Analysis describes codimension-one basic sets in closed manifolds, indicating their nature as attractors or repellers.
For flows satisfying Smale's Axiom A on closed manifolds of dimension n≥3 the structure of codimension-one basic sets is described, which are either expanding attractors or contracting repellers. For such nonmixing basic sets special trapping neighbourhoods with boundary components homeomorphic to Sⁿ⁻²× S¹ are constructed. This makes it possible to construct a compactification (casing) of the basin of a basic set, which is a locally trivial fibre bundle over a circle, and the extension of the original flow to the casing is a dynamical suspension and a structurally stable flow of attractor-repeller type. Bibliography: 57 titles.
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Zhuzhoma et al. (2025) studied this question.
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