On set-valued iteration groups generated by commuting functions

Let II be an open interval and f,g:I→If,g:I→I be commuting homeomorphisms such that ft} and Ft(x):=[f−t(x),f+t(x)], x∈Ix∈I. We show that the family {Ft:I→cc(I),t∈R}{Ft:I→cc(I),t∈R} is an iteration group in the sense of set-valued functions such that f(x)∈F1(x)f(x)∈F1(x) and g(x)∈Fs(x)g(x)∈Fs(x). If ff and gg are embeddable in an iteration group of continuous functions {ft:t∈R}{ft:t∈R}, then there exists an additive function γγ such that ft(x)∈Fγ(t)(x)ft(x)∈Fγ(t)(x), t∈Rt∈R. Several other properties of set-valued iteration groups {Ft:t∈R}{Ft:t∈R} as well as iteration groups {f−t:t∈R} and {f+t:t∈R} are given.