Distal functions and unique ergodicity

Salehi, Ebrahim

doi:10.1090/S0002-9947-1991-0986700-2

Distal functions and unique ergodicity
HTML articles powered by AMS MathViewer

by Ebrahim Salehi PDF

Trans. Amer. Math. Soc. 323 (1991), 703-713 Request permission

Abstract:

A. Knapp [5] has shown that the set, $D(S)$, of all distal functions on a group $S$ is a norm closed subalgebra of ${l^\infty }(S)$ that contains the constants and is closed under the complex conjugation and left translation by elements of $S$. Also it is proved that [7] for any $k \in \mathbb {N}$ and any $\lambda \in \mathbb {R}$ the function $f:\mathbb {Z} \to \mathbb {C}$ defined by $f(n) = {e^{i\lambda {n^k}}}$ is distal on $\mathbb {Z}$. Now let ${\mathbf {W}}$ be the norm closure of the algebra generated by the set of functions \[ \{ n \mapsto {e^{i\lambda {n^k}}}:k \in \mathbb {N},\;\lambda \in \mathbb {R}\} ,\] which will be called the Weyl algebra. According to the facts mentioned above, all members of the Weyl Algebra are distal functions on $\mathbb {Z}$. In this paper, we will show that any element of ${\mathbf {W}}$ is uniquely ergodic (Theorem 2.13) and that the set ${\mathbf {W}}$ does not exhaust all the distal functions on $\mathbb {Z}$ (Theorem 2.14). The latter will answer the question that has been asked (to the best of my knowledge) by P. Milnes [6]. The term Weyl algebra is suggested by S. Glasner. I would like to express my warmest gratitude to S. Glasner for his helpful advise, and to my advisor Professor Namioka for his enormous helps and contributions.

References

James R. Brown, Ergodic theory and topological dynamics, Pure and Applied Mathematics, No. 70, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0492177
Robert Ellis, Lectures on topological dynamics, W. A. Benjamin, Inc., New York, 1969. MR 0267561
H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math. 83 (1961), 573–601. MR 133429, DOI 10.2307/2372899
H. Furstenberg, The structure of distal flows, Amer. J. Math. 85 (1963), 477–515. MR 157368, DOI 10.2307/2373137
A. W. Knapp, Distal functions on groups, Trans. Amer. Math. Soc. 128 (1967), 1–40. MR 222873, DOI 10.1090/S0002-9947-1967-0222873-3

Minimal and distal functions on a semidirect products of groups

Isaac Namioka, Affine flows and distal points, Math. Z. 184 (1983), no. 2, 259–269. MR 716275, DOI 10.1007/BF01252861
I. Namioka, Right topological groups, distal flows, and a fixed-point theorem, Math. Systems Theory 6 (1972), 193–209. MR 316619, DOI 10.1007/BF01706088
John C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc. 58 (1952), 116–136. MR 47262, DOI 10.1090/S0002-9904-1952-09580-X
Hermann Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), no. 3, 313–352 (German). MR 1511862, DOI 10.1007/BF01475864