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On Partially Hyperbolic Diffeomorphisms and Regular Denjoy Type Homeomorphisms
Vyacheslav Z. Grines, Dmitrii I. Mints HSE University,
ul. Bolshaya Pecherskaya 25/12, 603155 Nizhny Novgorod, Russia
Аннотация:
In P.D. McSwiggen’s article, it was proposed Derived from Anosov type construction
which leads to a partially hyperbolic diffeomorphism of the 3-torus. The nonwandering set
of this diffeomorphism contains a two-dimensional attractor which consists of one-dimensional
unstable manifolds of its points. The constructed diffeomorphism admits an invariant onedimensional
orientable foliation such that it contains unstable manifolds of points of the
attractor as its leaves. Moreover, this foliation has a global cross section (2-torus) and defines
on it a Poincaré map which is a regular Denjoy type homeomorphism. Such homeomorphisms
are the most natural generalization of Denjoy homeomorphisms of the circle and play an
important role in the description of the dynamics of aforementioned partially hyperbolic
diffeomorphisms. In particular, the topological conjugacy of corresponding Poincaré maps
provides necessary conditions for the topological conjugacy of the restrictions of such partially
hyperbolic diffeomorphisms to their two-dimensional attractors. The nonwandering set of
each regular Denjoy type homeomorphism is a Sierpiński set and each such homeomorphism
is, by definition, semiconjugate to the minimal translation of the 2-torus. We introduce a
complete invariant of topological conjugacy for regular Denjoy type homeomorphisms that is
characterized by the minimal translation, which is semiconjugation of the given regular Denjoy
type homeomorphism, with a distinguished, no more than countable set of orbits.
Ключевые слова:
topological classification, Denjoy type homeomorphism, Sierpiński set, partial
hyperbolicity.
Поступила в редакцию: 14.01.2023 Принята в печать: 01.05.2023
Образец цитирования:
Vyacheslav Z. Grines, Dmitrii I. Mints, “On Partially Hyperbolic Diffeomorphisms and Regular Denjoy Type Homeomorphisms”, Regul. Chaotic Dyn., 28:3 (2023), 295–308
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/rcd1206 https://www.mathnet.ru/rus/rcd/v28/i3/p295
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