International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 67, Pages 3685-3693
doi:10.1155/S0161171204208031
The geometry of some natural conjugacies in ℂn dynamics
1Department of Mathematics, University of Michigan, 2074 East Hall, Ann Arbor 48109-1109, MI, USA
2Department of Mathematics and Statistics, Wichita State University, Wichita 67260-0033, KS, USA
Received 6 August 2002
Copyright © 2004 John W. Robertson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We show that under some simple conditions a topological conjugacy h between two holomorphic self-maps f1 and f2 of complex n-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.