Approximations by disjoint continua and a positive entropy conjecture

arXiv1910.00563MaRDI QIDQ6326349

David Sumner Lipham

Publication date: 1 October 2019

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.

Mathematics Subject Classification ID

Continua and generalizations (54F15) Topological entropy (37B40) Continua theory in dynamics (37B45) Counterexamples in general topology (54G20)

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