The shrinking target and recurrence problem for non-autonomous systems

Abstract: We investigate the shrinking target and recurrence set associated to non-autonomous measure-preserving systems on compact metric spaces, establishing zero-one criteria in the spirit of classical Borel-Cantelli results. Our first main theorem gives a quantitative shrinking target result for non-autonomous systems under a uniform mixing condition, providing asymptotics with an optimal error term. This general result is applicable to certain families of inner functions, yielding concrete applications such as digit patterns in multi-base expansions. Turning to recurrence, we establish new zero-measure laws for non-autonomous systems. In the autonomous case, we prove a zero-one criterion for recurrence sets of centred, one-component inner functions via Markov partitions and distortion estimates. Together, these results provide a unified framework for shrinking target and recurrence problems in both autonomous and non-autonomous dynamics.