The shrinking target and recurrence problem for non-autonomous systems

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 patterns of zeros in the multibase expansion. 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.