A pointwise ergodic theorem for the group of rational rotations

Dubins, Lester E.; Pitman, Jim

doi:10.1090/S0002-9947-1979-0531981-7

A pointwise ergodic theorem for the group of rational rotations
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by Lester E. Dubins and Jim Pitman PDF

Trans. Amer. Math. Soc. 251 (1979), 299-308 Request permission

Abstract:

Let f be a bounded, measurable function defined on the multiplicative group $\Omega$ of complex numbers of absolute value 1, and define \begin{equation}\tag {$(1)$}{{f_n}(\omega ) = \frac {1} {n}\sum \limits _{i = 1}^n {f(z_n^i\omega )} ,} \qquad \omega \in \Omega ,\end{equation} where ${z_n}$ is a primitive nth root of unity. The present paper generalizes this result of Jessen [1934]: if $n(k)$ is an increasing sequence of positive integers with $n(k)$ dividing $n(k’)$ whenever $k < k’$, then ${f_{n(k)}}$ converges almost surely as $k \to \infty$.

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Topological groups

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