Volume hyperbolicity and wildness
Volume hyperbolicity and wildness
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DOI：
10.1017/etds.2016.51
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发表时间：
2015-05
影响因子：
0.9
通讯作者：
C. Bonatti;Katsutoshi Shinohara
中科院分区：
文献类型：
--
作者：
C. Bonatti;Katsutoshi Shinohara
众所周知，体积双曲性（部分双曲性和极值丛中体积的均匀膨胀或收缩）是鲁棒传递性或鲁棒链递归的必要条件，因此也是驯服性的必要条件。在本文中，我们在任何 3$ 流形上构建了体积双曲且同时狂野的准吸引子示例。作为主要推论，我们看到，对于任何封闭的 $3$ -流形 $M$ ，空间 $\text{Diff}^{1}(M)$ 承认一个非空开集，其中每个 $C^{1}$ -通用微分同胚没有吸引子或排斥子。我们构建的主要工具是作者之前论文中引入的灵活周期点的概念。为了从准吸引子中弹出柔性点，我们使用本文介绍的部分双曲过滤马尔可夫分区的概念来控制准吸引子的拓扑。
It is known that volume hyperbolicity (partial hyperbolicity and uniform expansion or contraction of the volume in the extremal bundles) is a necessary condition for robust transitivity or robust chain recurrence and hence for tameness. In this paper, on any $3$ -manifold we build examples of quasi-attractors which are volume hyperbolic and wild at the same time. As a main corollary, we see that, for any closed $3$ -manifold $M$ , the space $\text{Diff}^{1}(M)$ admits a non-empty open set where every $C^{1}$ -generic diffeomorphism has no attractors or repellers. The main tool of our construction is the notion of flexible periodic points introduced in the authors’ previous paper. In order to eject the flexible points from the quasi-attractor, we control the topology of the quasi-attractor using the notion of partially hyperbolic filtrating Markov partitions, which we introduce in this paper.