分段光滑向量场的熵与Gurevich压力
Entropy and Gurevich Pressure for piecewise smooth vector fields
AI总结：
本文通过建立分段光滑向量场轨迹与单边移位空间的拓扑共轭，计算了非光滑向量场的Gurevich压力和熵，并用Lyapunov函数估计混合时间，还将无穷远处拓扑熵与逃逸回复点集的Hausdorff维数联系起来。
摘要：
本文在经典离散动力系统结果与由非光滑向量场支配的动力系统之间建立了一座桥梁。事实上，我们得到了一组分段光滑向量场轨迹，其时间一映射是良定义的。随后，我们在包含于该集合中的轨迹的行程与具有可数状态集合并配备离散拓扑的单边移位空间之间建立了一个拓扑共轭。然后我们利用这一事实来计算非光滑向量场的Gurevich压力和熵，并且我们还使用Lyapunov函数来估计混合时间。此外，我们将这些非光滑向量场的全局轨迹在无穷远处的拓扑熵与平均意义上逃逸的回复点集合的Hausdorff维数联系起来。最后，我们从余有限拓扑的角度来处理这一问题，从而得到相关例子。
英文摘要：
In this paper we provide a bridge between classical results concerning discrete dynamical systems and dynamical systems governed by nonsmooth vector fields. In fact, we obtain a set of piecewise smooth vector field trajectories where the time-one map is well defined. Afterwards, we obtain a topological conjugacy between the itinerary of a trajectory contained in this set and the one-sided shift space with a set of states countable with the discrete topology. We then use this fact to calculate the Gurevich pressure and entropy for non-smooth vector fields, and we also use the Lyapunnov function to estimate the mixing time. Furthermore, we relate the topological entropy at infinity of the global trajectories of these non-smooth vector fields to the Hausdorff dimension of the set of recurrent points that escape on average. Finally, we approach it from the point of view of cofinite topology, thus obtaining relevant examples.
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