  The goal of this paper is to study some basic properties of the Fatou and
Julia sets for a family of holomorphic endomorphisms of $\mathbb{C}^k,\; k \ge
2$. We are particularly interested in studying these sets for semigroups
generated by various classes of holomorphic endomorphisms of $\mathbb{C}^k,\; k
\ge 2.$ We prove that if the Julia set of a semigroup $G$ which is generated by
endomorphisms of maximal generic rank $k$ in $\mathbb{C}^k$ contains an
isolated point, then $G$ must contain an element that is conjugate to an upper
triangular automorphism of $\mathbb{C}^k.$ This generalizes a theorem of
Fornaess-Sibony. Secondly, we define recurrent domains for semigroups and
provide a description of such domains under some conditions.
