Almost reducibility, distortion and local perfection for diffeomorphisms of one-manifolds

TL;DR

This paper characterizes distortion elements in groups of smooth circle and real line diffeomorphisms, linking them to almost reducibility and local perfection, with implications for understanding their structure and dynamics.

Contribution

It establishes a precise equivalence between distortion and almost reducibility for diffeomorphisms, introducing new local perfection results and extending the theory to manifolds.

Findings

Distorted elements are exactly the almost reducible ones.

For fixed point diffeomorphisms, distortion corresponds to being a time-1 map of a non-hyperbolic vector field.

New local perfection results for diffeomorphisms of the real line.

Abstract

In this article, we characterize the distortion elements of the group of smooth diffeomorphisms of the circle and of the group of compactly supported smooth diffeomorphisms of the real line. More precisely, we prove that, in this context, an element is distorted if and only if it is almost reducible, that is if and only if it has conjugates arbitrarily close to an isometry. For diffeomorphisms with fixed points, we show that this is equivalent to being the time-1 map of a C 1 vector field without hyperbolic zero. The equivalence between distortion and almost reducibility relies on new more general results about distortion elements in groups of diffeomorphisms of manifolds and on a new local perfection result for the group of compactly supported smooth diffeomorphisms of the real line.