Rigidity of saddle loops

TL;DR

This paper classifies saddle loops in holomorphic foliations using Poincaré maps, establishes their formal rigidity under certain conditions, and lists all analytic classes of Liouville-integrable saddle loops, advancing understanding of their structure.

Contribution

It provides a classification of saddle loops via Poincaré maps, proves formal rigidity for multivalued maps, and enumerates all analytic classes of Liouville-integrable saddle loops.

Findings

Saddle loops are classified by their Poincaré return map.

Formal rigidity holds when the Poincaré map is multivalued.

Complete list of analytic classes of Liouville-integrable saddle loops.

Abstract

A saddle loop is a germ of a holomorphic foliation near a homoclinic saddle connection. We prove that they are classied by their Poincar{\'e} rst-return map. We also prove that they are formally rigid when the Poincar{\'e} map is multivalued. Finally, we provide a list of all analytic classes of Liouville-integrable saddle loops.