We study local and global trace formulae for $\mathcal{C}^{\infty}$ ($a\,priori$ non-analytic) hyperbolic diffeomorphisms, that is we compare the sum of its Ruelle resonances with the flat trace of its transfer operator. We explicit the link between trace formulae and dynamical determinants and give examples of systems with explicit dynamical determinants exhibiting interesting behaviours (in particular, trace formulae do not always hold). We also establish a bound on the growth of the dynamical determinant for Gevrey dynamics from which we deduce a control on the number of resonances outside of a disc centred at $0$. 查看全文>>