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Last update: 23/03/2003

 Houches March 2003

École de physique des Houches

G Gentile
Bryuno numbers and dynamical systems


Abstract:

We consider some simple analytic dynamical systems for which invariant curves can be shown to exist if the corresponding rotation numbers are Bryuno numbers, i.e. irrational numbers such that the corresponding Bryuno function introduced by Yoccoz is finite. Such systems include the Siegel problem, the semistandard map, and the standard map. For the latter we can show, by using also a result by Davie, that it is possible to give sort of an interpolation formula for the radius of convergence of the series expansion describing an invariant curve in terms of the Bryuno function itself. Generalizations and open problems will be briefly discussed.

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