2023/03/07 by Han, Bo, Wen, Xiao
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2303.03636
We introduce a new version of expansiveness similar to separating property for flows. Let M be a compact Riemannian manifold without boundary and X be a C1 vector field on M that generates a flow φt on M. We call X \it rescaling separating on a compact invariant set Λ of X if there is δ>0 such that, for any x,y∈ Λ, if d(φt(x), φt(y))≤ δ‖X(φt(x))‖ for all t∈ \mathbb R, then y∈\rm Orb(x). We prove that if X is rescaling separating on Λ and every singularity of X in Λ is hyperbolic, then for any C1 vector field Y, if the flow generated by Y is commuting with φt on Λ, then Y is collinear to X on Λ. As applications of the result, we show that the centralizer of a rescaling separating C1 vector field without nonhyperbolic singularity is quasi-trivial and there is is an open and dense set U\subsetX1(M) such that for any star vector field X\inU, the centralizer of X is collinear to X on the chain recurrent set of X.