On the iteration of holomorphic self-maps of ℂ*

Letf be a holomorphic self-map of the punctured plane ℂ*=ℂ\{0} with essentially singular points 0 and ∞. In this note, we discuss the setsI 0(f)={z ∈ ℂ*:f n (z) → 0,n → ∞} andI ∞(f)={z ∈ ℂ*:f n (z) → 0,n → ∞}. We try to find the relation betweenI 0(f),I ∞(t) andJ(f). It is proved that both the boundary ofI 0(f) and the boundary ofI ∞)f) equal toJ(f),I 0(f) ∩J(f) ≠ θ andI ∞(f) ∩J(f) ≠ θ. As a consequence of these results, we find bothI 0(f) andI ∞(f) are not doubly-bounded.