# Conjugacies between ergodic transformations and their inverses

Colloquium Mathematicae (2000)

- Volume: 84/85, Issue: 1, page 185-193
- ISSN: 0010-1354

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topGoodson, Geoffrey. "Conjugacies between ergodic transformations and their inverses." Colloquium Mathematicae 84/85.1 (2000): 185-193. <http://eudml.org/doc/210796>.

@article{Goodson2000,

abstract = {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.},

author = {Goodson, Geoffrey},

journal = {Colloquium Mathematicae},

keywords = {measure-preserving transformation; multiplicity function},

language = {eng},

number = {1},

pages = {185-193},

title = {Conjugacies between ergodic transformations and their inverses},

url = {http://eudml.org/doc/210796},

volume = {84/85},

year = {2000},

}

TY - JOUR

AU - Goodson, Geoffrey

TI - Conjugacies between ergodic transformations and their inverses

JO - Colloquium Mathematicae

PY - 2000

VL - 84/85

IS - 1

SP - 185

EP - 193

AB - 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.

LA - eng

KW - measure-preserving transformation; multiplicity function

UR - http://eudml.org/doc/210796

ER -

## References

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- [7] G. R. Goodson and V. V. Ryzhikov, z Conjugations, joinings, and direct products of locally rank one dynamical systems, J. Dynam. Control Systems 3 (1997), 321-341. Zbl0996.37002
- [8] P. R. Halmos, Introduction to Hilbert Space, Chelsea, New York, 1972.
- [9] A. del Junco and D. J. Rudolph, On ergodic actions whose self joinings are graphs, Ergodic Theory Dynam. Systems 7 (1987), 531-557. Zbl0646.60010
- [10] V. I. Oseledets, Two non-isomorphic dynamical systems with the same simple continuous spectrum, Funktsional. Anal. i Prilozhen. 5 (1971), no. 3, 75-79 (in Russian); English transl.: Functional Anal. Appl. 5 (1971), 233-236.
- [11] D. J. Rudolph, Fundamentals of Measurable Dynamics, Oxford Univ. Press, Oxford, 1990. Zbl0718.28008
- [12] V. V. Ryzhikov, Transformations having homogeneous spectra, J. Dynam. Control Systems 5 (1999), 145-148. Zbl0954.37007

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