Dynamics of symmetric holomorphic maps on projective spaces

TL;DR

This paper studies the complex dynamics of symmetric holomorphic maps on projective spaces, revealing their Fatou sets consist of attractive basins and confirming they satisfy Axiom A, with implications for understanding symmetric dynamical systems.

Contribution

It introduces a family of critically finite symmetric holomorphic maps on projective spaces and analyzes their dynamical properties, including Fatou sets and hyperbolicity.

Findings

Fatou set comprises attractive basins of superattracting points

All maps in the family satisfy Axiom A

Maps exhibit symmetric dynamical behavior

Abstract

We consider complex dynamics of a critically finite holomorphic map from P^k to P^k, which has symmetries associated with the symmetric group S_{k+2} acting on P^k, for each k \ge 1. The Fatou set of each map of this family consists of attractive basins of superattracting points. Each map of this family satisfies Axiom A.