 95161 Marmi S., Moussa P., Yoccoz J.C.
 Continued fraction transformations, Brjuno functions and BMO spaces
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Mar 24, 95

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Abstract. The small divisors problem which is raised by stability questions
in classical mechanics has an analog in holomorphic dynamical
systems, namely the existence of Siegel disks. The size of these
disks is well represented by the Brjuno functions. We analyse
the relation between these functions and the various continued
fraction transformations, and display the functional equation
which is fulfilled by these highly singular functions.
The analysis of this functional equation shows that the Brjuno
function belong to the BMO space, and that a regular
perturbation of this equation leads to a modification of the
singular function which is 1/2H\"older continuous. This leads
us to believe that the most singular part of the size of
the stability domains as function of the rotation number, is
`universal' up to a 1/2H\"older continuous function.
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