Vol. 87, No. 2, 1980

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Maps preserving translates of a function

Irving Leonard Glicksberg

Vol. 87 (1980), No. 2, 323–334

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 f1(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.

Mathematical Subject Classification 2000
Primary: 43A15
Secondary: 28D05, 43A60
Received: 29 December 1978
Revised: 29 March 1979
Published: 1 April 1980
Irving Leonard Glicksberg