The C1 generic diffeomorphism has trivial centralizer

TL;DR

This paper proves that in the space of C1 diffeomorphisms on a compact manifold, a generic subset has trivial centralizer, answering a question posed by Smale.

Contribution

It establishes that the set of C1 diffeomorphisms with trivial centralizer is residual, providing a significant genericity result in dynamical systems.

Findings

Residual subset of C1 diffeomorphisms with trivial centralizer

Answers Smale's question on centralizers

Advances understanding of generic properties in dynamical systems

Abstract

Answering a question of Smale, we prove that the space of C1 diffeomorphisms of a compact manifold contains a residual subset of diffeomorphisms whose centralizers are trivial.