Ergodic properties of multi-singular hyperbolic vector fields

Sylvain Crovisier, Xiaodong Wang, Dawei Yang, Jinhua Zhang

Published 2023-05-06Version 1

Bonatti and da Luz have introduced the class of multi-singular hyperbolic vector fields to characterize systems whose periodic orbits and singularities do not bifurcate under perturbation (called star vector fields). In this paper, we study the Sina\"{\i}-Ruelle-Bowen measures for multi-singular hyperbolic vector fields: in a $C^1$ open and $C^1$ dense subset of multi-singular hyperbolic vector fields, each $C^\infty$ one admits finitely many physical measures whose basins cover a full Lebesgue measure subset of the manifold. Similar results are also obtained for $C^1$ generic multi-singular hyperbolic vector fields.