Planar embeddings of chainable continua

Abstract

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.

Accessible points of planar embeddings of tent inverse limit spaces

“…With that result we partially answer a question posed by Boyland, de Carvalho and Hall in [8] asking for which embeddings of tent inverse limit spaces the natural shift homeomorphism can be extended to a planar homeomorphism. Note that recently authors together with Bruin [2] constructed larger class of embeddings of tent inverse limit spaces, for which it is yet unknown whether they can be extended to a planar homeomorphism. Because the embeddings from [1] cannot be extended to a planar homomorphism, we lack dynamical techniques as used in [8].…”