论文标题
非Trantsverse异二维循环：稳定和强大的角度
Nontransverse heterodimensional cycles: stabilisation and robust tangencies
论文作者
论文摘要
我们考虑具有同时具有异二维循环和与鞍 - - 奇怪相关的异二维的三维差异性。这些循环导致完全非主导的分叉设置。对于每$ r \ geqslant 2 $，我们都会展示一类这样的差异性，它们的异二维周期可以稳定，并且（同时）通过$ c^r $ $ c^r $ $ c^r $的同质性同质性固定性稳定。我们非主导的设置具有大量同质和异斜交集的复杂性，用于克服执行$ c^r $扰动的困难，$ r \ geqslant 2 $，这比$ c^1 $的难度非常困难。我们的证明是让人联想到帕利斯·塔基斯（Palis-Takens）的方法，以在表面差异性的同型层压层次上展开时，以无限多个下沉（Newhouse现象）获得表面差异。该证明涉及沿着非透明杂斜轨道的重新归一化方案，该轨道会融合到中心的不稳定的hénon家族，显示搅拌器 - horseshoes。一个关键的步骤是对这些搅拌机 - 霍斯的嵌入在非主导的环境中的分析。
We consider three-dimensional diffeomorphisms having simultaneously heterodimensional cycles and heterodimensional tangencies associated to saddle-foci. These cycles lead to a completely nondominated bifurcation setting. For every $r\geqslant 2$, we exhibit a class of such diffeomorphisms whose heterodimensional cycles can be $C^r$ stabilised and (simultaneously) approximated by diffeomorphisms with $C^r$ robust homoclinic tangencies. The complexity of our nondominated setting with plenty of homoclinic and heteroclinic intersections is used to overcome the difficulty of performing $C^r$ perturbations, $r\geqslant 2$, which are remarkably more difficult than $C^1$ ones. Our proof is reminiscent of the Palis-Takens' approach to get surface diffeomorphisms with infinitely many sinks (Newhouse phenomenon) in the unfolding of homoclinic tangencies of surface diffeomorphisms. This proof involves a scheme of renormalisation along nontransverse heteroclinic orbits converging to a center-unstable Hénon-like family displaying blender-horseshoes. A crucial step is the analysis of the embeddings of these blender-horseshoes in a nondominated context.
