Accessibility of countable sets in plane embeddings of arc-like continua
Abstract
We consider the problem of finding embeddings of arc-like continua in the plane for which each point in a given subset is accessible. We establish that, under certain conditions on an inverse system of arcs, there exists a plane embedding of the inverse limit for which each point of a given countable set is accessible. As an application, we show that for any Knaster continuum $K$, and any countable collection $\mathcal{C}$ of composants of $K$, there exists a plane embedding of $K$ in which every point in the union of the composants in $\mathcal{C}$ is accessible. We also exhibit new embeddings of the Knaster buckethandle continuum $K$ in the plane which are attractors of plane homeomorphisms, and for which the restriction of the plane homeomorphism to the attractor is conjugate to a power of the standard shift map on $K$.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2024
- DOI:
- 10.48550/arXiv.2407.16792
- arXiv:
- arXiv:2407.16792
- Bibcode:
- 2024arXiv240716792A
- Keywords:
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- Mathematics - General Topology;
- Mathematics - Dynamical Systems;
- Primary 54F15;
- 54C25;
- Secondary 54F50;
- 37B45;
- 37E30
- E-Print:
- 25 pages, 6 figures