David Sauzin and Stefano Marmi
Quasianalytic monogenic solutions of a cohomological equation
(1067K, Postscript)
ABSTRACT. We prove that the solutions of a cohomological equation of
complex dimension one and in the analytic category have a monogenic
dependence on the parameter, and we investigate the
question of their quasianalyticity.
This cohomological equation is the standard linearized conjugacy equation
for germs of holomorphic maps in a neighborhood of a fixed point.
The parameter is the eigenvalue of the linear part, denoted by~$q$.
Borel's theory of non-analytic monogenic functions has been first
investigated by Arnol'd and Herman in the related context of the problem
of linearization of analytic diffeomorphisms of the circle close to a
rotation.
Herman raised the question whether the solutions of the cohomological
equation had a quasianalytic dependence on the parameter~$q$.
Indeed they are analytic for $q\in\C\setminus\S^1$,
the unit circle~$\S^1$ appears as a natural boundary (because of
resonances, \ie roots of unity), but the solutions are still defined at
points of~$\S^1$ which lie ``far enough from resonances''.
We adapt to our case Herman's construction of an increasing sequence of
compacts which avoid resonances and prove that the solutions of our equation
belong to the associated space of monogenic functions;
some general properties of these monogenic functions and
particular properties of the solutions are then studied.
For instance the solutions are defined and admit asymptotic expansions at
the points of~$\S^1$ which satisfy some arithmetical condition, and the
classical Carleman Theorem allows us
to answer negatively to the question of quasianalyticity at these points.
But resonances (roots of unity) also lead to asymptotic expansions, for
which quasianalyticity is obtained as a particular case of \'Ecalle's
theory of resurgent functions.
And at constant-type points, where no quasianalytic Carleman class
contains the solutions, one can still recover the solutions from their
asymptotic expansions and obtain a special kind of quasianalyticity.
Our results are obtained by reducing the problem,
by means of Hadamard's product, to the study of a fundamental solution
(which turns out to be the so-called $q$-logarithm or ``quantum
logarithm'').
We deduce as a corollary of our work the proof of a conjecture of
Gammel on the monogenic and quasianalytic properties of a
certain number-theoretical Borel-Wolff-Denjoy series.