论文标题
由混沌和类似梯度的系统耦合产生的非均匀双曲系统
Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems
论文作者
论文摘要
我们研究了通过耦合两个地图获得的动态系统，其中一个是混乱的，用Anosov的差异形态示例，另一个是梯度类型的，并由圆圈的N杆到S杆映射示例。利用双曲线系统的几何和千古理论的利用技术，我们分析了以上两个地图耦合的三种不同方法。对于弱耦合，我们为现有理论提供了附录，表明当它通常不是双曲线时，吸引子几乎总是具有分形的几何形状。我们的主要结果是更强的耦合，其中Anosov差异图在圆形图上具有某些单调性能。在这些条件下，我们表明耦合系统具有不变的锥体，即使存在统一双曲线的真正障碍，也具有SRB措施。
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.
