Weak limits of powers, simple spectrum of symmetric products, and rank-one mixing constructions

“…Examples of transformations satisfying the assumptions of Theorem 2 are constructed using the class of transformations of rank 1 (for their definition, see, for example, [7]). It suffices to require that, for each N > 0, there exist an infinite sequence of stages at which transformations of rank 1 are constructed; at each such stage, the array of spacers must be of the form (0, 2N, N, 0, 2N, N, .…”

Section: Construction Of the Transformationmentioning

confidence: 99%

On the asymmetry of multiple asymptotic properties of ergodic actions

Ryzhikov

2014

Math Notes

View full text Add to dashboard Cite

show abstract

“…Recall that mixing means weak convergence T i → Θ, where Θ is the orthoprojection onto the space of constants in L 2 (X, μ). This result was announced in [9], and recently Tikhonov [12] used it in proving the existence of a mixing automorphism with homogeneous spectrum of multiplicity m > 2. We point out that for nonmixing transformations Rokhlin's problem on homogeneous spectrum for m > 2 was solved in [2].…”

mentioning

confidence: 98%

“…By making c p tend to 1 we obtain a limit mixing construction T with the required spectral property. The paper [9] contains a detailed description of the passage to the limit T p → T (as p → ∞) which ensures that the spectrum is simple for all symmetric powers T n . Our problem is solved in a similar fashion: using the same method we find that the spectrum of the products T ⊗ T 2 ⊗ .…”

mentioning

confidence: 99%

Simple spectrum of the tensor product of powers of a mixing automorphism

Ryzhikov

2013

Trans. Moscow Math. Soc.

View full text Add to dashboard Cite

It is proved that there exists a mixing automorphism of a Lebesgue space for which the tensor product of all its positive powers has simple spectrum. § 1. Introduction A measure-preserving invertible transformation T of a probability Lebesgue space (X, μ), often called an automorphism, induces a unitary operator U T , U T f (x) = f (T x), which acts in the space L 2 (X, μ). In what follows we denote a transformation and its corresponding operator in the same way. The aim of the paper is to describe the construction of a mixing automorphism T such that the tensor product T ⊗ T 2 ⊗ T 3 ⊗. .. has simple spectrum. Recall that mixing means weak convergence T i → Θ, where Θ is the orthoprojection onto the space of constants in L 2 (X, μ). This result was announced in [9], and recently Tikhonov [12] used it in proving the existence of a mixing automorphism with homogeneous spectrum of multiplicity m > 2. We point out that for nonmixing transformations Rokhlin's problem on homogeneous spectrum for m > 2 was solved in [2]. In the same paper, Ageev proved that the spectrum of tensor products of the powers of a typical nonmixing automorphism is simple. To find out about Rokhlin's problem and problems concerning realizing sets of multiplicities of the spectrum of a dynamical system we recommend Anosov's book [3] to the reader, and also the detailed survey by Danilenko [5], where not only are the results discussed but also the methods used to obtain them. An infinite product T 1 ⊗ T 2 ⊗ T 3 ⊗. .. has simple spectrum only in the case where the spectra of all the finite products T 1 ⊗ T 2 ⊗. .. ⊗ T n are simple. This property implies the mutual singularity not only of the spectral measures σ i of the automorphisms T i but also of various convolution products of them: for example, σ 1 * σ 2 and σ 2 * σ 3 * σ 5. One of the ways of proving that the spectrum of an operator is simple consists in producing a cyclic vector. In the paper we prove the existence of a cyclic vector for all finite products of the form T ⊗T 2 ⊗.. .⊗T n for some nonmixing automorphisms T. Also, the nonmixing property, and what is more, even the existence of nontrivial polynomials in the weak closure of powers will play an important role in proving that the spectrum is simple. Although this method does not work directly for mixing systems, it can be applied to nonmixing transformations that approximate some mixing transformation and ensure that the property of the spectrum being simple holds for the products T ⊗ T 2 ⊗. .. ⊗ T n. The plan of the paper is as follows. Constructions of rank 1 are described in § 2. In § 3 stochastic and staircase constructions are discussed, nonmixing modifications of which are considered in § § 4, 5. That the spectrum of tensor products of powers of special nonmixing 2010 Mathematics Subject Classification. Primary 37A30; Secondary 28D05, 47A35.

show abstract

Weak limits of powers, simple spectrum of symmetric products, and rank-one mixing constructions

Cited by 31 publications

References 15 publications

The local rank of an ergodic symmetric power T ⊙n does not exceed n!n −n

The local rank of an ergodic symmetric power T ⊙n does not exceed n!n −n

On the asymmetry of multiple asymptotic properties of ergodic actions

Simple spectrum of the tensor product of powers of a mixing automorphism

Contact Info

Product

Resources

About