Local and global trace formulae for smooth hyperbolic diffeomorphisms

TL;DR

This paper develops local and global trace formulae for smooth hyperbolic diffeomorphisms, highlighting differences from real-analytic cases and introducing Gevrey dynamics with ultradistribution spaces to establish trace class operators.

Contribution

It introduces a framework for trace formulae in Gevrey dynamics using anisotropic ultradistribution spaces, extending the understanding beyond real-analytic dynamics.

Findings

Counter-examples show less well-behaved dynamics for smooth systems.

Constructed anisotropic ultradistribution spaces for Gevrey dynamics.

Derived bounds on dynamical determinants and Ruelle resonances.

Abstract

We define and study local and global trace formulae for discrete-time uniformly hyperbolic weighted dynamics. We explain first why dynamical determinants are particularly convenient tools to tackle this question. Then we construct counter-examples that highlight that the situation is much less well-behaved for smooth dynamics than for real-analytic ones. This suggests to study this question for Gevrey dynamics. We do so by constructing an anisotropic space of ultradistributions on which a transfer operator acts as a trace class operator. From this construction, we deduce trace formulae for Gevrey dynamics, as well as bounds on the growth of their dynamical determinants and the asymptotics of their Ruelle resonances.