奇异流的可数划分及其在熵理论中的应用
A countable partition for singular flows, and its application on the entropy theory
摘要：
本文通过在每个奇点处引入新的截面，构造了具有双曲奇点流动的可数划分$\mathscr{A}$。这样的划分在奇点的邻域中形成了角谷塔，并且对于每个不变的概率测度都有有限的度量熵。此外，$\mathscr{A}^\infty$的每个元素将在缩放的管状邻域中停留任意长的时间。这种新的结构使我们能够研究熵理论的奇异流远离同宿切线，并表明熵函数是上半连续的不变测度和流量。
英文摘要：
In this paper, we construct a countable partition $\mathscr{A}$ for flows with hyperbolic singularities by introducing a new cross section at each singularity. Such partition forms a Kakutani tower in a neighborhood of the singularity, and turns out to have finite metric entropy for every invariant probability measure. Moreover, each element of $\mathscr{A}^\infty$ will stay in a scaled tubular neighborhood for arbitrarily long time. This new construction enables us to study entropy theory for singular flows away from homoclinic tangencies, and show that the entropy function is upper semi-continuous with respect to both invariant measures and the flows.
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