A maximum modulus theorem for functions admitting stokes phenomena, and specific cases of Dulac's theorem*

Abstract

We study large classes of real-valued analytic functions that naturally emerge in the understanding of Dulac's problem, which addresses the finiteness of limit cycles in planar differential equations. Building on a maximum modulus-type result, our main statement essentially follows. Namely, for any function belonging to these classes, the following dichotomy holds: either it has isolated fixed points or it coincides with the identity. As an application, we prove that the non-accumulation of limit cycles holds for vector fields around a specific class of the so-called superreal polycycles.