The Steep Nekhoroshev's Theorem

Revising Nekhoroshev's geometry of resonances, we provide a fully constructive and quantitative proof of Nekhoroshev's theorem for steep Hamiltonian systems proving, in particular, that the exponential stability exponent can be taken to be $1/ (2n \alpha_1\cdots\alpha_{n-2}$) ($\alpha_i$'s being Nekhoroshev's steepness indices and $n\ge 3$ the number of degrees of freedom).