Epsilon-neighborhoods of orbits of parabolic diffeomorphisms and cohomological equations

TL;DR

This paper investigates the analytic properties of epsilon-neighborhoods of orbits of parabolic diffeomorphisms, revealing a sectorially analytic principal part of the area that satisfies a cohomological equation, leading to new classification insights.

Contribution

It introduces a cohomological equation related to epsilon-neighborhoods of orbits and establishes conditions for its solutions, offering a novel classification framework for parabolic diffeomorphisms.

Findings

The principal part of the area is sectorially analytic in the initial point.

A cohomological equation characterizes the principal part and relates to analytic classification.

New classification types for diffeomorphisms are proposed based on the solutions to this equation.

Abstract

In this article, we study analyticity properties of (directed) areas of epsilon-neighborhoods of orbits of parabolic germs. The article is motivated by the question of analytic classification using epsilon-neighborhoods of orbits in the simplest formal class. We show that the coefficient in front of epsilon^2 term in the asymptotic expansion in epsilon, which we call the principal part of the area, is a sectorially analytic function of initial point of the orbit. It satisfies a cohomological equation similar to the standard trivialization equation for parabolic diffeomorphisms. We give necessary and sufficient conditions on a diffeomorphism f for the existence of globally analytic solution of this equation. Furthermore, we introduce new classification type for diffeomorphisms implied by this new equation and investigate the relative position of its classes with respect to the analytic classes.