Mixing properties of generalized $T, T^{-1}$ transformations

Abstract: We study mixing properties of generalized $T, T^{-1}$ transformation. We discuss two mixing mechanisms. In the case the fiber dynamics is mixing, it is sufficient that the driving cocycle is small with small probability. In the case the fiber dynamics is only assumed to be ergodic, one needs to use the shearing properties of the cocycle. Applications include the Central Limit Theorem for sufficiently fast mixing systems and the estimates on deviations of ergodic averages.