Küster Benjamin, Weich Tobias
Université Paris-Saclay, CNRS, Orsay France.
Universität Paderborn, Paderborn, Germany.
Commun Math Phys. 2020;378(2):917-941. doi: 10.1007/s00220-020-03793-2. Epub 2020 Jul 22.
Given a closed orientable hyperbolic manifold of dimension we prove that the multiplicity of the Pollicott-Ruelle resonance of the geodesic flow on perpendicular one-forms at zero agrees with the first Betti number of the manifold. Additionally, we prove that this equality is stable under small perturbations of the Riemannian metric and simultaneous small perturbations of the geodesic vector field within the class of contact vector fields. For more general perturbations we get bounds on the multiplicity of the resonance zero on all one-forms in terms of the first and zeroth Betti numbers. Furthermore, we identify for hyperbolic manifolds further resonance spaces whose multiplicities are given by higher Betti numbers.
给定一个(n)维闭可定向双曲流形，我们证明了测地线流在垂直一次形式上零处的波利科特 - 吕埃勒共振的重数与该流形的第一贝蒂数一致。此外，我们证明了在黎曼度量的小扰动以及在接触向量场类中测地线向量场的同时小扰动下，这种等式是稳定的。对于更一般的扰动，我们根据第一和零贝蒂数得到了所有一次形式上共振零的重数的界。此外，我们为双曲流形确定了更多共振空间，其重数由更高的贝蒂数给出。
