Normalization of strongly hyperbolic logarithmic transseries and complex Dulac germs

TL;DR

This paper develops normal forms for strongly hyperbolic logarithmic transseries and complex Dulac germs, using fixed point theorems and algorithmic limits, to establish their analytic normalizability within specific domains.

Contribution

It introduces a method to normalize strongly hyperbolic logarithmic transseries and complex Dulac germs, providing explicit algorithms and proving their analytic normalizability.

Findings

Normal forms for strongly hyperbolic logarithmic transseries are established.

Strongly hyperbolic complex Dulac germs are shown to be analytically normalizable.

Normalizations are obtained via fixed point theorems and Picard sequences.

Abstract

We give normal forms for strongly hyperbolic logarithmic transseries f = z^r + ... (r is a positive real number nonequal to 1), with respect to parabolic logarithmic normalizations. These normalizations are obtained using fixed point theorems, and are given algorithmically, as limits of Picard sequences in appropriate formal topologies. The results are applied to describe the supports of normalizations and to prove that the strongly hyperbolic complex Dulac germs are analytically normalizable on standard quadratic domains inside the class of complex Dulac germs.