Weak Closure of Infinite Actions of Rank 1, Joinings, and Spectrum

It is proved that the ergodic self-joining of an infinite transformation of rank $1$ is part of the weak limit of shifts of a diagonal measure. A continuous class of nonisomorphic transformations with polynomial closure is proposed. These transformations possess minimal self-joinings and certain unusual spectral properties. Thus, for example, the tensor products of the powers of transformations have both a singular and a Lebesgue spectrum, depending on the choice of the power.