Approximations by disjoint continua and a positive entropy conjecture

TL;DR

This paper explores the approximation of complex continua by disjoint subcontinua, addressing a positive entropy conjecture and revealing structural properties of indecomposable and Suslinian continua.

Contribution

It demonstrates new approximation techniques for indecomposable continua and resolves a conjecture related to positive entropy homeomorphisms.

Findings

Indecomposable semicontinua can be approximated by disjoint subcontinua.

No component of an indecomposable continuum can embed into a Suslinian continuum.

In hereditarily unicoherent Suslinian continua, dense subcontinua must intersect.

Abstract

E.D. Tymchatyn constructed a hereditarily locally connected continuum which can be approximated by a sequence of mutually disjoint arcs. We show the example re-opens a conjecture of G.T. Seidler and H. Kato about continua which admit positive entropy homeomorphisms. We prove that every indecomposable semicontinuum can be approximated by a sequence of disjoint subcontinua, and no composant of an indecomposable continuum can be embedded into a Suslinian continuum. We also prove that if $Y$ is a hereditarily unicoherent Suslinian continuum, then there exists $\varepsilon>0$ such that every two $\varepsilon$-dense subcontinua of $Y$ intersect.