Symmetries of smooth flows and their time-t maps

We study the discrete centralizers of smooth flows and their time-t maps. Centralizers describe the symmetries of the respective dynamics. Given a $C^1$ -flow $(\varphi_t)_t$ on a compact manifold preserving an ergodic probability measure $\mu$, we prove that there exists a Baire generic subset of values of $t \in \mathbb{R}$ such that the centralizer of the time-$t$ map coincides with the discrete centralizer of the flow on $supp(\mu)$. The result relies on the ergodicity of generic time-t maps, established by Pugh and Shub in the seventies. We derive several consequences, including for Anosov flows a generic volume preserving flows. Joint work with Jorge Rocha and Paulo Varandas