Epsilon-neighborhoods of orbits and formal classification of parabolic diffeomorphisms

TL;DR

This paper investigates the dynamics of parabolic diffeomorphisms near fixed points, revealing how formal classification can be derived from the asymptotic behavior of the orbit's epsilon-neighborhoods and their fractal properties.

Contribution

It introduces a novel method linking the formal classification of parabolic diffeomorphisms to the asymptotic analysis of directed areas of epsilon-neighborhoods of orbits.

Findings

Formal classification derived from epsilon-neighborhood asymptotics

Identification of fractal properties like box dimension and Minkowski content

Geometric interpretation of coefficients and constants

Abstract

In this article we study the dynamics generated by germs of parabolic diffeomorphisms f : (C; 0)->(C; 0) tangent to the identity. We show how formal classification of a given parabolic diffeomorphism can be deduced from the asymptotic development of what we call directed area of the epsilon-neighborhood of any orbit near the origin. Relevant coefficients and constants in the development have a geometric meaning. They present fractal properties of the orbit, namely its box dimension, Minkowski content and what we call residual content.