Let ϕ be a umiformly
continuous function on the locally compact, non-compacl abelian group G.
Denote the dual of G by Γ. Then ϕ is ergodic on E ⊂ rif ϕγ has a unique
(translation-invariant) mean value for each γ ∈ E. Theorem: If E is closed and
scattered, then there is a sequence {pn} of ffigonometric polynomials and a sequence
{gn}⊂ L1(G) with suppĝn∩ E = Φ such that g∗ϕ + pn→ ϕ uniformly. Now
suppose E = Γ∖{γ} and let {pβ} be any net of almost periodic functions on G with
bounded L1 (G)- norm. Theorem: All cluster points 20 {Mypβ(⋅− y)ϕ(y)}β in
the topoloy of uniform convergence on compact subsets of G are ergodic.
Here M is any invariant mean on G. A compact set E ⊂ Γ is ergodic if
each ϕ ∈ Φ(E), the L∞(G)-weak* closure of span E, is (equal a.e. to) a
function ergodic on Γ. Theorem: If E ergodic and S is compact scattered,
then E ∪ S is ergodic. For E’s which are Helson sets this implies that the
elements of Φ(E ∪ S) decompose into μ+ u where μ ∈ Mc(E) and u is almost
periodic.
