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 Arnold 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 $q\in\mathbb{C}\setminus\mathbb{S}^1$, 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
$\mathbb{S}^1$ 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 $\mathbb{S}^1$ 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 Écalle'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.

Readership

Graduate students and research mathematicians interested in
differential equations.