Commuting pairs of endomorphisms of P^2

TL;DR

This paper investigates pairs of commuting holomorphic endomorphisms of the complex projective plane, classifying their structure when their degrees align after some iterations, revealing they are either Lattès maps or certain polynomial maps.

Contribution

It provides a classification of commuting pairs of endomorphisms of P^2 with coinciding degrees after some iterations, identifying specific types of maps involved.

Findings

Such pairs are either commuting Lattès maps or commuting homogeneous polynomial maps.

The classification clarifies the structure of commuting endomorphisms with degree coincidence.

The results extend understanding of dynamical systems on P^2.

Abstract

We consider commuting pairs of holomorphic endomorphisms of P^2 with disjoint sequence of iterates. The remaining case to be studied is when their degrees coincide after some number of iterations. We show in this case that they are either commuting Latt\`es maps or commuting homogeneous polynomial maps of C^2 inducing a Latt\`es map on the hyperplane at infinity.