Commuting pairs of endomorphisms of

Abstract

We consider commuting pairs of holomorphic endomorphisms of $\mathbb{P}^{2}$ with disjoint sequence of iterates. The case that has not been completely studied is when their degrees coincide after some number of iterations. We show in this case that they are either commuting Lattès maps or commuting homogeneous polynomial maps of $\mathbb{C}^{2}$ inducing a Lattès map on the line at infinity.