The Fatou set for critically finite maps

Rong, Feng

doi:10.1090/S0002-9939-08-09358-1

The Fatou set for critically finite maps
HTML articles powered by AMS MathViewer

by Feng Rong PDF

Proc. Amer. Math. Soc. 136 (2008), 3621-3625 Request permission

Abstract:

It is a classical result in complex dynamics of one variable that the Fatou set for a critically finite map on $\mathbf {P}^1$ consists of only basins of attraction for superattracting periodic points. In this paper, we deal with critically finite maps on $\mathbf {P}^k$. We show that the Fatou set for a critically finite map on $\mathbf {P}^2$ consists of only basins of attraction for superattracting periodic points. We also show that the Fatou set for a $k-$critically finite map on $\mathbf {P}^k$ is empty.

References

Araceli M. Bonifant and Marius Dabija, Self-maps of ${\Bbb P}^2$ with invariant elliptic curves, Complex manifolds and hyperbolic geometry (Guanajuato, 2001) Contemp. Math., vol. 311, Amer. Math. Soc., Providence, RI, 2002, pp. 1–25. MR 1940161, DOI 10.1090/conm/311/05444
John Erik Fornæss and Nessim Sibony, Complex dynamics in higher dimension. I, Astérisque 222 (1994), 5, 201–231. Complex analytic methods in dynamical systems (Rio de Janeiro, 1992). MR 1285389
Mattias Jonsson, Some properties of $2$-critically finite holomorphic maps of $\textbf {P}^2$, Ergodic Theory Dynam. Systems 18 (1998), no. 1, 171–187. MR 1609475, DOI 10.1017/S0143385798097521
John Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309
Thurston, W.; On the combinatorics and dynamics of rational maps, preprint.
Tetsuo Ueda, Critical orbits of holomorphic maps on projective spaces, J. Geom. Anal. 8 (1998), no. 2, 319–334. MR 1705160, DOI 10.1007/BF02921645