Pollicott-Ruelle共振态的拓扑
Topology of Pollicott-Ruelle resonant states
摘要：
本文证明了光滑流形上平坦向量场的扭曲De Rham上同调同构于Anosov流和Morse-Smale流的不变Pollicott-Ruelle共振态的上同调。因此，我们得到了这类流的广义Morse不等式。在Morse-Smale流的情况下，我们将虚轴上的共振与Fried在他的关于Reidemeister扭转的工作中所用的扭曲的Fuller测量联系起来。特别是，当V是一个nonsingularMorse-Smale流，我们表明，Reidemeister扭转可以恢复从onlyknowledge的动力学共振的虚轴上的表示为azeta正则化的这些共振的无穷乘积的扭转。
英文摘要：
We prove that the twisted De Rham cohomology of a flat vector bundleover some smooth manifold is isomorphic to the cohomology of invariant Pollicott--Ruelleresonant states associated with Anosov and Morse--Smale flows. As a consequence, weobtain generalized Morse inequalities for such flows. In the case of Morse--Smale flows,we relate the resonances lying on the imaginary axis with the twisted Fuller measuresused by Fried in his work on Reidemeister torsion. In particular, when V is a nonsingularMorse-Smale flow, we show that the Reidemeister torsion can be recovered from the onlyknowledge of dynamical resonances on the imaginary axis by expressing the torsion as azeta regularized infinite product of these resonances.
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