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

Abstract

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:

$$\frac{1}{n}\sum\limits_{k = 0}^{n - 1} {f \circ T^k \to \int_\Omega {f\;dP,{ a}{.s as }n \to \infty } } { (0}{.1)}$$

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

$$M_n = \mathop {\max }\limits_{k\; \leqslant \;n} f \circ T^k \uparrow {ess}\;\sup f,{ a}{.s as }n \to \infty {(0}{.2)}$$

Series criteria are provided which characterize the a.s. maximal and minimal growth rates of the sequence of partial maxima.