ABSTRACT We prove the saturation of a generalized partially hyperbolic attractor of a map. As a consequence, we show that any generalized partially hyperbolic horseshoe-like attractor of a -generic diffeomorphism has zero volume. In contrast, by modification of the Poincaré cross section of Lorenz geometric model, we build a -diffeomorphism with a partially hyperbolic horseshoe-like attractor of positive volume.

中文翻译：

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广义部分双曲吸引子的饱和度

摘要 我们证明了地图的广义部分双曲线吸引子的饱和度。因此，我们证明了泛微分同胚的任何广义部分双曲马蹄形吸引子的体积为零。相比之下，通过修改 Lorenz 几何模型的 Poincaré 横截面，我们构建了具有正体积的部分双曲马蹄形吸引子的 微分同胚。