Local holomorphic dynamics

seminar: Monogenic regularity of linearizations

speaker: Stefano Marmi (Scuola Normale Superiore)

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. 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 $q\inC\setminus S1$, the unit circle $S1$ appears as a natural boundary (because of resonances, \ie roots of unity), but the solutions are still defined at points of $S1$ 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.

timetable:
Mon 22 Jan, 10:00 - 11:00, Aula Dini
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