Conjugations, joinings, and direct products of locally rank one dynamical systems

Goodson, Geoffrey R.; Ryzhikov, V. V.

doi:10.1007/bf02463255

Cited by 8 publications

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“…If T has minimal self-joinings, then it was shown in [7] that R(x, y) = (y, T x) is prime with trivial centralizer (this also follows from the work of Rudolph [11]). This gives an explicit example of the situation where ST = T −1 S, with S prime and T weakly mixing (just replace S by R and T by T × T −1 ).…”

Section: P R O O F (I) the σ-Algebramentioning

confidence: 96%

“…Although S 2 is not ergodic, if such a T were to exist, the only allowable eigenvalue for both S and T would be −1. In fact, if ST = T −1 S for T ergodic, having finite uniform rank, S has to have finite order (from [7]). …”

Section: P R O O F (I) the σ-Algebramentioning

confidence: 99%

“…It is known that if T has simple spectrum and ST = T −1 S, then S 2 = I, the identity automorphism [5]. Similar results for finite rank maps are known [7]. Generally, it is known that S can take any even order, it can be aperiodic or weakly mixing.…”

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confidence: 91%

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Conjugacies between ergodic transformations and their inverses

Goodson

2000

Colloq. Math.

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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.0. Introduction. Let S and T be invertible measure preserving transformations (automorphisms) defined on a Lebesgue probability space (X, F , µ). It was shown in [6] that if ST = T −1 S, where S and T are automorphisms, then T has an even multiplicity function on the orthogonal complement of the subspaceIn this paper we are interested in the form a conjugating map S between an ergodic transformation T and its inverse T −1 can take. It is known that if T has simple spectrum and ST = T −1 S, then S 2 = I, the identity automorphism [5]. Similar results for finite rank maps are known [7]. Generally, it is known that S can take any even order, it can be aperiodic or weakly mixing. We give examples with S and T ergodic but not weakly mixing and we observe that in this case −1 is necessarily the unique eigenvalue of both S and T . Our main theorem, which restricts considerably the form the conjugating map S can take, is: Theorem 1. If S and T are automorphisms with ST = T −1 S, then S 2 has an even multiplicity function on the orthogonal complement of the subspace {f ∈ L 2 (X, µ) : f (T 2 ) = f }.

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Section: P R O O F (I) the σ-Algebramentioning

confidence: 96%

Section: P R O O F (I) the σ-Algebramentioning

confidence: 99%

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Conjugacies between ergodic transformations and their inverses

Goodson

2000

Colloq. Math.

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“…(ii) Katok and Stepin pointed out that there is a three interval exchange transformation T (in fact an automorphism of the half-circle, induced by some rotation of the unit circle) possessing the property of the (n, n+1)-type approximation (see [7], [5] for the definition). This property implies, for some sequence h(i) → ∞ and any p > 0, the weak convergence…”

Section: The Casementioning

confidence: 99%

Homogeneous spectrum, Disjointness of Convolutions, and Mixing Properties of Dynamical Systems

Ryzhikov

2012

Preprint

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In connection with Rokhlin's question on an automorphism with a homogeneous nonsimple spectrum, we indicate a class of measure-preserving maps T such that T × T has a homogeneous spectrum of multiplicity 2. The automorphisms in question satisfy the condition σ * σ ⊥ σ, where σ is the spectral measure of T . We also show that there is a mixing automorphism possessing the above properties and their higher order analogs.

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“…The following theorem and corollary were given in [2] (see also [3] and [4] for related results). We use I to denote both the identity operator, and the identity automorphism.…”

Section: §0 Introductionmentioning

confidence: 98%

The converse of the Inverse-Conjugacy Theorem for unitary operators and ergodic dynamical systems

Goodson

1999

Proc. Amer. Math. Soc.

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Abstract. We show that the converse to the main theorem of Ergodic transformations conjugate to their inverses by involutions, by Goodson et al. (Ergodic Theory and Dynamical Systems 16 (1996), 97-124), holds in the unitary category. Specifically it is shown that if U is a unitary operator defined on an L 2 space which preserves real valued functions, and if U −1 S = SU implies S 2 = I whenever S is another such operator, then U has simple spectrum. The corresponding result for measure preserving transformations is shown to be false. The counter-example we have involves Gaussian automorphisms. We show that a Gaussian automorphism is always conjugate to its inverse, so that the Inverse-Conjugacy Theorem is applicable to such maps having simple spectrum. Furthermore, there are Gaussian automorphisms having non-simple spectrum for which every conjugation of T with T −1 is an involution. §0. Introduction Let U : H → H be a unitary operator defined on a separable Hilbert space H. U is said to have simple spectrum if there exists an h ∈ H for which Z(h) = H, where Z(h) is the closed linear span of the set {U n h : n ∈ Z}. We are mainly interested in the case where the Hilbert space is a function space. Let (X, F , µ) denote a standard Borel probability space, and let T : X → X be an invertible measure preserving transformation (automorphism). To say that the automorphism T has simple spectrum means that the unitary operator, induced by T , has simple spectrum. It is known that if T has simple spectrum, then T is ergodic, and that unitary operators preserving real valued functions are conjugate to their inverses.The following theorem and corollary were given in [2] (see also [3] and [4] for related results). We use I to denote both the identity operator, and the identity automorphism.

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Conjugations, joinings, and direct products of locally rank one dynamical systems

Cited by 8 publications

References 13 publications

Conjugacies between ergodic transformations and their inverses

Conjugacies between ergodic transformations and their inverses

Homogeneous spectrum, Disjointness of Convolutions, and Mixing Properties of Dynamical Systems

The converse of the Inverse-Conjugacy Theorem for unitary operators and ergodic dynamical systems

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