Conjugacies between ergodic transformations and their inverses

We study certain symmetries that arise when automorphisms S and T defined on a Lebesgue probability space (X, ℱ, μ) satisfy the equation ${\displaystyle ST=T^{-1}S}$. In an earlier paper [6] it was shown that this puts certain constraints on the spectrum of T. Here we show that it also forces constraints on the spectrum of ${\displaystyle S^2}$. In particular, ${\displaystyle S^2}$ has to have a multiplicity function which only takes even values on the orthogonal complement of the subspace ${\displaystyle \{f\in L^2(X,ℱ,\mu ):f\left(T^{2}x\right)=f\left(x\right)\}}$. For S and T ergodic satisfying this equation further constraints arise, which we illustrate with examples. As an application of these results we give a general method for constructing weakly mixing rank one maps T for which ${\displaystyle T^2}$ has non-simple spectrum.