On set-valued iteration groups generated by commuting functions

Let I be an open interval and f , g : I → I be commuting homeomorphisms such that f < i d , g < i d and inf { n m : n , m ∈ N , f n < g m } = : s ∉ Q . Define f + t : = sup { f n ∘ g − m : n , m ∈ N , n − s m < t } , f − t : = inf { f n ∘ g − m : n , m ∈ N , n − s m > t } and F t ( x ) : = [ f − t ( x ) , f + t ( x ) ] , x ∈ I . We show that the family { F t : I → c c ( I ) , t ∈ R } is an iteration group in the sense of set-valued functions such that f ( x ) ∈ F 1 ( x ) and g ( x ) ∈ F s ( x ) . If f and g are embeddable in an iteration group of continuous functions { f t : t ∈ R } , then there exists an additive function γ such that f t ( x ) ∈ F γ ( t ) ( x ) , t ∈ R . Several other properties of set-valued iteration groups { F t : t ∈ R } as well as iteration groups { f − t : t ∈ R } and { f + t : t ∈ R } are given.