On iteration groups of C1C1-diffeomorphisms with two fixed points

Let I=[0,1]I=[0,1] and let f:I→If:I→I satisfy the following general hypothesis: f∈Diff1[I]f∈Diff1[I], f(0)=0f(0)=0, f(1)=1f(1)=1, f(x)≠xf(x)≠x, x∈IntI and the condition (A): f′(x)=s+0(xδ)f′(x)=s+0(xδ), x→0x→0, and f′(x)=M+0(δ|x−1|)f′(x)=M+0(|x−1|δ), x→1x→1, with s<1