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We study mixing properties of generalized $T, T^{-1}$ transformation. Wediscuss two mixing mechanisms. In the case the fiber dynamics is mixing, it issufficient that the driving cocycle is small with small probability. In thecase the fiber dynamics is only assumed to be ergodic, one needs to use theshearing properties of the cocycle. Applications include the Central LimitTheorem for sufficiently fast mixing systems and the estimates on deviations ofergodic averages.

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