Quantitative recurrence for $T,T^{-1}$ tranformation

Abstract

We are interested in the study of the asymptotic behaviour of return times in small balls for the $T,T^{-1}$-transformation. We exhibit different asymptotic behaviours (different scaling, different limit point process) depending on the respective dimensions of the measures of the two underlying dynamical systems. It behaves either as for the direct product of the underlying systems, or as for the Z-extension of the driving system (also studied in this article), or as a more sophisticated process.