Embeddability of homeomorphisms of the circle in set-valued iteration groups

Let F:S1→S1F:S1→S1 be a homeomorphism without periodic points. It is known that F is embeddable in a continuous iteration group if and only if F is minimal. We deal with F which is not minimal. In this case, F satisfying some additional assumptions can be embedded but only in a nonmeasurable iteration groups. There are infinitely many such nonmeasurable groups. We propose here a new approach to the problem of embeddability. For a given homeomorphism F   without periodic points we construct some substitute of an iteration group, namely the unique special set-valued iteration group {Ft:S1→cc[S1],t∈R}{Ft:S1→cc[S1],t∈R}, which is regular in a sense and in which F   can be embedded i.e. F(x)∈F1(x)F(x)∈F1(x). We also determine a maximal countable and dense subgroup T⊂RT⊂R such that {Ft:S1→cc[S1],t∈T}{Ft:S1→cc[S1],t∈T} has a continuous selection {ft:S1→S1,t∈T}{ft:S1→S1,t∈T} being the best regular embedding of F  . If there exists a nonmeasurable embedding {ft:S1→S1,t∈R}{ft:S1→S1,t∈R} of F  , then there exists an additive function γ:R→Tγ:R→T such that ft(z)∈Fγ(t)(z),t∈Rft(z)∈Fγ(t)(z),t∈R. We determine a unique maximal subgroup T with this property.