Sur les endomorphismes holomorphes permutables de $$\mathbb{P}^k$$

Abstract

Let f1, f2 be holomorphic endomorphisms of \(\mathbb{P}^k\) of degrees d1 ≥ 2, d2 ≥ 2. Assume that \(f_1 \circ f_2 = f_2 \circ f_1\) and that \(d_1^{n_1 } \ne d_2^{n_2 }\) for all integers n1, n2. We then show that fj are critically finite. Moreover there is an orbifold (\(\mathbb{P}^k\), n) such that f1, f2 are coverings of (\(\mathbb{P}^k\), n). In \(\mathbb{P}^k\) case we give the list of commuting pairs satisfying the above conditions. We also show that any endomorphism is approximable by endomorphisms commuting only with their iterates.