Centralizers of C^1-generic diffeomorphisms

TL;DR

This paper investigates the structure of centralizers in C^1-generic diffeomorphisms, showing that typically they are trivial in some spaces but can be large in others, like circle and sphere diffeomorphisms.

Contribution

It establishes that in certain spaces, C^1-generic diffeomorphisms have trivial centralizers, while in others, they can have centralizers containing continuous subgroups.

Findings

Residual subsets with trivial centralizers in symplectomorphisms and volume-preserving diffeomorphisms.

Dense subsets with large centralizers in circle and sphere diffeomorphisms.

Abstract

On the one hand, we prove that the spaces of C^1 symplectomorphisms and of C^1 volume-preserving diffeomorphisms both contain residual subsets of diffeomorphisms whose centralizers are trivial. On the other hand, we show that the space of C^1 diffeomorphisms of the circle and a non-empty open set of C^1 diffeomorphisms of the two-sphere contain dense subsets of diffeomorphisms whose centralizer has a sub-group isomorphic to R.