We relate the Mather invariant of diffeomorphisms of the (closed) interval to their asymptotic distortion. For maps with only parabolic fixed points, we show that the former is trivial if and only if the latter vanishes. As a consequence, we obtain that such a diffeomorphism of the interval with no fixed point in the interior contains the identity in the closure of its $C^{1+bv}$ conjugacy class if and only if it is the time-1 map of a $C^1$ vector field. A corollary of this is that diffeomorphisms that do not arise from vector fields are undistorted in the whole group of interval diffeomorphisms. Several related results in other regularity classes are obtained, and many open questions are addressed.

我们将（闭）区间的微分同胚的 Mather 不变量与它们的渐近失真联系起来。对于只有抛物线不动点的地图，我们证明前者是微不足道的，当且仅当后者消失。结果，我们得到这样一个在内部没有不动点的区间的微分同胚在它的闭包中包含恒等式。$C^{1+乙v}$ 共轭类当且仅当它是一个的时间 1 映射 $C^1$矢量场。一个推论是，不是由向量场产生的微分同胚在整个区间微分同胚组中是不失真的。获得了其他规律类中的几个相关结果，并解决了许多开放性问题。