ABSTRACT:

I will report on some results obtained in two separate papers, one in collaboration with Carlo Carminati (University of Pisa) and the other in collaboration with David Sauzin (IMCCE, Paris). The main goal will be to show that linearization of germs of holomorphic maps with a fixed point in one complex dimension and in the analytic category have a monogenic dependence on the parameter. The same is true for the linearization of (local) diffeomorphisms of the circle as Herman proved long ago. Monogenic refers to Borel's theory of monogenic funtions, an extension to non-open domains of the notion of holomorphic functions. I will also discuss the quasianalytic properties of various spaces of functions suitable for one-dimensional small divisor problems. These spaces are formed of functions C^1-holomorphic on certain compact sets K of the Riemann sphere (in the Whitney sense), as is the solution of a linear or non-linear small divisor problem when viewed as a function of the multiplier (the intersection of K with the unit circle is defined by a Diophantine-type condition, so as to avoid the divergence caused by roots of unity). It turns out that a kind of generalized analytic continuation through the unit circle is possible under suitable conditions on the set K.