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## Colloquium Mathematicum

2000 | 84/85 | 1 | 185-193
Tytuł artykułu

### Conjugacies between ergodic transformations and their inverses

Autorzy
Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
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Rocznik
Tom
Numer
Strony
185-193
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-06-21
poprawiono
1999-09-24
Twórcy
autor
• Department of Mathematics, Towson University, Towson, MD 21252, USA.
Bibliografia
• [1] O. N. Ageev, On ergodic transformations with homogeneous spectrum, J. Dynam. Control Systems 5 (1999), 149-152.
• [2] H. El Abdalaoui, Étude spectrale des transformations d'Ornstein, Ph.D. thesis, Université de Rouen, 1998.
• [3] H. El Abdalaoui, La singularité mutuelle presque sûre du spectre des transformations d'Ornstein, Israel J. Math., to appear.
• [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.
• [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.
• [6] G. R. Goodson and M. Lemańczyk, Transformations conjugate to their inverses have even essential values, Proc. Amer. Math. Soc. 124 (1996), 2703-2710.
• [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.
• [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.
• [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.
• [12] V. V. Ryzhikov, Transformations having homogeneous spectra, J. Dynam. Control Systems 5 (1999), 145-148.
Typ dokumentu
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