On the notions of singular domination and(multi-)singular hyperbolicity  

作　　者：Sylvain Crovisier Adriana da Luz Dawei Yang Jinhua Zhang 

机构地区：[1]Laboratoire de Mathematiques d'Orsay,CNRS,Universite Paris-Sud,Orsay 91405,France [2]Beijing International Center for Mathematical Research,Peking University,Beijing 100871,China [3]School of Mathematical Sciences,Soochow University,Suzhou 215006,China [4]School of Mathematical Sciences,Beihang University,Beijing 100191,China

出　　处：《Science China Mathematics》2020年第9期1721-1744,共24页中国科学：数学（英文版）

基　　金：The first author was supported by the European Research Council(Grant No.692925);The third author was supported by National Natural Science Foundation of China(Grant Nos.11671288,11822109 and 11790274);The fourth author was supported by the starting grant from Beihang University and the European Research Council(Grant No.692925).

摘　　要：The properties of uniform hyperbolicity and dominated splitting have been introduced to study the stability of the dynamics of diffeomorphisms.One meets difficulties when trying to extend these definitions to vector fields and Shantao Liao has shown that it is more relevant to consider the linear Poincare flow rather than the tangent flow in order to study the properties of the derivative.In this paper,we define the notion of singular domination,an analog of the dominated splitting for the linear Poincar´e flow which is robust under perturbations.Based on this,we give a new definition of multi-singular hyperbolicity which is equivalent to the one recently introduced by Bonatti and da Luz(2017).The novelty of our definition is that it does not involve the blow-up of the singular set and the rescaling cocycle of the linear flows.

关 键 词：multi-singular hyperbolicity singular domination star vector field linear Poincar´e flow 

分 类 号：O186.11[理学—数学]