no. 3
Permutations preserving Cesàro mean, densities of natural numbers and uniform distribution of sequences
Annales de l'Institut Fourier, Tome 41 (1991) no. 3, pp. 665-678

We are interested in permutations preserving certain distribution properties of sequences. In particular we consider μ-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(N) 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 ∞ (N) is an extension of Cesàro means. Finally we prove that there exist 𝒢 -invariant extensions of Cesàro mean to all of l ∞ (N).

Nous considérons les permutations de N qui conservent la μ-répartition des suites ou la densité des parties de 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 ∞ (N) 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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     author = {Bl\"umlinger, M. and Obata, N.},
     title = {Permutations preserving {Ces\`aro} mean, densities of natural numbers and uniform distribution of sequences},
     journal = {Annales de l'Institut Fourier},
     pages = {665--678},
     year = {1991},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {41},
     number = {3},
     doi = {10.5802/aif.1269},
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     mrnumber = {92j:43002},
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}
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Blümlinger, M.; Obata, N. Permutations preserving Cesàro mean, densities of natural numbers and uniform distribution of sequences. Annales de l'Institut Fourier, Tome 41 (1991) no. 3, pp. 665-678. doi: 10.5802/aif.1269

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