Wescley Bonomo | Searcharxiv

arXiv2019

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

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.

math.DS↗

arXiv2017

The centralizer of Komuro-expansive flows and expansive R^d-actions

In this paper we study the centralizer of flows and $\mathbb R^d$-actions on compact Riemannian manifolds. We prove that the centralizer of every $C^\infty$ Komuro-expansive flow with non-ressonant singularities is trivial, meaning it is the smallest possible, and deduce there exists an open and dense subset of geometric Lorenz attractors with trivial centralizer. We show that $\mathbb R^d$-actions obtained as suspension of $\mathbb Z^d$-actions are expansive if and only if the same holds for the $\mathbb Z^d$-actions. We also show that homogeneous expansive $\mathbb R^d$-actions have quasi-trivial centralizers, meaning that it consists of orbit invariant, continuous linear reparametrizations of the $\mathbb R^d$-action. In particular, homogeneous Anosov $\mathbb R^d$-actions have quasi-trivial centralizer.

math.DS↗