SRB MEASURES FOR PARTIALLY HYPERBOLIC SYSTEMS WITH ONE-DIMENSIONAL CENTER SUBBUNDLES

For a partially hyperbolic attractor with a center bundle splitting in a dominated way into one-dimensional subbundles we show that for Lebesgue almost every point x in the topological basin U there is an empirical measure from x with an SRB component. Moreover, if all SRB measures are assumed to be hyperbolic, then there are only finitely many ergodic SRB measures and their basins cover U Lebesgue almost everywhere. This gives another proof of the existence of SRB measures in this context, which was established firstly by Yongluo Cao, Zeya Mi, and Dawei Yang [Comm. Math. Phys. 391 (2022), pp. 1271–1306] by using random perturbations. Furthermore this generalizes results of Sylvain Crovisier, Dawei Yang, and Jinhua Zhang [Comm. Math. Phys. 375 (2020), pp. 725–764] and Yongxia Hua, Fan Yang, and Jiagang Yang [Trans. Amer. Math. Soc. 373 (2020), pp. 385–417] which deal with a single one-dimensional center subbundle.