95-161 Marmi S., Moussa P., Yoccoz J.-C.

Continued fraction transformations, Brjuno functions and BMO spaces (121K, TeX) Mar 24, 95

Abstract , Paper (src), View paper (auto. generated ps), Index of related papers

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 1/2-H\"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 1/2-H\"older continuous function.

Files: 95-161.tex