A-flows with codimension one basic sets

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Abstract

For flows satisfying Smale's Axiom A on closed manifolds of dimension $n\ge3$ the structure of codimension-one basis sets is described, which are either extending attractors or contracting repellers. For such nonmixing basis sets special trapping neighbourhoods with boundary components homeomorphic to ${\mathbb S}^{n-2}\times{\mathbb S}^1$ are constructed. This makes it possible to construct a compactification (support) of the basin of a basis set. which is a locally trivial fibre bundle over the circle, and the extension of the original flow to the support is a dynamical suspension and a structurally stable flow of attractor-repeller type.

About the authors

Evgenii Viktorovich Zhuzhoma

National Research University Higher School of Economics, Nizhnii Novgorod, Russia

Email: zhuzhoma@mail.ru
ORCID iD: 0000-0001-8682-7591
Doctor of physico-mathematical sciences, Professor

Vladislav Sergeevich Medvedev

National Research University Higher School of Economics, Nizhnii Novgorod, Russia

Email: medvedev-1942@mail.ru
ORCID iD: 0000-0001-6369-0000
Candidate of physico-mathematical sciences

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