Geodesic
Conjugacies between ergodic transformations and their inverses
Colloquium Mathematicum, Tome 84 (2000) no. 1, pp. 185-193.

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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.
DOI : 10.4064/cm-84/85-1-185-193

Geoffrey Goodson 1

1 
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Geoffrey Goodson. Conjugacies between ergodic transformations and their inverses. Colloquium Mathematicum, Tome 84 (2000) no. 1, pp. 185-193. doi : 10.4064/cm-84/85-1-185-193. http://geodesic.mathdoc.fr/articles/10.4064/cm-84/85-1-185-193/

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