# Global trace formula for ultra-differentiable Anosov flows

## TL;DR

This paper proves a trace formula for a class of Anosov flows with regularity between Gevrey and smooth functions, using anisotropic Hilbert spaces to analyze the flow's generator.

## Contribution

It extends the trace formula conjecture to less regular Anosov flows by constructing specialized anisotropic Hilbert spaces.

## Key findings

- Trace formula holds for certain regularity classes of Anosov flows.
- Construction of anisotropic Hilbert spaces with discrete spectrum.
- Flow generator has discrete spectrum on these spaces.

## Abstract

Adapting tools that we introduced in [19] to study Anosov flows, we prove that the trace formula conjectured by Dyatlov and Zworski in [12] holds for Anosov flows in a certain class of regularity (smaller than $\mathcal{C}^\infty$ but larger than the class of Gevrey functions). The main ingredient of the proof is the construction of a family of anisotropic Hilbert spaces of generalized distributions on which the generator of the flow has discrete spectrum.