On the Centralizers of Rescaling Separating Differentiable Vector Fields

In this paper, we introduce a new concept of expansiveness, similar to the separating property. Specifically, we consider a compact Riemannian manifold M without boundary and a C¹ vector field X on M, which generates a flow φ_t on M. We say that X is rescaling separating on a compact invariant set Λ of X if there is a constant δ > 0 such that, for any x, y ∈ Λ, if d(φ_t(x), φ_t(y)) ≤ δ∥X (φ_t(x))∥ for all t ∈ ℝ, then y ∈ Orb(x). We prove that if X is rescaling separating on Λ and every singularity of X in Λ is hyperbolic, then any C¹ vector field Y, whose flow commutes with φ_t on Λ, must be collinear to X on Λ. As applications of this result, we show that the centralizer of a rescaling separating C¹ vector field without nonhyperbolic singularity is quasi-trivial. We also proved that there is an open and dense set \({\cal U} \subset {{\cal X}^{1}}(M)\) $$\mathcal{U}\subset {\mathcal{X}}^1(M)$$ such that for any star vector field \(X \in {\cal U}\) $$X\in \mathcal{U}$$ , the centralizer of X is collinear to X on the chain recurrent set of X.