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 one-dimensional 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.

在 PD McSwiggen 的文章中，提出了 Derived from Anosov type construction 这导致了 3-torus 的部分双曲微分同胚。此微分同胚的非漫游集包含一个二维吸引子，该吸引子由其点的一维不稳定流形组成。构造的微分同胚允许不变的一维可定向叶状结构，因此它包含吸引子点的不稳定流形作为其叶子。此外，该叶状结构具有全局横截面（2-环面）并在其上定义了一个 Poincaré 映射，该映射是常规 Denjoy 型同胚。这种同胚是圆的 Denjoy 同胚最自然的推广，在描述上述部分双曲微分同胚的动力学方面起着重要作用。尤其，相应的庞加莱映射的拓扑共轭为这种部分双曲微分同胚的限制条件的拓扑共轭为其二维吸引子提供了必要条件。每个正则 Denjoy 类型同胚的非漫游集是一个 Sierpiński 集，并且根据定义，每个这样的同胚都是半共轭到 2-torus 的最小平移。我们为正则 Denjoy 型同胚引入拓扑共轭的完全不变量，其特征是最小平移，它是给定正则 Denjoy 型同胚的半共轭，具有明显的、不超过可数的轨道集。