On asymmetry of the future and the past for limit self-joinings

Let Delta(T) be an off-diagonal joining of a transformation T. We construct a non-typical transformation having asymmetry between limit sets of Delta(Tn) for positive and negative powers of T. It follows from a correspondence between subpolymorphisms and positive operators, and from the structure of limit polynomial operators. We apply this technique to find all polynomial operators of degree 1 in the weak closure (in the space of positive operators on L-2) of powers of Chacon's automorphism and its generalizations.