We investigate dynamical systems obtained by coupling two maps, one of which is chaotic and is exemplified by an Anosov diffeomorphism, and the other is of gradient type and is exemplified by a N-pole-to-S-pole map of the circle. Leveraging techniques from the geometric and ergodic theories of hyperbolic systems, we analyze three different ways of coupling together the two maps above. For weak coupling, we offer an addendum to existing theory showing that almost always the attractor has fractal-like geometry when it is not normally hyperbolic. Our main results are for stronger couplings in which the action of the Anosov diffeomorphism on the circle map has certain monotonicity properties. Under these conditions, we show that the coupled systems have invariant cones and possess SRB measures even though there are genuine obstructions to uniform hyperbolicity.

中文翻译：

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混沌系统和梯度系统耦合产生的非均匀双曲系统

我们研究了通过耦合两个图获得的动力学系统，其中一个是混沌的，以Anosov微分态为例，另一个是梯度类型的，以该圆的N极到S极图为例。利用双曲线系统的几何和遍历理论，我们分析了将以上两个图耦合在一起的三种不同方式。对于弱耦合，我们对现有理论进行了补充，表明当吸引子通常不是双曲线时，几乎总是具有分形的几何形状。我们的主要结果是得到了更强的耦合，其中圆映射上的Anosov微分同质作用具有一定的单调性。在这种情况下