Birational conjugacies between endomorphisms on the projective plane

TL;DR

This paper proves that in characteristic 0, conjugacy of dominant endomorphisms of the projective plane by birational maps implies conjugacy by automorphisms, with counterexamples provided in positive characteristic.

Contribution

It establishes a rigidity result for conjugacies in characteristic 0 and presents counterexamples in positive characteristic.

Findings

In characteristic 0, birational conjugacy implies automorphic conjugacy.

Counterexamples exist in positive characteristic.

Provides a clear distinction between behaviors in different characteristics.

Abstract

We proved that, in characteristic 0, if two dominant endomorphisms of the projective plane of degree at least 2 are conjugate by some birational transformation, then they are conjugate by an automorphism. We also gave counterexamples in positive characteristic.