Pseudoarcs, pseudocircles, Lakes of Wada and generic maps on

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.