×

The geometry of some natural conjugacies in \(\mathbb{C}^n\) dynamics. (English) Zbl 1068.37029

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
PDFBibTeX XMLCite
Full Text: DOI EuDML