[Paper Review] Planar embeddings of chainable continua

Ana Anušić, Henk Bruin+1 more|arXiv (Cornell University)|Jun 13, 2018

Advanced Topology and Set Theory19 references3 citations

TL;DR

This paper establishes that every nondegenerate indecomposable chainable continuum admits uncountably many planar embeddings that are not strongly equivalent, using a geometric construction based on nested discs and permutations of bonding map branches. It partially resolves Nadler and Quinn's 1972 question by showing that points in chainable continua with finitely many zigzag coordinates are accessible in some planar embeddings, advancing the understanding of planar embeddings and accessibility in continuum theory.

ABSTRACT

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

Motivation & Objective

• To investigate the existence of uncountably many nonequivalent planar embeddings for chainable continua.
• To address Nadler and Quinn's 1972 question on whether every point in a chainable continuum can be made accessible in some planar embedding.
• To develop a geometric method for constructing planar embeddings using nested discs and permutations of interval map branches.
• To distinguish between strong equivalence and standard equivalence of embeddings in the context of inverse limit spaces.
• To explore the topological and dynamical constraints on self-homeomorphisms and composant structures in chainable continua.

Proposed method

• Constructing planar embeddings via nested intersections of discs that serve as tubular neighborhoods of polygonal lines derived from bonding maps.
• Using permutations of branches in graphs of linear interval maps to generate distinct embedding structures.
• Defining and analyzing 'zigzags' in the graph of a bonding map to identify points that may not be accessible.
• Applying Lemma 9.16 to select subintervals and sequences of indices to generate uncountably many subcontinua with accessible endpoints.
• Extending embeddings of subcontinua to the full chainable continuum using inverse limit constructions.
• Proving strong nonequivalence by showing that no homeomorphism of the plane can extend the map between embeddings.

Experimental results

Research questions

• RQ1 Can every point in a chainable continuum be made accessible in some planar embedding?
• RQ2 Are there uncountably many nonequivalent planar embeddings for every nondegenerate indecomposable chainable continuum?
• RQ3 Does the existence of uncountably many strongly nonequivalent embeddings hold for hereditarily decomposable chainable continua?
• RQ4 Can the group of self-homeomorphisms of a chainable continuum up to pseudo-isotopy be at most countable?
• RQ5 Do inverse limit spaces of unimodal maps with positive entropy admit uncountably many nonequivalent planar embeddings?

Key findings

• Every nondegenerate indecomposable chainable continuum admits uncountably many planar embeddings that are not strongly equivalent.
• For any chainable continuum and any point with finitely many zigzag coordinates, there exists a planar embedding in which that point is accessible.
• The construction yields uncountably many embeddings by choosing sequences of branch indices from at least four surjective intervals in each bonding map.
• The method applies to inverse limit spaces of unimodal maps that are not hereditarily decomposable, and in such cases, the result holds under both standard and strong equivalence.
• The paper provides a geometric alternative to symbolic techniques used in prior work, enabling direct analysis of accessibility and embedding equivalence.
• The results generalize previous findings on unimodal inverse limit spaces and extend the understanding of planar embeddings in continuum theory.