Rigidity of saddle loops

TL;DR

The paper develops a comprehensive framework for the analytic classification and rigidity of saddle loops via complexification, Dulac germs, and holonomy data. It shows that the analytic type of a saddle loop is determined by its Poincaré first return map, and that in the real setting the loop germ and Poincaré map are rigidly linked under fibered equivalence. By introducing prepared saddles, corner and regular transitions, monodromy/holo­nomy data, and the Dulac group, the authors establish both formal and topological rigidity results, and provide a complete list of Liouville-integrable loop germs, including linear, Bernoulli, and flow-embedded models. The work also connects saddle loops to birational foliation theory and Liouville-type integrable systems in symplectic geometry, offering invariants and a rigorous structure for future classifications. Overall, the results yield precise correspondences between loop germs, their Poincaré maps, and integrability properties with broad implications for complex dynamics and foliation theory.

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{é} rst-return map. We also prove that they are formally rigid when the Poincar{é} map is multivalued. Finally, we provide a list of all analytic classes of Liouville-integrable saddle loops.

Figures (15)

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Theorems & Definitions (141)

• Remark 1.1
• Corollary
• Remark 1.2
• Remark 1.3
• Proposition 2.1: see IlyaDu
• Proposition 2.2
• Proposition 2.3
• proof
• Remark 2.4
• Proposition 2.5: Ramified classification Per