Common fixed points of commuting holomorphic maps in the unit ball of ℂⁿ

Abstract

Let B n \mathbb {B}^n be the unit ball of C n \mathbb {C}^n ( n > 1 n>1 ). We prove that if f , g ∈ H o l ( B n , B n ) f,g \in \mathrm {Hol}(\mathbb {B}^n, \mathbb {B}^n) are holomorphic self-maps of B n \mathbb {B}^n such that f ∘ g = g ∘ f f \circ g = g \circ f , then f f and g g have a common fixed point (possibly at the boundary, in the sense of K K -limits). Furthermore, if f f and g g have no fixed points in B n \mathbb {B}^n , then they have the same Wolff point, unless the restrictions of f f and g g to the one-dimensional complex affine subset of B n \mathbb {B}^n determined by the Wolff points of f f and g g are commuting hyperbolic automorphisms of that subset.