A connectedness result for commuting diffeomorphisms of the interval

Abstract

AbstractLet 𝒟r+[0,1], r≥1, denote the group of orientation-preserving 𝒞r diffeomorphisms of [0,1]. We show that any two representations of ℤ2 in 𝒟r+[0,1], r≥2, are connected by a continuous path of representations of ℤ2 in 𝒟1+[0,1] . We derive this result from the classical works by G. Szekeres and N. Kopell on the 𝒞1 centralizers of the diffeomorphisms of [0,1) that are at least 𝒞2 and fix only 0 .