Pseudoarcs, Pseudocircles, Lakes of Wada and Generic Maps on S

We prove a Bruckner-Garg type theorem for the fiber structure of a generic map from a continuum X into the unit interval I. We also study the specific case of X = S2. We show that each nondegenerate component of each fiber of a generic map in C(S2, I) is figure-eight-like. This together with a result by Krasinkiewicz and Levin gives that each nondegenerate component of each fiber of a generic map in C(S2, I) is hereditarily indecomposable and figure-eight-like. We also show that pseudoarcs, pseudocircles and Lakes of Wada appear in abundance in fibers of a generic map in C(S2, I). We also exhibit a general method for proving when a P -like hereditarily indecomposable continuum is Q-like when Q is a certain graph containing P .