Abstract
For any locally compact groupG, we show that any locally tight homomorphism from a real directed semigroup intoM 1 (G) (semigroup of probability measures onG) has a ‘shift’ which extends to a continuous one-parameter semigroup. IfG is ap-adic algebraic group then the above holds even iff is not locally tight. These results are applied to give sufficient conditions for embeddability of some translate of limits of sequences of the form {v kn n } and μ∈M 1 (G) such that τ(μ)=μM, for somek>1 and τ∈AutG (cf. Theorems 2.1, 2.4, 3.7).
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