Decomposable Continuum Research Articles

Some theorems on decomposable continua

We prove some theorems on decomposable continua. In particular, we prove;(i) the property of being a Wilder continuum is not a Whitney reversible property,(ii) inverse limits of D⁎⁎-continua with surjective monotone upper semi-continuous bonding functions are D⁎⁎, and(iii) there exists a D⁎⁎-continuum which contains neither Wilder continua nor D⁎-continua.Also, we show the existence of a Wilder continuum containing no D⁎-continua and a D⁎-continuum containing no Wilder continua.

Nonblockers for hereditarily decomposable continua with the property of Kelley

Given a continuum X, let NB(F1(X)) be the hyperspace of nonblockers of F1(X). In this paper, we show that if X is hereditarily decomposable with the property of Kelley such that NB(F1(X)) is a continuum, then X is a simple closed curve. Thus, we characterize the simple closed curve as the unique hereditarily decomposable continuum with the property of Kelley X such that its hyperspace NB(F1(X)) is a continuum.

Some decomposable continua and Whitney levels of their hyperspaces

We introduce the new class of continua; D⁎⁎-continua. The classes of Wilder continua and D⁎-continua are strictly contained in the class of D⁎⁎-continua. Also, the class of D-continua properly contains the class of D⁎⁎-continua. Using D⁎⁎-continua, we give the negative answer to [2, Question 2]. Furthermore, we prove that being a Wilder continuum, being a D-continuum, being a D⁎-continuum and being a D⁎⁎-continuum are Whitney properties.

Formula omitted]-Homogeneous n-fold hyperspace suspensions

We study 12-homogeneity of the n-fold hyperspace suspension of continua. We prove that if X is a decomposable continuum and its n-fold hyperspace suspension is 12-homogeneous, then X must be continuum chainable. We also characterize 12-homogeneity of the hyperspace suspension for several classes of continua including: continua containing a free arc, atriodic and decomposable continua, decomposable irreducible continua about a finite set.

1\over 2-Homogeneous hyperspace suspensions

We continue the study of $1\over 2$-homogeneity of the hyperspace suspension of continua. We prove that if $X$ is a decomposable continuum and its hyperspace suspension is $1\over 2$-homogeneous, then $X$ must be continuum chainable. We also characterize

Any counterexample to Makienko’s conjecture is an indecomposable continuum

AbstractMakienko’s conjecture, a proposed addition to Sullivan’s dictionary, can be stated as follows: the Julia set of a rational function R:ℂ∞→ℂ∞ has buried points if and only if no component of the Fatou set is completely invariant under the second iterate of R. We prove Makienko’s conjecture for rational functions with Julia sets that are decomposable continua. This is a very broad collection of Julia sets; it is not known if there exists a rational function whose Julia set is an indecomposable continuum.

On [formula omitted]-homogeneous continua

A continuum is 1 2 -homogeneous provided there are exactly two orbits for the action of the group of homeomorphisms of the continuum onto itself. In this paper we study some relations between 1 2 -homogeneous continua and their set of cut points. We also prove that if X is a hereditarily decomposable continuum whose proper, nondegenerate subcontinua are arc-like, then X is 1 2 -homogeneous if and only if X is an arc. Suitable examples and counterexamples are given.

Mapping chainable continua onto dendroids

We prove that every chainable continuum can be mapped into a dendroid such that all point-inverses consist of at most three points. In particular, it follows that there exists a finite-to-one map from a hereditarily indecomposable continuum (the pseudoarc) onto a hereditarily decomposable continuum. This answers a question by J. Krasinkiewicz.

Higher dimensional aposyndetic decompositions

Let $X$ be a homogeneous, decomposable continuum that is not aposyndetic. The Aposyndetic Decomposition Theorem yields a cell-like decomposition of $X$ into homogeneous continua with quotient space $Y$ being an aposyndetic, homogeneous continuum. Assume the dimension of $X$ is greater than one. About 20 years ago the author asked the following questions: Can this aposyndetic decomposition raise dimension? Can it lower dimension? We answer these questions by proving the following theorem. Theorem. The dimension of the quotient space $Y$ is one.

Building blocks for quadratic Julia sets

We obtain results on the structure of the Julia set of a quadratic polynomial P with an irrationally indifferent fixed point zo in the iterative dynamics of P. In the Cremer point case, under the assumption that the Julia set is a decomposable continuum, we obtain a building block structure theorem for the corresponding Julia set J = J(P): there exists a nowhere dense subcontinuum B C J such that P(B) = B, B is the union of the impressions of a minimally invariant Cantor set A of external rays, B contains the critical point, and B contains both the Cremer point zo and its preimage. In the Siegel disk case, under the assumption that no impression of an external ray contains the boundary of the Siegel disk, we obtain a similar result. In this case B contains the boundary of the Siegel disk, properly if the critical point is not in the boundary, and B contains no periodic points. In both cases, the Julia set J is the closure of a skeleton S which is the increasing union of countably many copies of the building block B joined along preimages of copies of a critical continuum C containing the critical point. In addition, we prove that if P is any polynomial of degree d > 2 with a Siegel disk which contains no critical point on its boundary, then the Julia set J(P) is not locally connected. We also observe that all quadratic polynomials which have an irrationally indifferent fixed point and a locally connected Julia set have homeomorphic Julia sets.

Each locally one-to-one map from a continuum onto a tree-like continuum is a homeomorphism

In 1977 T. Maćkowiak proved that each local homeomorphism from a continuum onto a tree-like continuum is a homeomorphism. Recently, J. Rogers proved that each locally one-to-one (not necessarily open) map from a hereditarily decomposable continuum onto a tree-like continuum is a homeomorphism, and this paper removes “hereditarily decomposable" from the hypothesis of Rogers’ theorem.

A note on subcontinua of 𝛽[0,∞)-[0,∞)

Let M = ⊕ n ∈ ω I n M = { \oplus _{n \in \omega }}{I_n} be the topological sum of countably many copies of the unit interval I I . For any ultrafilter u ∈ ω ∗ u \in {\omega ^{\ast }} , we let M u = ∩ { cl β M ( ∪ { I n : n ∈ A } ) : A ∈ u } {M^u} = \cap \{ {\operatorname {cl} _{\beta M}}( \cup \{ {I_n}:n \in A\} ):A \in u\} . It is well known that M u {M^u} is a decomposable continuum with a very nice internal structure. In this paper, we show: (1) every nondegenerate subcontinuum of β [ 0 , ∞ ) − [ 0 , ∞ ) \beta [0,\infty ) - [0,\infty ) contains a copy of M u {M^u} for some u ∈ ω ∗ u \in {\omega ^{\ast }} . (2) there is no nontrivial simple point in Laver’s model for the Borel conjecture. The second answers a question posed by Baldwin and Smith negatively.

Continua in R∗

R ∗ is the Stone–Čech remainder of the real line. We prove that every decomposable continuum in R ∗ is a section of a standard continuum. Every indecomposable continuum in R ∗ is the union of a family of standard continua. About the general structures of continua in R ∗, the following statements are true: 1. (1) Let C and D be continua in R ∗. If one of them is indecomposable, then C ⊂ D, D ⊂ C or C ∩ D = Ø. 1. (2) R ∗ is hereditarily unicoherent, i.e., any intersection of continua in R ∗ is a continuum. Moreover, any intersection of indecomposable continua in R ∗ is indecomposable. 1. (3) The closure of the union of a chain of indecomposable continua in R ∗ is an indecomposable continuum. 1. (4) A point x of R ∗ is not a sub cutpoint iff { x} is the intersection of a maximal chain of nondegenerate indecomposable continua in R ∗. 1. (5) There are no Q-points in ω ∗ iff every composant of β[0, ∞)−[0, ∞) is the union of a strictly increasing sequence of proper indecomposable subcontinua. 1. (6) The principal NCF is equivalent to the statement that β[0, ∞)−[0, ∞) is the union of a strictly increasing sequence of proper indecomposable subcontinua. Now we know in ZFC that there are nine different continua in R ∗.

$2$-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.

β( X−{ x}) for X not locally connected

Let X be a metric continuum, xϵX, and Y = β( X −{ x}) − X. We prove that if Y is a continuum then Y is a decomposable continuum with one composant whenever X is not connected im kleinen at x. We give a sufficient condition for Y to be indecomposable. Finally we prove that Y is never a hereditarily indecomposable continuum although Y contains hereditarily indecomposable continua whenever X is hereditarily indecomposable.

𝛽([0,∞)) does not contain nondegenerate hereditarily indecomposable continua

Bellamy has shown that if A = [ 0 , ∞ ) A = \left [ {0,\infty } \right ) , then β A − A \beta A - A is an indecomposable continuum and every nondegenerate subcontinuum of β A − A \beta A - A can be mapped onto every metric continuum. Thus, it follows that every nondegenerate subcontinuum of β A − A \beta A - A contains a nondegenerate indecomposable continuum. We show, however, that no nondegenerate subcontinuum of β A − A \beta A - A is hereditarily indecomposable. Thus, every nondegenerate subcontinuum of β A − A \beta A - A contains a decomposable continuum as well as a nondegenerate indecomposable continuum.

Cell-Like Decompositions of Homogeneous Continua

Certain decompositions of homogeneous continua are shown to be cell-like. In particular, the aposyndetic decomposition described by F. B. Jones of a homogeneous, decomposable continuum is cell-like, and we prove that any homogeneous decomposable continuum admits a continuous decomposition into mutually homeomorphic, indecomposable, homogeneous, cell-like terminal continua so that the quotient space is an aposyndetic homogeneous continuum.

Cell-like decompositions of homogeneous continua

Certain decompositions of homogeneous continua are shown to be cell-like. In particular, the aposyndetic decomposition described by F. B. Jones of a homogeneous, decomposable continuum is cell-like, and we prove that any homogeneous decomposable continuum admits a continuous decomposition into mutually homeomorphic, indecomposable, homogeneous, cell-like terminal continua so that the quotient space is an aposyndetic homogeneous continuum.

Subcontinua with degenerate tranches in hereditarily decomposable continua

A hereditarily decomposable, irreducible, metric continuum M M admits a mapping f f onto [ 0 , 1 ] [0,1] such that each f − 1 ( t ) {f^{ - 1}}(t) is a nowhere dense subcontinuum. The sets f − 1 ( t ) {f^{ - 1}}(t) are the tranches of M M and f − 1 ( t ) {f^{ - 1}}(t) is a tranche of cohesion if t ∈ { 0 , 1 } t \in \{ 0,1\} or f − 1 ( t ) = C1 ( f − 1 ( [ 0 , t ) ) ) ∩ C1 ( f − 1 ( ( t , 1 ] ) ) {f^{ - 1}}(t) = {\text {C1}}({f^{ - 1}}([0,t))) \cap {\text {C1}}\,({f^{ - 1}}((t,1])) . The following answer a question of Mahavier and of E. S. Thomas, Jr. Theorem. Every hereditarily decomposable continuum contains a subcontinuum with a degenerate tranche. Corollary. If in an irreducible hereditarily decomposable continuum each tranche is nondegenerate then some tranche is not a tranche of cohesion. The theorem answers a question of Nadler concerning arcwise accessibility in hyperspaces.

𝜆 connected plane continua

A continuum M is said to be λ {\mathbf {\lambda }} connected if any two distinct points of M can be joined by a hereditarily decomposable continuum in M. Recently this generalization of arcwise connectivity has been related to fixed point ptoblems in the plane. In particular, it is known that every λ {\mathbf {\lambda }} connected nonseparating plane continuum has the fixed point property. The importance of arcwise connectivity is, to a considerable extent, due to the fact that it is a continuous invariant. To show that λ {\mathbf {\lambda }} connectivity has a similar feature is the primary purpose of this paper. Here it is proved that if M is a λ {\mathbf {\lambda }} connected continuum and f is a continuous function of M into the plane, then f ( M ) f(M) is λ {\mathbf {\lambda }} connected. It is also proved that every semiaposyndetic plane continuum is λ {\mathbf {\lambda }} connected.