On the algebraic properties of exponentially stable integrable Hamiltonian systems (Q2103477)

In a series of papers beginning with [Russ. Math. Surv. 32 No. 6, 1--65 (1977; Zbl 0389.70028)] \textit{N. Nekhoroshev} proved the stability of a nearly integrable Hamiltonian system for an open set of initial conditions for a time that is exponentially long in the inverse of the size of the perturbation \(\epsilon\) as long as the Hamiltonian is analytic and that its integrable part satisfies a generic transversality property called ``steepness''. In action-angle coordinates \((I, \vartheta\)) in \(\mathbb{R}^n \times \mathbb{T}^n\), such a Hamiltonian has the form \(H(I,\vartheta) = h(I) + \epsilon f(I, \vartheta)\). The original definition of steepness was not constructive, so it is difficult to establish directly if a given function is steep. \textit{N. Nekhoroshev} in [loc. cit.] did provide a scheme that allows one to identify steepness from algebraic properties using derivatives up to an arbitrary order. In this paper the author develops a scheme to establish steepness, as proposed in [\textit{G. Schirinzi} and \textit{M. Guzzo}, J. Math. Phys. 54, No. 7, 072702, 22 p. (2013; Zbl 1302.58002)]. It uses new explicit conditions for steepness using the five-jet of the given function.