Let f be a holomorphic self-map of the punctured plane C = C\{0} with
essentially singular points 0 and infinity. In this note, we discuss
the sets I-0(f) = {z is an element of C : f(n)(z) --> 0,n --> infinit
y} and I-infinity(f) = {z is an element of C : f(n)(z) --> infinity,n
--> infinity}.We try to find the relation between I-0(f),I-infinity(f
) and J(f). It is proved that both the boundary of I-0(f) and the boun
dary of I-infinity(f) equal to J(f), I-0(f) boolean AND J(f) not equal
0 and I-infinity(f) boolean AND J(f) not equal 0. As a consequence of
these results, we find both <(I-0(f))over bar> and I-infinity(f) are
not doubly-bounded.