# Pollicott-Ruelle resonant states and Betti numbers

## TL;DR

This paper establishes a precise relationship between Pollicott-Ruelle resonances of geodesic flows and Betti numbers of hyperbolic manifolds, showing stability under perturbations and extending to higher Betti numbers.

## Contribution

It proves the equality of resonance multiplicity and first Betti number for hyperbolic manifolds and demonstrates stability under small metric and vector field perturbations.

## Key findings

- Resonance multiplicity at zero equals the first Betti number.
- Stability of this equality under small perturbations.
- Extension to higher Betti numbers for certain resonance spaces.

## Abstract

Given a closed orientable hyperbolic manifold of dimension $\neq 3$ 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.