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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 Sinaï-Ruelle-Bowen measures for multi-singular hyperbolic vector fields: in a C 1 C¹ open and C 1 C¹ dense subset of multi-singular hyperbolic vector fields, each C ∞ C^ 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 C¹ generic multi-singular hyperbolic vector fields.
Crovisier et al. (Fri,) studied this question.