向量场的预解式与莱夫谢茨数
Resolvent of vector fields and Lefschetz numbers
AI总结：
本文研究向量场预解式与动力级数的关联，给出适用于任意光滑流的一般形式公式，引入新型动力级数并计算其零点值及与拓扑不变量的关系，拓展了相关研究的适用范围。
摘要：
Ruelle zeta函数等动力级数已成为双曲流研究中的常用工具，通常通过将其与向量场的预解式关联来进行分析。本文给出了这类关联的一般形式，涉及上述预解式的核与积分流的相交。我们的公式实际上适用于任意光滑流，而非仅限于双曲流。作为应用，我们引入了此前未出现过的若干动力级数。最后，我们计算了它们在零点处的值，以及它们与拓扑不变量的关系。
英文摘要：
Dynamical series such as the Ruelle zeta function have become a staple in the study of hyperbolic flows. They are usually analyzed by relating them to the resolvent of the vector field. In this paper we give the general form of such relations, which involves the intersection of the kernel of said resolvent with integration currents. Our formula is actually valid for any smooth flow, not necessarily hyperbolic. As an application, we introduce certain dynamical series that had not appeared before. Finally, we compute their value at zero, and their relation with topological invariants.
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