Recurrence of transformations with absolutely continuous invariant measures

摘要

Let X=[0,1]. If an ergodic transformation T:X→X preserves an absolutely continuous probability measure ρ(x)dx with ρ(x)>0, then it is shown that for almost every x∈X,liminfn→∞n·|Tnx−x|⩽1ρ(x).Define the kth first return time Rk(x)=min{s⩾1:|Tsx−x|⩽1/2k} and the kth recurrence error by ϵk(x)=|TRk(x)x−x|. Then it is shown thatliminfk→∞Rk(x)ϵk(x)⩽2ρ(x).