EXPONENTIAL DECAY OF RANDOM CORRELATIONS FOR RANDOM ANOSOV SYSTEMS MIXING ON FIBERS

The paper proves Theorem 1 for Anosov-on-fibers systems that are topologically mixing on fibers, constructing fiberwise Birkhoff cones and showing a uniform contraction (after a uniform block length N coming from mixing on fibers) in the Hilbert projective metric. This yields a unique random SRB family μ_ω via the weak-* limit of Ln_{θ^{-n}ω}1·m and uniform quenched exponential decay of past and future random correlations for Hölder observables with exponents μ,ν satisfying μ+ν<ν0; see the theorem statement and the proof outline, including the construction of N in (4.40)–(4.41), the uniform finite-diameter/contraction of LN_ω, the construction of μ_ω via (4.65), and the correlation bounds (4.79) and (4.84) . The candidate solution reaches the same conclusions but proposes a different functional-analytic route: a cocycle of transfer operators on anisotropic Banach spaces with a Lasota–Yorke inequality and a cone/Doeblin–Fortet structure leading to Birkhoff projective contraction and a random rank-one bundle, plus multiplicative bounds for Hölder observables ensuring the μ+ν<ν0 threshold. While this is consistent with known technology, the outline assumes several nontrivial uniformities (space construction, multiplier bounds, and aperiodicity from fiber mixing) without proof. The paper’s argument is self-contained and complete within its cone framework. Hence, both are correct, but by different methods.