# A criterion for the triviality of the centralizer for vector fields and applications

## TL;DR

This paper provides a criterion to determine when the centralizer of vector fields and flows is trivial, with applications to various classes including homoclinic classes, generic volume-preserving, Hamiltonian, and certain non-hyperbolic flows.

## Contribution

It introduces a new criterion for triviality of the $C^1$-centralizer and applies it to diverse classes of flows, expanding understanding of flow symmetries.

## Key findings

- Triviality of the $C^1$-centralizer at homoclinic classes.
- Generic volume-preserving flows have trivial $C^1$-centralizer.
- Certain non-hyperbolic vector fields with Lorenz attractors also have trivial centralizer.

## Abstract

In this paper we establish a criterion for the triviality of the $C^1$-centralizer for vector fields and flows. In particular we deduce the triviality of the centralizer at homoclinic classes of $C^r$ vector fields ($r\ge 1$). Furthermore, we show that the set of flows whose $C^1$-centralizer is trivial include: (i) $C^1$-generic volume preserving flows, (ii) $C^2$-generic Hamiltonian flows on a generic and full Lebesgue measure set of energy levels, and (iii) $C^1$-open set of non-hyperbolic vector fields (that admit a Lorenz attractor). We also provide a criterion for the triviality of the $C^0$-centralizer of continuous flows.