$C^1$-generic sectional Axiom A flows have only trivial symmetries

Abstract

The space of continuous time symmetries of a flow $(X_t{)}_{t\in \mathbb{R}}$ consists of the space of flows commuting with $(X_t{)}_{t\in \mathbb{R}}$. We prove that $C^1$-generic singular Axiom A flows on three-dimensional manifolds, $C^1$-generic Axiom A flows in arbitrary dimension and $C^1$-generic flows on surfaces have trivial symmetries, that is, any flow commuting with any of the latter is a constant reparametrization of it.