Distal functions and unique ergodicity
Ebrahim Salehi · Transactions of the American Mathematical Society · 1991
A. Knapp [5] has shown that the set, D ( S ) D(S) , of all distal functions on a group S S is a norm closed subalgebra of l ∞ ( S ) {l^\infty }(S) that contains the constants and is closed under the complex conjugation and left translation by elements of S S . Also it is proved that [7] for any k ∈ N k \in \mathbb {N} and any λ ∈ R \lambda \in \mathbb {R} the function f : Z → C f:\mathbb {Z} \to \mathbb {C} defined by f ( n ) = e i λ n k f(n) = {e^{i\lambda {n^k}}} is distal on Z \mathbb {Z} . Now let W {\mathbf {W}} be the norm closure of the algebra generated by the set of functions \[