P , h ′ P,\,h’ is a homeomorphism from P ′ P’ onto itself, and Θ \Theta is an open set in P P that contains P ′ P’ , then there is a homeomorphism h h from P P onto itself such that h | P ′ = h ′ h|P’ = h’ and h ( x ) = x h(x) = x for x ∉ Θ x \notin \Theta .

Extremal continua: A class of non-separating subcontinua

The notion of a terminal continuum, as defined by D.E. Bennett and J.B. Fugate, is used to introduce extremal continua, a class of non-separating subcontinua of a continuum. An extremal continuum can be characterized as a proper subcontinuum Y of a metric continuum X with the property that Y contains a point of irreducibility of each irreducible subcontinuum of X that meets Y. If Y is an extremal subcontinuum of X, then Y does not separate any subcontinuum of X containing Y; moreover, if Y is a proper subcontinuum of X and Y does not cut any subcontinuum of X containing Y, then Y is extremal in X.

Extension theorems for some classes of continua

It is proved that every mapping from a proper subcontinuum of a hereditarily unicoherent continuum X onto the Knaster's indecomposable continuum (onto a cone over a zerodimensional compact metric set) can be extended to a mapping defined on X. Similarly, every mapping from a proper subcontinuum of a hereditarily indecomposable continuum onto a pseudoarc can be extended to a mapping defined on the whole space. Both of the above results are generalizations of the author's earlier results to the nonmetric case. As a consequence it is obtained that a pseudoarc is continuously n-homogeneous.

Monotone decompositions of IUC continua

For the class of hereditarily unicoherent metric continua a spectrum of monotone decompositions has been developed by several authors which "improves" the quotient spaces. This spectrum is developed for a broader class of continua, namely continua with property IUC. A metric continuum M M has property IUC provided each proper subcontinuum of M M with interior is unicoherent. One important result which develops is that semiaposyndetic IUC continua are hereditarily arcwise connected. Also the notion of smoothness is studied for IUC continua.

A Nonmetric Indecomposable Homogeneous Continuum Every Proper Subcontinuum of which is an ARC

A Nonmetric Indecomposable Homogeneous Continuum Every Proper Subcontinuum of which is an ARC

A nonmetric indecomposable homogeneous continuum every proper subcontinuum of which is an arc

We construct a nonmetric indecomposable homogeneous continuum with the property that each of its proper subcontinua is an arc.