Distal functions and unique ergodicity

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.