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

Birkhoffʹs 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: 1/n (sigma)f(ring operator)T^k-(integral)(omega)f dp, a.s. as n-(infinity) In this article, we discuss the analogous result for sequences of partial maxima: given a measurable f, if T is measure-preserving and ergodic then M n =max k<=n f (ring operator) T ^k |ess sup f , a.s. as n-(infinity) Series criteria are provided which characterize the a.s. maximal and minimal growth rates of the sequence of partial maxima.