Hedgehogs in higher dimensions and their applications

TL;DR

This paper explores the dynamics of holomorphic diffeomorphisms in higher-dimensional complex spaces with a neutral eigenvalue, establishing quasiconformal conjugacy to holomorphic maps and extending one-dimensional results to multiple dimensions.

Contribution

It introduces a method to relate higher-dimensional dynamics to one-dimensional cases via quasiconformal conjugacy, enabling transfer of known results.

Findings

Local center manifold maps are quasiconformally conjugate to holomorphic maps.

Results from one complex dimension are extended to higher dimensions.

Provides new tools for analyzing neutral fixed points in complex dynamics.

Abstract

In this paper we study the dynamics of germs of holomorphic diffeomorphisms of $(\mathbb{C}^{n},0)$ with a fixed point at the origin with exactly one neutral eigenvalue. We prove that the map on any local center manifold of $0$ is quasiconformally conjugate to a holomorphic map and use this to transport results from one complex dimension to higher dimensions.