COLLOQUIUM - SMALL DIVISORS, CONTINUED FRACTIONS AND STABILITY OF ORBITS - STEFANO MARMI - SCUOLA NORMALE SUPERIORE DI PISA
Abstract
The problems linked with the so-called “small divisors”, i. e. the near resonances of frequencies
in a quasiperiodic motion, have been known and studied since the 19th century. Poincaré was
not certain about the possibility of overcoming them and only beginning with Siegel’s Theorem
in 1942 a statisfactory theory of stability of quasiperiodic motions has been developed. In a
few cases Yoccoz proved that it’s even possible to introduce purely arithmetical functions, built
using the continued fraction development of the frequencies, which allow to compute the size
of the stability domain of the orbits.
I problemi con i cosiddetti “piccoli divisori”, ovvero quasi risonanze tra le frequenze in un moto quasi-periodico,
sono conosciuti e studiati fin dal 19esimo secolo. Poincaré non era certo della possibilità di risolverli e solo a
partire dal teorema di Siegel nel 1942 si è sviluppata una teoria soddisfacente della stabilità dei moti quasiperiodici.
In alcuni casi, come mostrato da Yoccoz, è persino possibile introdurre funzioni puramente aritmetiche,
costruite a partire dallo sviluppo in frazione continua delle frequenze, che permettono di calcolare la grandezza
del dominio di stabilità delle orbite.
Bio
Stefano Marmi is Full Professor of Dynamical Systems at Scuola Normale Superiore di Pisa since 2003. Previously
he was Associate Professor at Università di Udine and Researcher at Università di Firenze. Prof. Marmi studied
Physics in Bologna, obtaining a PhD in Theoretical Physics in 1990. He spent research periods, among other
universities, at Freie Universität Berlin, Université Paris Sud (Orsay), and Princeton University. He is the author of
two books and many important research articles. He received the ISAAC Award in 1999.
La conferenza sarà online, al link Zoom
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