The Fatou Set for Critically Finite Maps

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.

5 pages

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