The topology of continua that are approximated by disjoint subcontinua

Mouron, Christopher

doi:10.1016/j.topol.2008.08.010

Cited by 6 publications

(9 citation statements)

References 4 publications

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“…(3) φ : Q −→ D where Q is a closed φ-mixing set for X and D is infinite; (4) : X −→ X/φ is the natural quotient map. Define Q 0 = {Y ∈ C(X ) | there exists n ∈ N such that (h n (Y ) ∩ Q) = D}.…”

Section: Resultsmentioning

confidence: 99%

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Mixing sets, positive entropy homeomorphisms and non-Suslinean continua

Mouron¹

2015

Ergod. Th. Dynam. Sys.

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Let $\{h_{n}\}_{n\in \mathbb{N}}$ be a sequence of self maps on a metric space $X$. We say that $Q\subset X$ is a mixing set on $\{h_{n}\}_{n\in \mathbb{N}}$ if for every $V\subset Q$ such that $\text{int}_{Q}(V)\not =\emptyset$ and every $\unicode[STIX]{x1D716}>0$ there exists $N=N(V,\unicode[STIX]{x1D716})$ such that $\text{d}_{H}(Q,h_{n}(V))<\unicode[STIX]{x1D716}$ for all $n\geq N$, where $\text{d}_{H}$ is the Hausdorff metric. It is shown that if $Q$ is a non-degenerate mixing set for a sequence of homeomorphisms on a continuum, then the continuum must be non-Suslinean. This is generalized to the notion of a $\unicode[STIX]{x1D719}$-mixing set. As a corollary, it is shown that a continuum must be non-Suslinean in order to admit a positive entropy homeomorphism.

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Section: Resultsmentioning

confidence: 99%

“…PROPOSITION 12. [4] Suppose that f : X −→ Y and g : Y −→ Z are maps. Let U be a subset of Z , V be a component of g −1 (U ) and C be component of f −1 (V ).…”

Section: Non-suslinean Continuamentioning

confidence: 99%

Mixing sets, positive entropy homeomorphisms and non-Suslinean continua

Mouron¹

2015

Ergod. Th. Dynam. Sys.

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“…Suppose X is a singular dense meager composant of the graph-like continuum Y . To show Y is indecomposable, by [11,Theorem 32] it suffices to show there is a sequence (X i ) of pairwise disjoint continua in Y such that d H (X i , Y ) → 0, where d H is the Hausdorff distance induced by a metric d on Y . It is unknown whether Y must have infinitely many meager composants (see Question 1 in Section 7), but in any case the continua X i can be selected from X.…”

Section: Proof Of Corollary 13mentioning

confidence: 99%

Singularities of meager composants and filament composants

Lipham

2019

Topology and its Applications

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Suppose Y is a continuum, x ∈ Y , and X is the union of all nowhere dense subcontinua of Y containing x. Suppose further that there exists y ∈ Y such that every connected subset of X limiting to y is dense in X. And, suppose X is dense in Y . We prove X is homeomorphic to a composant of an indecomposable continuum, even though Y may be decomposable. An example establishing the latter was given by Christopher Mouron and Norberto Ordoñez in 2016. If Y is chainable or, more generally, an inverse limit of identical topological graphs, then we show Y is indecomposable and X is a composant of Y . For homogeneous continua we explore similar problems which are related to a 2007 question of Janusz Prajs and Keith Whittington. (David Sumner Lipham) 1 Whenever Y is a space of which X is a subspace, and A ⊆ X, then we write A for the closure of A in X, and A for the closure of A in Y .

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“…In this section, we present [15, Example 3], and show it is a counterexample to [12,Theorem 30]. More precisely, we show the example is a Suslinian continuum which can be approximated by a sequence of disjoint arcs. "…”

Section: Tymchatyn's Counterexamplementioning

confidence: 99%

“…Tymchatyn of a Suslinian continuum which can be approximated by a sequence of mutually disjoint arcs. The example shows that [12,Theorem 30] and its generalization [11,Theorem 17] are false. Seidler's conjecture thus remains an open problem.…”

Section: Introductionmentioning

confidence: 99%

Approximations by disjoint continua and a positive entropy conjecture

Lipham

2019

Preprint

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In 1983, E.D. Tymchatyn constructed a hereditarily locally connected continuum which is the closure of a first category ray. We show the example re-opens a conjecture of G.T. Seidler and H. Kato on positive entropy homeomorphisms. Additionally, we show that every indecomposable semi-continuum can be approximated by a sequence of disjoint subcontinua, and no composant of an indecomposable continuum can be embedded into a Suslinian continuum.

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The topology of continua that are approximated by disjoint subcontinua

Cited by 6 publications

References 4 publications

Mixing sets, positive entropy homeomorphisms and non-Suslinean continua

Mixing sets, positive entropy homeomorphisms and non-Suslinean continua

Singularities of meager composants and filament composants

Approximations by disjoint continua and a positive entropy conjecture

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