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Transformation groups of strong characteristic 0

It is shown that a transformation group (X, T, Π) is of strong characteristic 0 if and only if it is of P-strong characteristic 0 for some replete semigroup P in the phase group, provided that all orbit closures are compact. It is shown also that, under certain conditions, (X, T, Π) is of P-strong characteristic 0 if and only if (X × X, T, Π × Π) is Liapunov stable.