Abstract We show that if ( X , f ) is a locally connected Smale space (e.g., a basic set of an Axiom-A diffeomorphism) such that the local product structure on X can be lifted by a covering with virtually nilpotent group of deck transformations to a global direct product, then ( X , f ) is topologically conjugate to a hyperbolic infra-nilmanifold automorphism. We use this result to give a generalization to Smale spaces of a theorem of M. Brin and A. Manning on Anosov diffeomorphisms with pinched spectrum, and to show that every locally connected codimension one Smale space is topologically conjugate to a hyperbolic automorphism of a torus.

中文翻译：

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局部连接的 Smale 空间、收缩频谱和下半流形

摘要 我们证明如果 ( X , f ) 是一个局部连通的 Smale 空间（例如，Axiom-A 微分同胚的一个基本集合），使得 X 上的局部积结构可以被具有几乎幂零的甲板变换群的覆盖提升到全局直接乘积，则 ( X , f ) 拓扑共轭到双曲下半折自同构。我们使用这个结果对 M. Brin 和 A. Manning 的定理在具有收缩谱的 Anosov 微分同胚上的 Smale 空间进行推广，并证明每个局部连接的余维一个 Smale 空间在拓扑上共轭到环面的双曲自同构.