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In this note the proof of a slight generalization of the Maximal Ergodic Inequality is a bit simplified and it is shown that from this generalized inequality Birkhoff’s Pointwise Ergodic Theorem follows almost immediately.
Summary Let F q be a finite field with q elements. We consider formal Laurent series of F q -coefficients with their continued fraction expansions by F q -polynomials. We prove some arithmetic properties for almost every formal Laurent series with respect to the Haar measure. We construct a group extension of the non-archimedean continued fraction transformation and show its ergodicity. Then we get some results as an application of the individual ergodic theorem. We also discuss the convergence rate for limit behaviors.
