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Quasianalytic Monogenic Solutions of a Cohomological Equation
S. Marmi, University of Udine, Italy, and D. Sauzin, Astronomie et Systems Dynamiques, Paris, France
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Memoirs of the American Mathematical Society
2003; 83 pp; softcover
Volume: 164
ISBN-10: 0-8218-3325-1
ISBN-13: 978-0-8218-3325-4
List Price: US$57 Individual Members: US$34.20
Institutional Members: US\$45.60
Order Code: MEMO/164/780

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.

Graduate students and research mathematicians interested in differential equations.