Heterodimensional tangencies
Heterodimensional tangencies
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异维相切
DOI：
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发表时间：
2006
期刊：
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通讯作者：
E. Pujals
中科院分区：
文献类型：
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作者：
L. Díaz;A. Nogueira;E. Pujals
我们考虑在三维流形上定义的 C1-微分同胚 f，具有一对鞍点 Pf 和 Qf（不稳定指数一和二），其同宿类持续重合。我们证明，如果 Pf 的二维稳定流形和 Qf 的二维不稳定流形具有某种非横向交集（异维相切），则这种相切的展开会导致微分同胚 h，使得 Qh 的同宿类（Qf 对于 h 的延续）是鲁棒非支配的。这导致了无限多个汇或源的（C1-局部泛型）共存现象，并且在某些相关情况下，导致无限多个最小康托集的共存。我们给出了先前动态配置发生的示例，提供了从部分双曲线到鲁棒非支配动态的自然过渡。
We consider C1-diffeomorphisms f defined on three-dimensional manifolds having a pair of saddles Pf and Qf (of unstable indices one and two) whose homoclinic classes coincide persistently. We prove that if the two-dimensional stable manifold of Pf and the two-dimensional unstable manifold of Qf have some non-transverse intersection (a heterodimensional tangency) the unfolding of such a tangency leads to diffeomorphisms h such that the homoclinic class of Qh (the continuation of Qf for h) is robustly non-dominated. This leads to the phenomena of (C1-locally generic) coexistence of infinitely many sinks or sources and, in some relevant cases, to the coexistence of infinitely many minimal Cantor sets. We give examples where the previous dynamical configuration occurs, providing a natural transition from partially hyperbolic to robustly non-dominated dynamics.