On the centralizer of vector fields: criteria of triviality and genericity results
Abstract
In this paper, we investigate the question of whether a typical vector field on a compact connected Riemannian manifold Md has a `small' centralizer. In the C1 case, we give two criteria, one of which is C1-generic, which guarantees that the centralizer of a C1-generic vector field is indeed small, namely collinear. The other criterion states that a C1 separating flow has a collinear C1-centralizer. When all the singularities are hyperbolic, we prove that the collinearity property can actually be promoted to a stronger one, refered as quasi-triviality. In particular, the C1-centralizer of a C1-generic vector field is quasi-trivial. In certain cases, we obtain the triviality of the centralizer of a C1-generic vector field, which includes C1-generic Axiom A (or sectional Axiom A) vector fields and C1-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 trivial (as small as it can be), and we show that in higher regularity, collinearity and triviality of the Cd-centralizer are equivalent properties for a generic vector field in the Cd topology. We also obtain that in the non-uniformly hyperbolic scenario, with regularity C2, the C1-centralizer is trivial.
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