Spectral multiplicities of infinite measure preserving transformations

Danilenko, Alexandre I.; Ryzhikov, V. V.

doi:10.1007/s10688-010-0021-2

Cited by 9 publications

(13 citation statements)

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“…Theorem 7.1 [DaRy1]. Given E ⊂ N, there is an ergodic rigid infinite measure preserving transformation T such that M(T ) = E.…”

Section: Spectral Multiplicities Of Infinite Measure Preserving Systemsmentioning

confidence: 99%

“…In order to obtain ergodic systems with various spectral multiplicities Robinson introduces in [Ro1] and [Ro2] a preliminary step which we call an algebraic realization of subsets of positive integers. This preliminary step played an important role in subsequent papers [Go-Li], [KwiLe], [KaLe], [Da3], [Da4], [DaRy1] on spectral multiplicities. This appendix is devoted completely to algebraic realizations.…”

Section: Appendix Algebraic Realizationsmentioning

confidence: 99%

“…Since L 2 (X, µ) does not contain constants if µ(X) = ∞, the Koopman operator U T is considered in the entire L 2 (X, µ). Quite surprisingly, (Pr1) can be solved completely in this case by combining (Tec1) and (Tec2) [DaRy1]. Moreover, this result was refined in a subsequent paper [DaRy2]: for each E ⊂ N, there exists a mixing (or zero type) ergodic multiply recurrent infinite measure preserving transformation 6 T with M(T ) = E. Thus, we conclude that the main obstacle to realizations of spectral multiplicities in the probability preserving case is the presence of constants in L 2 (X, µ).…”

Section: Introductionmentioning

confidence: 99%

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A survey on spectral multiplicities of ergodic actions

Danilenko

2011

Ergod. Th. Dynam. Sys.

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Given a transformation T of a standard measure space (X, µ), let M(T ) denote the set of spectral multiplicities of the Koopman operator U T defined in L 2 (X, µ)⊖C by U T f := f •T . It is discussed in this survey paper which subsets of N∪ {∞} are realizable as M(T ) for various T : ergodic, weakly mixing, mixing, Gaussian, Poisson, ergodic infinite measure preserving, etc. The corresponding constructions are considered in detail. Generalizations to actions of Abelian locally compact second countable groups are also discussed.The corresponding unitary isomorphism conjugates U T with the operator of multiplication by the independent variable z on T. 4 In fact, an extra condition on the realizable subsets was imposed in [KaLe]. However, as was shown later in [Da4], this condition is satisfied for all finite subsets of N. 5 Gaussian transformations are never of rank one [dR]. 6 In the infinite case, mixing does not imply ergodicity and ergodicity does not imply recurrence.

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“…Theorem 7.1 [DaRy1]. Given E ⊂ N, there is an ergodic rigid infinite measure preserving transformation T such that M(T ) = E.…”

Section: Spectral Multiplicities Of Infinite Measure Preserving Systemsmentioning

confidence: 99%

Section: Appendix Algebraic Realizationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A survey on spectral multiplicities of ergodic actions

Danilenko

2011

Ergod. Th. Dynam. Sys.

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“…More generally, for each n > 1, Ageev built mixing realizations for {2, 3, . We also mention that (Pr1) and (Pr2) are solved completely in the category of infinite measure-preserving transformations (see [DaR1,DaR2] respectively). .…”

Section: A I Danilenkomentioning

confidence: 99%