Conjugacy for Orientation-preserving Diffeomorphisms that fix only the Ends of an Interval

Given a group G, the conjugacy problem in G is the problem of giving an effective procedure for determining whether or not two given elements f, g ∈ G are conjugate. This paper is about the conjugacy problem in the groupG = D(I) of all orientation-preserving diffeomorphisms of an interval I ⊂ R. The kernel of this problem is the special case in which the diffeomorphisms f, g ∈ D(I) are fixed-point-free on the interior J of I. The problem is trivial if I is open. We present an effective way to approach it when I is half-open or compact