On continuum-wise minimality

Alfonso Artigue Fundamenta Mathematicae MSC: Primary 37B05; Secondary 37B45. DOI: 10.4064/fm155-3-2022 Published online: 19 May 2022

Abstract

We say that a homeomorphism of a compact metric space is cw-minimal if all the proper closed invariant subsets have dimension zero. This concept was previously considered by H. Kato. We explore this notion and provide examples. We give sufficient conditions for the existence of cw-minimal subsets and we prove several characterizations. We show that cw-minimal systems are transitive and either minimal or sensitive if the space is locally connected. A subset is said to be mindual if it intersects every minimal subset. We show that every cw-minimal subset contains a closed, zero-dimensional mindual set.

Authors

  • Alfonso ArtigueDepartamento de Matemática y Estadística del Litoral
    Centro Universitario Regional Litoral Norte
    Universidad de la República
    Gral. Rivera 1350
    50000 Salto, Uruguay
    e-mail

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