Quasianalytic monogenic solutions of a cohomological equation

TL;DR

This paper investigates the monogenic and quasianalytic properties of solutions to a cohomological equation in complex dynamics, revealing conditions under which solutions are uniquely determined by their asymptotic expansions and extending previous conjectures.

Contribution

It establishes the monogenic dependence and quasianalyticity of solutions to a fundamental cohomological equation, connecting complex dynamics, asymptotic analysis, and number theory.

Findings

Solutions are monogenic functions with asymptotic expansions at points on the unit circle.

Quasianalyticity fails at generic points but holds at constant-type points under certain conditions.

The results confirm a conjecture of Gammel regarding Borel-Wolff-Denjoy series.

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 equation is the standard linearized conjugacy equation for germs of holomorphic maps in a neighborhood of a fixed point. The parameter is the eigenvalue $q$ of the linear part. This problem has been first investigated by Arnol'd and Herman. Herman raised the question whether the solutions of the cohomological equation had a quasianalytic dependence on the parameter. Indeed they are analytic outside $\S^1$ which is a natural boundary 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 belong to the associated space of monogenic functions. The solutions admit asymptotic expansions at the points of $\S^1$ which satisfy some arithmetical condition, and Carleman's Theorem allows us to answer negatively to the question of quasianalyticity at these points. But resonances (roots of unity) lead to asymptotic expansions, for which quasianalyticity is obtained as a particular case of \'Ecalle's theory of resurgent functions. At constant-type points 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 to the study of a fundamental solution (which is the ``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.