Robertson, John W. The geometry of some natural conjugacies in \(\mathbb{C}^n\) dynamics. (English) Zbl 1068.37029 Int. J. Math. Math. Sci. 2004, No. 65-68, 3685-3693 (2004). Summary: We show that under some simple conditions a topological conjugacy \(h\) between two holomorphic selfmaps \(f_1\) and \(f_2\) of complex \(n\)-dimensional projective space \(\mathbb{P}^n\) lifts canonically to a topological conjugacy \(H\) between the two corresponding polynomial selfmaps of \(\mathbb{C}^{n+1}\), and this conjugacy relates the two Green functions of \(f_1\) and \(f_2\). These conjugacies are interesting because their geometry is not inherited entirely from the geometry of the conjugacy on \(\mathbb{P}^n\). 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. MSC: 37F50 Small divisors, rotation domains and linearization in holomorphic dynamics 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables 32U05 Plurisubharmonic functions and generalizations Keywords:topological conjugacy; holomorphic selfmaps; polynomial selfmaps; Green functions; complex-valued function; Fatou set; Fatou varieties PDFBibTeX XMLCite \textit{J. W. Robertson}, Int. J. Math. Math. Sci. 2004, No. 65--68, 3685--3693 (2004; Zbl 1068.37029) Full Text: DOI EuDML