On the rate of convergence in the individual ergodic theorem for the action of a semigroup

We consider the individual ergodic theorem for the action of a semigroup of measure-preserving mappings. We estimate the rate of convergence using estimates for the probability of large deviations for the ergodic averages with an essentially bounded averaging function. We find estimates for the rate of convergence of the ergodic averages in the cases of Benedicks–Carleson quadratic mappings, expanding mappings of Pomeau–Manneville type with a neutral point, and multidimensional shifts.