Commuting analytic functions

Abstract

Let f f and g g (not conformal automorphisms of the unit disk) be analytic mappings of the unit disk into itself. We say f f and g g commute if f ∘ g = g ∘ f f \circ g = g \circ f . This paper characterizes those functions g g that commute with a given function f f . Several corollaries of this characterization give qualitative information about g g given similar information about f f , and examples are given in each case to show the limitations of the conclusions. Some of the qualitative properties considered are univalence, fixed point sets, and whether two such g g must commute with each other.