A topological characterisation of holomorphic parabolic germs in the plane

TL;DR

This paper characterizes the topological dynamics of holomorphic parabolic germs in the plane by linking the negation of a property introduced by Gambaudo and Pécou to their dynamics, enabling a new rotation set concept.

Contribution

It establishes a topological characterization of holomorphic parabolic germs through the negation of the linking property, extending the understanding of their dynamics.

Findings

Negation of Gambaudo-Pécou property characterizes holomorphic parabolic germs.

A rotation set for surface homeomorphisms around a fixed point is defined.

The rotation set is non-trivial for all but countably many conjugacy classes.

Abstract

Gambaudo and P\'ecou introduced the ``linking property'' to study the dynamics of germs of planar homeomorphims and provide a new proof of Naishul theorem in their paper "A topological invariant for volume preserving diffeomorphisms" (Ergodic Theory Dynam. Systems 15 (1995), no. 3, 535--541). In this paper we prove that the negation of Gambaudo-P\'ecou property characterises the topological dynamics of holomorphic parabolic germs. As a consequence, a rotation set for germs of surface homeomorphisms around a fixed point can be defined, and it will turn out to be non trivial except for countably many conjugacy classes.