For a point x in the inverse limit space X with a single unimodal bonding map we construct, with the use of symbolic dynamics, a planar embedding such that x is accessible. It follows that there are uncountably many non-equivalent planar embeddings of X.
In this paper we study a class of embeddings of tent inverse limit spaces. We introduce techniques relying on the Milnor-Thurston kneading theory and use them to study sets of accessible points and prime ends of given embeddings. We completely characterize accessible points and prime ends of standard embeddings arising from the Barge-Martin construction of global attractors. In the other studied embeddings we find phenomena which do not occur in the standard embeddings. Furthermore, for the class of studied non-standard embeddings we prove that shift homeomorphism can not be extended to a planar homeomorphism.
We study the properties of folding points and endpoints of unimodal inverse limit spaces. We distinguish between non-end folding points and three types of end-points (flat, spiral and nasty) and give conditions for their existence and prevalence. Additionally, we give a characterisation of tent inverse limit spaces for which the set of folding points equals the set of endpoints.
We study inverse limit spaces of tent maps, and the Ingram Conjecture, which states that the inverse limit spaces of tent maps with different slopes are non-homeomorphic. When the tent map is restricted to its core, so there is no ray compactifying on the inverse limit space, this result is referred to as the Core Ingram Conjecture. We prove the Core Ingram Conjecture when the critical point is non-recurrent and not preperiodic.
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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