Connections between the approximative and spectral properties of metric automorphisms

Stëpin, A. M.

doi:10.1007/bf01155664

Cited by 8 publications

(9 citation statements)

References 2 publications

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“…(See [10] where it is proved that locally rank one actions have finite spectral multiplicity. The method of the proof was given for a similar situation by Stepin in [17]. ) We recall that an automorphism S which admits a linked approximation of type ( n, n + 1,..., n + r) has to possess a sequence {~i} of partitions of X such that ~i -* ~ and for all i, ~i has the following form: )).…”

Section: N>2)mentioning

confidence: 97%

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

Goodson

Ryzhikov

1997

Journal of Dynamical and Control Systems

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ABSTRACT. An automorphism T : (X, ~', p) --* (X, ~c,/~) on a standard Borel probability space is said to have uniform rank n, if it has rank n and the columns at each stage can be chosen of equal height. We show that for such an automorphism T, if S is an automorphism conjugating T to its inverse (i.e., ST = T-1S), then S 2m -.~ I for some positive integer rn ~ n (I denotes the identity automorphism). Related results for transformations having local rank one (also called positive ~-rank maps) with no partial rigidity are given. Particular attention is given to the question of whether the Cartesian square T • T can have finite rank, and this question is answered in certain cases. The transformations R(z, y) = (y, Tz) and the symmetric Cartesian square T | are shown to have some analogous properties. INTRODUCTIONLet T : (X, ~',/~) --, (X, ~', p) be an invertible measure preserving tr~n~ formation (automorphism) defined on a standard Borel nonatomic probability space. It is known that if T has simple spectrum and S is an automorphism satisfying ST = T-1S, then S 2 = I (see [4]). We note also that there are automorphisms with simple continuous spectrum which are not conjugate to their inverses; see [13].It is natural to conjecture that if the maximal spectral multiplicity of T is equal to n, then S 2n = I (or S 2m = I for some m ~_ n). However, we shall give examples which show this to be false. Consider instead the analogous question regarding finite rank transformations. Suppose that T has rank 1; then, in fact, T has simple spectrum, so if ST = T-1S, then S 2 = I. Now consider the class of --iform rank n maps. We show that the natural generalization of the rank 1 situation is true, i.e., we prove 1991 Mathematics Subject Classification. 28D05.

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Section: N>2)mentioning

confidence: 97%

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

Goodson

Ryzhikov

1997

Journal of Dynamical and Control Systems

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“…;(«)} with £(n) -{C t (n): i = 1, ...,q(n)} such that (i) £(«) -> s as «-+ oo, (ii) rfC t {n)) q ( n ) l (Hi) £ n(TC i (n)&C i+l (n))<f{q(")). Stepin [9] has shown that if T admits an approximation with speed 6/n, 0 < 2m/(m+\), then T has spectral multiplicity at most m, and Baxter [1] has shown that if T admits a simple approximation then T has simple spectrum.…”

Section: Preliminariesmentioning

confidence: 99%

“…It is known (Stepin [9]) that if an invertible measure-preserving transformation T admits an approximation (in the sense of Katok and Stepin [6]) with speed 0/n, 9 < 2m/(m+l), then the spectral multiplicity of T is at most m. The purpose of this paper is to give a bound on the spectral multiplicity of the finite skew product of those transformations that admit approximations in the sense of Katok and Stepin. Sufficient conditions are also given for the two-point skew product of a transformation admitting a simple approximation to have simple spectrum, and also to have continuous spectrum.…”

Section: Introductionmentioning

confidence: 99%

Approximation and the Spectral Multiplicity of Finite Skew Products

Goodson

1976

Journal of the London Mathematical Society

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“…We note that we could have obtained similar results with other types of approximations. The proofs of Theorems 3 and 4 following are variations of proofs due to Katok and Stepin [5] and Stepin [10].…”

Section: Cyclic Approximationsmentioning

confidence: 99%

“…A recent result of Stepin [10] is that an automorphism which admits a cyclic approximation with speed 0/n, where 6 < 2 -2/m, has spectral multiplicity at most m -1. It follows then that the automorphisms we have been considering here have bounded multiplicities.…”

Section: /=Imentioning

confidence: 99%

On Automorphisms with No Mixing Factors

Newton

1974

Journal of the London Mathematical Society

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Connections between the approximative and spectral properties of metric automorphisms

Cited by 8 publications

References 2 publications

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

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

Approximation and the Spectral Multiplicity of Finite Skew Products

On Automorphisms with No Mixing Factors

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