Exponent of Convergence of a Sequence of Ergodic Averages

For a sequence of ergodic averages, we consider its exponent of convergence, which is a numerical characteristic of two-sided power-law estimates of the rate of pointwise convergence of this sequence. Criteria for the boundary values 1 and $\infty$ of the exponent of convergence are given. Functions cohomologous to zero with a given the exponent of convergence are also described.