Conjugacies between ergodic transformations and their inverses

Volume 84 / 2000

Geoffrey Goodson Colloquium Mathematicum 84 (2000), 185-193 DOI: 10.4064/cm-84/85-1-185-193


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 ${ f ∈ L^{2}(X, ℱ, μ): f(T^{2}x) = f(x) }$. 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 $T^{2}$ has non-simple spectrum.


  • Geoffrey Goodson

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