Centralizer of expansive flows and $\mathbb R^d$-actions

In seminal papers Smale conjectured that most diffeomorphisms should have trivial centralizer. That is, maps commuting with the original dynamical system are necessarily a power of it. Also, a similar conjecture  about centralizers of smooth flows can be formulated. In this context, we present a result obtained with Paulo Varandas (UFBA) and Jorge Rocha (UP) about  the existence of an open and dense subset of Komuro-expansive fields with singularities whose elements have centralizer unstable. We also indroduce the concept of expansiveness to $\mathbb R^d$-actions and describe the centralizer of this actions and present some examples and applications of our main results.   This work is related to my PhD thesis realized in UFBA - Brazil and supervised by Professor Paulo Varandas.