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.
