SRB measures and the fiber leaf condition

In this paper, we study (Sinai–Ruelle–Bowen) SRB measures for random dynamical systems and their associated skew-product transformations. Extending Burguet’s variational method [SRB measures for C ∞ C^∞ upper C Superscript infinity surface diffeomorphisms. Invent. Math. 235 (3) (2024), 1019–1062], we establish the existence of random SRB measures under a fiber leaf condition, which ensures recurrent hyperbolic behavior along unstable leaves within generic fibers. The construction is based on geometric-time techniques and Følner averaging, leading to invariant measures that satisfy the relative Pesin entropy formula. We further identify a spectral-gap condition relating the Lyapunov spectra of the base and fiber dynamics, under which the random SRB measure coincides with an SRB measure for the smooth skew-product system.