Embeddings of the pseudo-arc in<i>E</i><sup>2</sup>

In this paper we will generalize the following well-known result. Suppose that I is an arc in the complex sphere C * and h : I → C * is an embedding. Then there exists an orientation-preserving homeomorphism H : C * → C * such that H I = h. It follows that h is isotopic to the identity. Suppose X ⊂ C * is an arbitrary, in particular not necessarily locally connected, continuum. In this paper we give necessary and sufficient conditions on an embedding h : X → C * to be extendable to an orientation-preserving homeomorphism of the entire sphere. It follows that in this case h is isotopic to the identity. The proof will make use of partitions of complementary domains U of X, into hyperbolically convex subsets, which have limited distortion under the conformal map ϕ U : D → U on the unit disk.

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“…no proper subcontinuum X of P contains more than one accessible point); cf. [13]. Now let P be that continuum and U = C \ P ∪ {∞}.…”

Section: Lemma 51 the Mazurkiewicz Metric ρ U Is A Metric On Smentioning

confidence: 99%

Characterizing isotopic continua in the sphere

Oversteegen

Valkenburg

2011

Proc. Amer. Math. Soc.

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“…Specifically, Theorem 9.14 proves that the pseudo-arc has uncountably many non-equivalent embeddings in the strong sense. Lewis [15], has already proven this with respect to the weaker version of equivalence, by carefully constructing embeddings with different prime end structures. Some planar embeddings of the pseudo-arc were constructed earlier by Brechner in [8].…”

Section: Now Fix a Sequence (εmentioning

confidence: 93%

“…It is well-known that every chainable continuum can be embedded in the plane, see [7]. In this paper we develop methods to study non-equivalent planar embeddings, similar to the methods used by Lewis in [15] and Smith in [26] for the study of planar embeddings of the pseudo-arc. Following Bing's approach from [7] (see Lemma 3.1), we construct nested intersections of discs which are small tubular neighbourhoods of polygonal lines obtained from the bonding maps.…”

Section: Introductionmentioning

confidence: 99%

Planar embeddings of chainable continua

Anušić¹,

Bruin²,

Činč³

2018

Preprint

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We prove that for a chainable continuum X and every non-zigzag x ∈ X there exists a planar embedding ϕ : X → ϕ(X) ⊂ R 2 such that ϕ(x) is accessible, partially answering the question of Nadler and Quinn from 1972. Two embeddings ϕ, ψ : X → R 2 are called strongly equivalent if ϕ•ψ −1 : ψ(X) → ϕ(X) can be extended to a homeomorphism of R 2 . We also prove that every indecomposable chainable continuum can be embedded in the plane in uncountably many strongly non-equivalent ways.

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“…Similar observations can be made for several other embeddings. Michel Smith [9] and the second author [5] have independently constructed a variety of embeddings of the pseudo-arc in the plane. Extendability of homeomorphisms for various embeddings is an area fresh for further studies.…”

Section: Further Observationsmentioning

confidence: 99%

An orientation reversing homeomorphism of the plane with invariant pseudo-arc

Bellamy¹,

Lewis²

1992

Proc. Amer. Math. Soc.

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Abstract.Using the universal cover of the Möbius band, we construct an example as described in the title which answers a question of K. Kuperberg.Krystyna Kuperberg has recently asked whether there exists an orientation reversing homeomorphism of the plane with an invariant pseudo-arc. We provide a positive answer to this question. Her question was apparently motivated by her recent work [4] on fixed points and orientation reversing homeomorphisms.Most embeddings of the pseudo-arc in the plane appear to admit very few homeomorphism which extend to homeomorphisms of the plane. For many embeddings, e.g., the "standard" embedding with two accessible composants [7], one can show that every extendable homeomorphism must preserve orientation of the plane. Thus our example depends on choosing the correct embedding of the pseudo-arc.Since there has been significant recent interest in the dynamics of homeomorphisms of the pseudo-arc and in which homeomorphisms of the pseudo-arc extend to homeomorphisms of the plane, it is worth noting that the example we construct has fairly elementary dynamics. There are two fixed points of the pseudo-arc under our homeomorphism.Every other point of the psuedo-arc is repelled from one of these fixed points and attracted to the other. One can construct examples with more interesting dynamics.We conclude with some observations about prime ends, accessible points, and extendable homeomorphisms. The constructionOur main result is the following Theorem. There exists an orientation reversins homeomorphism of the plane with an invariant pseudo-arc.Our proof is constructive. The construction can be outlined in the following few steps and observations. -

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Embeddings of the pseudo-arc inE²

Cited by 7 publications

References 5 publications

Characterizing isotopic continua in the sphere

Characterizing isotopic continua in the sphere

Planar embeddings of chainable continua

An orientation reversing homeomorphism of the plane with invariant pseudo-arc

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Embeddings of the pseudo-arc inE2

Cited by 7 publications

References 5 publications

Characterizing isotopic continua in the sphere

Characterizing isotopic continua in the sphere

Planar embeddings of chainable continua

An orientation reversing homeomorphism of the plane with invariant pseudo-arc

Contact Info

Product

Resources

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Embeddings of the pseudo-arc inE²