The geometry of some natural conjugacies in ℂn dynamics

We show that under some simple conditions a topological conjugacy h between two holomorphic self-maps f1 and f2 of complexn-dimensional projective space Pn lifts canonically to a topological conjugacy H between the two corresponding polynomial self-maps of Cn+1, and this conjugacy relates the two Green functions of f1 and f2. These conjugacies are interesting because their geometry is not inherited entirely from the geometry of the conjugacy on Pn. Part of the geometry of such a conjugacy is given (locally) by a complex-valued function whose absolute value is determined by the Green functions for the two maps, but whose argument seems to appear out of thin air. We work out the local geometry of such conjugacies over the Fatou set and over Fatou varieties of the original map.