### Conjugacies between ergodic transformations and their inverses

Geoffrey Goodson (2000)

Colloquium Mathematicae

Similarity:

We study certain symmetries that arise when automorphisms S and T defined on a Lebesgue probability space (X, ℱ, μ) satisfy the equation $ST={T}^{-1}S$. 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 ${S}^{2}$. In particular, ${S}^{2}$ has to have a multiplicity function which only takes even values on the orthogonal complement of the subspace $f\in {L}^{2}(X,\mathcal{F},\mu ):f\left({T}^{2}x\right)=f\left(x\right)$. For S and T ergodic satisfying this equation further constraints...