International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 19, Pages 1233-1240
doi:10.1155/S0161171203208048

Fatou maps in ℙn dynamics

John W. Robertson1,2

1Department of Mathematics, University of Michigan, 2074 East Hall, Ann Arbor 48109-1109, MI, USA
2Department of Mathematics, University of Toronto, Toronto M5S 3G3, Ontario, Canada

Received 7 August 2002

Copyright © 2003 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 study the dynamics of a holomorphic self-map f of complex projective space of degree d>1 by utilizing the notion of a Fatou map, introduced originally by Ueda (1997) and independently by the author (2000). A Fatou map is intuitively like an analytic subvariety on which the dynamics of f are a normal family (such as a local stable manifold of a hyperbolic periodic point). We show that global stable manifolds of hyperbolic fixed points are given by Fatou maps. We further show that they are necessarily Kobayashi hyperbolic and are always ramified by f (and therefore any hyperbolic periodic point attracts a point of the critical set of f). We also show that Fatou components are hyperbolically embedded in ℙn and that a Fatou component which is attracted to a taut subset of itself is necessarily taut.