Abstract
Let f be a non-Möbius self-map of C *. Write J *(f) for the Julia Set of f in Ĉ and J(f) for the closure of J *(f) in Č. Then we have one of the three cases (i)J(f), J *(f) are connected, (ii)J(f) is connected, J *(f) has infinitely many components, (iii)J(f) has two components, J *(f) has infinitely many components. All three cases can occur. It is known that F(f) has at most one multiply-connected domain A whose connectivity must in fact be two. If in addition A is relatively compact in nC *, then either (i) A is a Herman ring, (ii) A is pre-periodic but nor periodic, or (iii) A is a wandering component Example of all three cases are constructed.
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