On approximative embeddability of diffeomorphisms in C1-flows

A function f:I→I is said to be C1-embeddable if there exists a C1-flow (iteration group) {ft:I→I,t∈R} such that f1=f. The C1-embeddability on a compact interval I is a rare property. It is known that even C∞-diffeomorphisms with two hyperbolic fixed points need not be C1-embeddable. However, every Cr-diffeomorphism, for r≥2, with one hyperbolic fixed point is uniquely embeddable in a Cr-flow. We consider the problem how to correct a given diffeomorphism with two hyperbolic fixed points making it C1-embeddable. We prove that if f∈Diff20,1, 0 0 and every diffeomorphism g such that supp f−g⊂a−ϵ,a+ϵ, ga=fa and g′a=f′aθf for a suitable chosen θf, there exists a unique C1-embeddable function f˜ such that f=f˜ in 0,1∖a−ϵ,f−1a and f˜=g in a−ϵ,a. We determine the coefficient θf and we give a necessary and sufficient condition for the best C1-embeddable approximation of f that is such that g = f.