Calculus of Variations and Geometric Measure Theory

C^1-holomorphic functions and complex dynamics

Stefano Marmi

created by abbondand on 23 Jan 2007

15 feb 2007

Abstract.

I will report on some results obtained in collaboration with Carlo Carminati (University of Pisa).

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. Here monogenic refers to Borel's theory of monogenic funtions, an extension to non-open domains of the notion of holomorphic functions. The parameter is the eigenvalue of the linear part, denoted by $q$. The linearization is analytic for $q$ in $C \setminus S^1$, the unit circle $S^1$ appears as a natural boundary (because of resonances, i.e. roots of unity), but the solutions are still defined at points of $S^1$ which lie ``far enough from resonances''. One can construct an increasing sequence of compacts which avoid resonances and prove that the linearization belongs to the associated space of monogenic functions. Among the consequences of these results, one can prove that the linearizations are defined and admit asymptotic expansions of Gevrey type at the points of $S1$ which satisfy some arithmetical condition. Despite the fact that they do not seem to belong to any quasianalytic Carleman class, the associated space of monogenic functions has some quasinaliticity property.