International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 67, Pages 3685-3693

The geometry of some natural conjugacies in n dynamics

John W. Robertson1,2

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.


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 n lifts canonically to a topological conjugacy H between the two corresponding polynomial self-maps of n+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 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.