KAM, $\alpha$-Gevrey regularity and the $\alpha$-Bruno-Rüssmann condition

We prove a new invariant torus theorem, for α-Gevrey smooth Hamiltonian systems , under an arithmetic assumption which we call the α-Bruno-Russmann condition , and which reduces to the classical Bruno-Russmann condition in the analytic category. Our proof is direct in the sense that, for analytic Hamiltonians, we avoid the use of complex extensions and, for non-analytic Hamiltonians, we do not use analytic approximation nor smoothing operators. Following Bessi, we also show that if a slightly weaker arithmetic condition is not satisfied, the invariant torus may be destroyed. Crucial to this work are new functional estimates in the Gevrey class.