Quasianalytic Ilyashenko algebras

TL;DR

This paper constructs a quasianalytic Hardy field of functions with logarithmic power series as asymptotics, closed under differentiation and composition, containing key transition maps for planar vector fields.

Contribution

It introduces a new quasianalytic field of germs at infinity with specific closure properties and includes all transition maps of hyperbolic saddles.

Findings

Constructed a quasianalytic field with logarithmic asymptotics.

Proved the field is closed under differentiation and composition.

Contained all transition maps of hyperbolic saddles.

Abstract

I construct a quasianalytic field $\mathcal F$ of germs at $+\infty$ of real functions with logarithmic generalized power series as asymp\-totic expansions, such that $\mathcal F$ is closed under differentiation and $\log$-composition; in particular, $\mathcal F$ is a Hardy field. Moreover, the field $\mathcal F \circ (-\log)$ of germs at $0^+$ contains all transition maps of hyperbolic saddles of planar real analytic vector fields.