Local and global trace formulae for smooth hyperbolic diffeomorphisms

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.

v6: electronic copy of final peer-reviewed manuscript accepted for publication.small mistake in the proof of Proposition 4.3, see footnote 6 p.9 of the v2 of arXiv:1901.09576 for the fix. First published in: Malo Jézéquel, Local and global trace formulae for smooth hyperbolics diffeomorphisms. J. Spectr. Theory 10(2020), 185-249. doi:10.4171/JST/290. \c{opyright} European Mathematical Society