On the centralizer of vector fields: criteria of triviality and genericity results
arXiv:1810.05085 · doi:10.1007/s00209-020-02511-x
Abstract
In this paper, we investigate the question of whether a typical vector field on a compact connected Riemannian manifold has a `small' centralizer. In the case, we give two criteria, one of which is -generic, which guarantees that the centralizer of a -generic vector field is indeed small, namely \textit{collinear}. The other criterion states that a \textit{separating} flow has a collinear -centralizer. When all the singularities are hyperbolic, we prove that the collinearity property can actually be promoted to a stronger one, refered as \textit{quasi-triviality}. In particular, the -centralizer of a -generic vector field is quasi-trivial. In certain cases, we obtain the triviality of the centralizer of a -generic vector field, which includes -generic Axiom A (or sectional Axiom A) vector fields and -generic vector fields with countably many chain recurrent classes. For sufficiently regular vector fields, we also obtain various criteria which ensure that the centralizer is \textit{trivial} (as small as it can be), and we show that in higher regularity, collinearity and triviality of the -centralizer are equivalent properties for a generic vector field in the topology. We also obtain that in the non-uniformly hyperbolic scenario, with regularity , the -centralizer is trivial.
This is the final version, accepted in Mathematische Zeitschrift. New introduction and some proofs where rewritten and/or expanded, according to referee's suggestion. Also, a new appendix was added