Mather invariant, distortion, and conjugates for diffeomorphisms of the interval

TL;DR

This paper explores the relationship between the Mather invariant and asymptotic distortion of interval diffeomorphisms, revealing conditions under which certain maps are conjugate to the identity or arise from vector fields.

Contribution

It establishes a link between the Mather invariant and asymptotic distortion for interval diffeomorphisms, characterizes maps conjugate to the identity, and discusses distortion properties in various regularity classes.

Findings

Mather invariant is trivial iff asymptotic distortion vanishes for maps with parabolic fixed points.

Diffeomorphisms with no interior fixed points are limits of conjugates of the identity iff they are time-1 maps of C^1 vector fields.

Diffeomorphisms not arising from vector fields are undistorted in the group of interval diffeomorphisms.

Abstract

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 interval diffeomorphisms. Several related results in other regularity classes are obtained, and many open questions are addressed.