Conjugacies between ergodic transformations and their inverses

Geoffrey Goodson

Colloquium Mathematicae (2000)

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

Abstract

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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 S T = 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.

How to cite

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Goodson, 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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  1. [1] O. N. Ageev, On ergodic transformations with homogeneous spectrum, J. Dynam. Control Systems 5 (1999), 149-152. Zbl0943.37005
  2. [2] H. El Abdalaoui, Étude spectrale des transformations d'Ornstein, Ph.D. thesis, Université de Rouen, 1998. 
  3. [3] H. El Abdalaoui, La singularité mutuelle presque sûre du spectre des transformations d'Ornstein, Israel J. Math., to appear. 
  4. [4] G. R. Goodson, A survey of recent results in the spectral theory of ergodic dynamical systems, J. Dynam. Control Systems 5 (1999), 173-226. Zbl0987.37004
  5. [5] G. R. Goodson, A. del Junco, M. Lemańczyk and D. Rudolph, Ergodic transformations conjugate to their inverses by involutions, Ergodic Theory Dynam. Systems 24 (1995), 95-124. Zbl0846.28008
  6. [6] G. R. Goodson and M. Lemańczyk, Transformations conjugate to their inverses have even essential values, Proc. Amer. Math. Soc. 124 (1996), 2703-2710. Zbl0867.28008
  7. [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. [8] P. R. Halmos, Introduction to Hilbert Space, Chelsea, New York, 1972. 
  9. [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. [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. [11] D. J. Rudolph, Fundamentals of Measurable Dynamics, Oxford Univ. Press, Oxford, 1990. Zbl0718.28008
  12. [12] V. V. Ryzhikov, Transformations having homogeneous spectra, J. Dynam. Control Systems 5 (1999), 145-148. Zbl0954.37007

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