Pollicott-Ruelle Resonant States and Betti Numbers

Benjamin Küster,Tobias Weich

doi:10.1007/s00220-020-03793-2

Abstract

Given a closed orientable hyperbolic manifold of dimension ne 3 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.

Highlights

Pollicott-Ruelle resonances have been introduced in the 1980’s in order to study mixing properties of hyperbolic flows and can nowadays be understood as a discrete spectrum of the generating vector field
The latter are those k-forms on the unit co-sphere bundle S∗M that vanish upon contraction with X
The resonance zero has no Jordan block and if n ≥ 3, zero is the unique leading resonance and there is a spectral gap.3. We prove these statements using the general framework of vector-valued quantumclassical correspondence developed by the authors [KW19] as well as a Poisson transform of Gaillard [Gai86]

Summary

Introduction

Pollicott-Ruelle resonances have been introduced in the 1980’s in order to study mixing properties of hyperbolic flows and can nowadays be understood as a discrete spectrum of the generating vector field (see Sect. 1.2 for a definition and references). Very recently it has been discovered that in certain cases some particular Pollicott-Ruelle resonances have a topological meaning Let us recall these results: In [DZ17] Dyatlov and Zworski prove that on a closed orientable surface M of negative curvature the Ruelle zeta function at zero vanishes to the order |χ (M)|, where χ (M) is the Euler characteristic of M, generalizing a result of Fried in constant curvature [Fri86].1. Without any further effort these ingredients provide additional examples of resonance multiplicities related to the first but to all Betti numbers, see Proposition 2.3 The latter result shows that the p-th Betti number of a closed orientable hyperbolic manifold can be recovered as the dimension of the space of some particular resonant p-forms in the kernel of a so-called horocycle operator

Pollicott-Ruelle Resonances for Geodesic Flows

Multiplicities on Constant Curvature Manifolds

Non-Constant Curvature Perturbations