On the existence of time averages for time-varying dynamical systems

Abstract

For general sequences of measure preserving transformations on a measure space, the ergodic averages considered in Birkhoff's pointwise ergodic theorem do not, in general, converge almost everywhere. The paper provides an example where the following situation occurs: {/spl Phi/1/t} is a sequence for which the ergodic averages converge a.e. and {/spl Phi/2/t} is a sequence converging to {/spl Phi/1/t} in the strong Rokhlin-type metric. However, the ergodic averages do not converge a.e. for {/spl Phi/1/t}. Two types of conditions are given to ensure the convergence of the ergodic averages for {/spl Phi/2/t}. One of them is of topological type and the other requiring sufficient speed in the convergence. Convergence conditions along the ergodicity of the limit transformation are used in proving the recurrence theorem and the mean ergodic theorem for sequences.