Topological equivalence of finitely determined real analytic plane-to-plane map-germs

TL;DR

This paper provides a comprehensive topological classification of finitely determined real analytic plane-to-plane map germs by leveraging the classification of smooth stable circle mappings, revealing their equivalence to cone mappings.

Contribution

It introduces a complete topological classification framework for finitely determined real analytic map germs based on stable circle mappings.

Findings

Finitely determined real analytic map germs are topologically equivalent to cones of circle mappings.

Complete classification of smooth stable circle mappings is achieved.

The classification facilitates understanding of the topological types of plane-to-plane map germs.

Abstract

Generic smooth plane-to-plane map germs are topologically equivalent to cones of mappings of the circle. We carry out a complete topological classification of smooth stable mappings of the circle and show how this classification leads, via the result mentioned above, to a topological classification of finitely determined real analytic plane-to-plane map germs.