C1-generic conservative diffeomorphisms have trivial centralizer

TL;DR

This paper proves that generic C1 symplectomorphisms and volume-preserving diffeomorphisms on connected manifolds have trivial centralizers, indicating a form of generic simplicity in their symmetry groups.

Contribution

It establishes that residual subsets of these diffeomorphism spaces have trivial centralizers, extending understanding of generic properties in conservative dynamical systems.

Findings

Residual sets with trivial centralizer in C1 symplectomorphisms

Residual sets with trivial centralizer in C1 volume-preserving diffeomorphisms

Trivial centralizer is a generic property in these spaces

Abstract

We prove that the spaces of C1 symplectomorphisms and of C1 volume-preserving diffeomorphisms of connected manifolds both contain residual subsets of diffeomorphisms whose centralizers are trivial. (Les diff\'eomorphismes conservatifs C1-g\'en\'eriques ont un centralisateur trivial. Nous montrons que l'espace des symplectomorphismes de classe C1 et l'espace des diff\'eomomorphismes de classe C1 pr\'eservant une forme volume contiennent tous deux des sous-ensembles r\'esiduels de diff\'eomorphismes dont le centralisateur est trivial.)