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

Abstract Let F : S 1 → S 1 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 { F t : S 1 → cc [ S 1 ] , t ∈ R } , which is regular in a sense and in which F can be embedded i.e. F ( x ) ∈ F 1 ( x ) . We also determine a maximal countable and dense subgroup T ⊂ R such that { F t : S 1 → cc [ S 1 ] , t ∈ T } has a continuous selection { f t : S 1 → S 1 , t ∈ T } being the best regular embedding of F . If there exists a nonmeasurable embedding { f t : S 1 → S 1 , t ∈ R } of F , then there exists an additive function γ : R → T such that f t ( z ) ∈ F γ ( t ) ( z ) , t ∈ R . We determine a unique maximal subgroup T with this property.