Let f be a bounded continuous
function on a topological group G and 𝒪(f) the set of all (left) translates fx of f.
One might ask what self maps ϕ of G have 𝒪(f) ∘ ϕ ⊂𝒪(f) (so fx∘ ϕ is
always another translate fy of f). Since this says ϕ maps each translate of
a set of constancy f−1(c) into another translate of the set, and indeed a
translate independent of c, unless f is very special one would expect ϕ to
be quite rigid, and in fact almost a translation, perhaps on a quotient of
G.
When G is a compact connected abelian group this is, in essence, the situation if
ϕ maps G onto itself; alternatively one can take f ∈ Lp(G),1 ≦ p < ∞, and assume
ϕ is measure preserving and arrive at the same conclusions. In §1 we determine when
𝒪(f) ∘ϕ ⊂𝒪(f) and in §3 when the distorted orbit 𝒪(f) ∘ϕ coincides with another,
𝒪(g), along with some related results. Section 2 is devoted to analogous results on
(weakly) almost periodic functions.
