The asymptotic behavior of Frobenius–Perron operator with local lower-bound function

Let (X,Σ,μ) be a σ-finite measure space, S:X→X be a nonsingular transformation and PS:L1→L1 be the Frobenius–Perron operator associated with S. It is proved that if PS satisfies the local lower-bound function condition then for every f D the sequence {PSnf} converges strongly to a stationary density of PS as n→∞. The statistical stability of S is also concerned via the local lower-bound function method.