Abstract: In frictionless phenomena, such as planetary dynamics, regular motions (periodic and quasi-periodic) have been observed and studied since ancient times (Ptolemy). In particular, in the nineteenth century, mathematicians and astronomers engaged in an intense debate over the convergence of formal power-series expansions of quasi-periodic solutions of nearly integrable systems (for example, Poincaré was convinced of their generic divergence, while Weierstrass argued in favour of convergence).
A crucial step was taken by Kolmogorov in 1954, when he proved convergence of such series (under suitable assumptions) and stated (without proof) that the set of quasi-periodic solutions of a general nearly integrable Hamiltonian system (the “Kolmogorov set”) fills compact regions of its phase space up to a small exceptional set, whose Lebesgue measure tends to zero as the perturbation parameter goes to zero.
The main question then becomes: what is, generically, the asymptotic measure of the Kolmogorov set (as the perturbation parameter goes to zero)?
In this Tè di matematica seminar, I will discuss, in general terms, some of these issues, from their (modern) origins up to the research frontier.
Il seminario si svolgerà in presenza presso il Dipartimento di Matematica e Fisica, Lungotevere Dante 476, aula M1.
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