Tree-like Continuum Research Articles

Meager composants of tree-like continua

A subset M of a continuum X is called a meager composant if M is maximal with respect to the property that every two of its points are contained in a nowhere dense subcontinuum of X. Motivated by questions of Bellamy, Mouron and Ordoñez, we show that no tree-like continuum has a proper open meager composant, and that every tree-like continuum has either 1 or 2ℵ0 meager composants. We also prove a decomposition theorem: If X is tree-like and every indecomposable subcontinuum of X is nowhere dense, then the partition of X into meager composants is upper semi-continuous and the space of meager composants is a dendrite.

Tree-like continua and dendrites on set-valued inverse limits

The purpose of this paper is to give conditions under which, the inverse limit with set-valued functions is a tree-like continuum and in some cases, a dendrite.

Mixing homeomorphisms and indecomposability

In this paper we show that if h:X→X is a mixing homeomorphism on a G-like continuum, then X must be indecomposable and if X is finitely cyclic, then X must be 1n-indecomposable for some natural number n. Furthermore, we give an example of a mixing homeomorphism on a hereditary decomposable tree-like continuum.

Every continuous action of a compact group on a uniquely arcwise connected continuum has a fixed point

We are dealing with the question whether every group or semigroup action (with some additional property) on a continuum (with some additional property) has a fixed point. One of such results was given in 2009 by Shi and Sun. They proved that every nilpotent group action on a uniquely arcwise connected continuum has a fixed point. We are seeking for this type of results with, e.g., commutative, compact, or torsion groups and semigroups acting on dendrites, dendroids, $$\lambda $$ -dendroids and uniquely arcwise connected continua. We prove that every continuous action of a compact or torsion group on a uniquely arcwise connected continuum has a fixed point. We also prove that every continuous action of a compact and commutative semigroup on a uniquely arcwise connected continuum or on a tree-like continuum has a fixed point.

A fixed-point-free map of a tree-like continuum induced by bounded valence maps on trees

Towards attaining a better working understanding of fixed points of maps of tree-like continua, Oversteegen and Rogers constructed a tree-like continuum with a fixed-point-free self-map, described explicitly in terms of inverse limits. Specifically, they developed a sequence of trees $T_n$, $n \in \mathbb{N}$ and maps $f_n$ and $g_n$ from $T_{n+1}$ to $T_n$ for each $n$, such that the $g_n$ maps induce a fixed-point-free self-map of the inverse limit space $\varprojlim (T_n,f_n)$. The complexity of the trees and the valences of the maps in their example all grow exponentially with $n$, making it difficult to visualize and compute with their space and map. We construct another such example, in which the maps $f_n$ and $g_n$ have uniformly bounded valence, and the trees $T_n$ have a simpler structure.

An uncountable family of copies of a non-chainable tree-like continuum in the plane

A well-known theorem of R. L. Moore states that the plane does not contain an uncountable family of pairwise disjoint triods. In 1974, Ingram demonstrated that the same is not true for non-chainable tree-like continua. The continua in Ingram’s family are not pairwise homeomorphic, making the example less applicable to the study of homogeneous continua in the plane. In this paper, we construct a non-chainable tree-like continuum X X such that the product of X X with the Cantor set can be embedded in the plane.

Fixed points in indecomposable 𝑘-junctioned tree-like continua

LetMMbe an indecomposablekk-junctioned tree-like continuum. Letffbe a map ofMMthat sends each composant ofMMinto itself. Using an argument of O. H. Hamilton, we prove thatffhas a fixed point.

On Non-Metric Confluently Tree-Like Continua

It is proved, using inverse systems, that if X is a non-metric confuently tree-like continuum, then it is confuently sigma-tree-representable and has the fixed point property.

On lifting properties for confluent mappings

Known results about lifting of paths for covering, light open and light confluent mappings are in some sense extended for all confluent mappings with the domain being a continuum having the arc property of Kelley. As an application we prove that each confluently tree-like continuum has the fixed point property.

Hereditarily Unicoherent Continua and Their Absolute Retracts

We investigate absolute retracts for classes of hereditarily unicoherent continua, tree-like continua, λdendroids, dendroids and some other related ones. The main results are: (1) the inverse limits of trees with confluent bonding mappings are absolute retracts of hereditarily unicoherent continua; (2) each tree-like continuum is embeddable in a special way in a tree-like absolute retract for the class of hereditarily unicoherent continua; (3) a dendroid is an absolute retract for hereditarily unicoherent continua if and only if it can be embedded as a retract into the Mohler-Nikiel universal smooth dendroid.

Tree-like continua and 2-to-1 maps

It is not known if there is a 2-to-1 map from a continuum onto a tree-like continuum. In fact, it is not known if there is a 2-to-1 map onto a hereditarily decomposable tree-like continuum. We show that the domain of such a map would have to contain an indecomposable continuum.

Exactly 𝑘-to-1 maps and hereditarily indecomposable tree-like continua

In 1947, W.H. Gottschalk proved that no dendrite is the continuous, exactly k k -to-1 image of any continuum if k ≥ 2 k \geq 2 . Since that time, no other class of continua has been shown to have this same property. It is shown that no hereditarily indecomposable tree-like continuum is the continuous, exactly k k -to-1 image of any continuum if k ≥ 2 k \geq 2 .

No arc-connected treelike continuum is the 2-to-1 image of a continuum

In 1940, O. G. Harrold showed that no arc can be the exactly 2-to-1 continuous image of a metric continuum, and in 1947 W. H. Gottschalk showed that no dendrite is a 2-to-1 image. In 2003 we show that no arc-connected treelike continuum is the 2-to-1 imag

Tree-like continua do not admit expansive homeomorphisms

A homeomorphism h: X → X is called expansive provided that for some fixed c > 0 and every x, y ∈ X there exists an integer n, dependent only on x and y, such that d(h n (x), h n (y)) > c. It is shown that if X is a tree-like continuum, then h cannot be expansive.

A self-map of a tree-like continuum with no invariant indecomposable subcontinuum

The Bellamy's construction of a fixed-point-free map on a tree-like continuum is modified in such a way that no indecomposable subcontinuum is mapped into itself.

A hereditarily indecomposable tree-like continuum without the fixed point property

A hereditarily indecomposable tree-like continuum without the fixed point property is constructed. The example answers a question of Knaster and Bellamy.

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 non-treelike continuum that is not the 2-to-1 image of any continuum

Some thirteen years ago S. B. Nadler, Jr. and L. E. Ward, Jr., asked if any treelike continuum could be the 2-to-1 image of a continuum. In fact, it has been conjectured that the property of being treelike characterizes those continua that are not the 2-to-1 image of any continuum. But the characterization must be something else; this paper shows that many pseudo-solenoids are not the 2-to-1 image of any continuum.

A Periodic Point Free Homeomorphism of a Tree-Like Continuum

An example of a homeomorphism without periodic points is constructed on a tree-like continuum.

On uncountable collections of continua and their span

We prove that if the Euclidean plane $ℝ^2$ contains an uncountable collection of pairwise disjoint copies of a tree-like continuum X, then the symmetric span of X is zero, sX = 0. We also construct a modification of the Oversteegen-Tymchatyn example: for