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Singular-Hyperbolic Connecting Lemma
Qualitative Theory of Dynamical Systems ( IF 1.4 ) Pub Date : 2020-08-03 , DOI: 10.1007/s12346-020-00412-2
S. Bautista , Y. Sánchez , V. Sales

Bautista and Morales (Ergod Theory Dyn Syst 30(2):339–359, 2010), present the sectional-Anosov connecting lemma as a property of a singular-hyperbolic attracting set \(\Lambda \) on a compact three-dimensional manifold M; this property says that given any two points p and q in \(\Lambda \), such that for every \(\epsilon >0\), there is a trajectory from a point \(\epsilon \)-close to p to a point \(\epsilon \)-close to q, and p has non-singular \(\alpha \)-limit set, then there is a point in M whose \(\alpha \)-limit is that of p and whose \(\omega \)-limit is either a singularity or that of q. In this paper, we prove a generalization of this result, for singular-hyperbolic sets that contain the unstable manifolds of their hyperbolic subsets, although these are not necessarily attracting sets, also, we extend the result to compact manifolds of dimension greater or equal to 3 when the singular-hyperbolic set is of codimension one.

中文翻译:

奇异双曲连接引理

包蒂斯塔和莫拉莱斯(Ergod Theory Dyn Syst 30(2):339–359,2010),提出了截面-Anosov连接引理,它是一个紧凑的三维流形上奇异双曲线吸引集\(\ Lambda \)的性质M ; 此属性表示,给定的任意两点pq\(\ LAMBDA \),使得对于每\(\小量> 0 \) ,存在来自点的轨迹\(\小量\) -close到p到一个点\(\ epsilon \)-接近q,并且p具有非奇异\(\ alpha \)- limit集,那么在M中有一个点,其\(\ alpha \)- limit是p的值,而\(\ omega \)- limit是奇数或q的值。在本文中,我们证明了此结果的推广,对于包含其双曲子集的不稳定流形的奇异双曲集,尽管它们不一定吸引集,而且,我们将结果扩展到尺寸大于或等于的紧凑流形3,当奇异双曲集是余维一时。
更新日期:2020-08-03
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