Series Criteria for Growth Rates of Partial Maxima of Iterated Ergodic Map Values

Birkhoff's well-known ergodic theorem states that the simple averages of a sequence of real (integrable) function values on successive iterates of a measure-preserving mapping T converge a.s. to the conditional expected value of the function conditioned on the invariant sigma-field. If the mapping is in addition ergodic, then the limit is simply the unconditional expected value:In this article, we discuss the analogous result for sequences of partial maxima: given a measurable f, if T is measure-preserving and ergodic thenSeries criteria are provided which characterize the a.s. maximal and minimal growth rates of the sequence of partial maxima.