On the Centralizers of Rescaling Separating Differentiable Vector Fields  

作　　者：Bo HAN Xiao WEN 

机构地区：[1]School of Mathematical Sciences,Beihang University,Beijing,100191,P.R.China [2]LMIB,Institute of Artificial Intelligence&School of Mathematical Sciences,Beihang University,Beijing,100191,P.R.China [3]Beijing Zhongguancun Laboratory,Beijing,100094,P.R.China

出　　处：《Acta Mathematica Sinica,English Series》2024年第11期2671-2683,共13页数学学报（英文版）

基　　金：Supported by National Key R&D Program of China(Grant No.2022YFA1005801);National Natural Science Foundation of China(Grant No.12071018);the Fundamental Research Funds for the Central Universities。

摘　　要：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 C^(1)vector field X on M,which generates a flowφ_(t)on M.We say that X is rescaling separating on a compact invariant setΛof X if there is a constantδ>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 onΛand every singularity of X inΛis hyperbolic,then any C^(1)vector field Y,whose flow commutes withφ_(t)onΛ,must be collinear to X onΛ.As applications of this result,we show that the centralizer of a rescaling separating C^(1)vector field without nonhyperbolic singularity is quasi-trivial.We also proved that there is an open and dense set u⊂χ^(1)(M)such that for any star vector fieldχ∈u,the centralizer of X is collinear to X on the chain recurrent set of X.

关 键 词：CENTRALIZER EXPANSIVENESS HYPERBOLICITY 

分 类 号：O186.12[理学—数学]