Permutations preserving Cesàro mean, densities of natural numbers and uniform distribution of sequences

Blümlinger, M.; Obata, N.

Annales de l'Institut Fourier, Volume 41 (1991) no. 3, pp. 665-678.

We are interested in permutations preserving certain distribution properties of sequences. In particular we consider $\mu$-uniformly distributed sequences on a compact metric space $X$, 0-1 sequences with densities, and Cesàro summable bounded sequences. It is shown that the maximal subgroups, respectively subsemigroups, of $Aut\left(\mathbf{N}\right)$ leaving any of the above spaces invariant coincide. A subgroup of these permutation groups, which can be determined explicitly, is the Lévy group $𝒢$. We show that $𝒢$ is big in the sense that the Cesàro mean is characterized by its invariance under the Lévy group. As a result, any $𝒢$ -invariant positive normalized linear functional on $l^{\infty }\left(\mathbf{N}\right)$ is an extension of Cesàro means. Finally we prove that there exist $𝒢$ -invariant extensions of Cesàro mean to all of $l^{\infty }\left(\mathbf{N}\right)$.

Nous considérons les permutations de $\mathbf{N}$ qui conservent la $\mu$-répartition des suites ou la densité des parties de $\mathbf{N}$ ou la somme de Cesàro des suites sommables, et montrons que le groupe (resp. semi-groupe) de ces permutations sont les mêmes. Il est prouvé qu’il y a des fonctionnelles de $l^{\infty }\left(\mathbf{N}\right)$ qui sont invariantes sous l’action du groupe de Lévy et que toutes ces fonctionnelles sont des extensions de la somme de Cesàro.

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