Cocycles on the Circle. II

Helson, Henry; Merrill, Kathy D.

doi:10.1007/978-3-0348-9164-6_9

Cited by 25 publications

(35 citation statements)

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“…We prove the result without restriction, as a particular case of a more general theorem which deals with cocycle extension (THEOREM 5.4) : the "maximal group" for a cocycle associated with a group of translations on T is an eigenvalue group. This contains some results by H. HELSON and K. MERRILL ( [16], [17]). …”

mentioning

confidence: 81%

“…But we get rid of any ergodicity condition and we will prove moreover a general theorem on the extension of cocycles, which provides a new class of saturated subgroups of the circle containing the groups H(^i) TOME 119 -1991 -?1 for every measure fi G M(T). The problem of extension of a multiplicative cocycle has been paid attention by many authors, specially for our concern by HELSON and MERRILL [16], [17].…”

Section: For Any Positive Measure Fi E M(j) H{ii) Is a Saturated Submentioning

confidence: 99%

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Non singular transformations and spectral analysis of measures

Host¹,

Mela²,

Parreau³

1991

Bul. Soc. Math. France

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mentioning

confidence: 81%

Section: For Any Positive Measure Fi E M(j) H{ii) Is a Saturated Submentioning

confidence: 99%

Non singular transformations and spectral analysis of measures

Host¹,

Mela²,

Parreau³

1991

Bul. Soc. Math. France

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“…It is known that both U and V have full spectrum, that is, where [y] := y modulo 1. Then, U and V satisfy the commutation relation (3.1), and the spectrum of U is either purely punctual, purely singularly continuous or purely Lebesgue [11,Thm. 3] (the precise nature of the spectrum highly depends on properties of the pair (f; ), see for example [14,15,17]).…”

Section: Cocycles Over Irrational Rotationsmentioning

confidence: 99%

Commutator methods for unitary operators

Fernández¹,

Richard²,

Aldecoa³

2013

J. Spectr. Theory

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We present an improved version of commutator methods for unitary operators under a weak regularity condition. Once applied to a unitary operator, the method typically leads to the absence of singularly continuous spectrum and to the local niteness of point spectrum. Large families of locally smooth operators are also exhibited. Half of the paper is dedicated to applications, and a special emphasize is put on the study of cocycles over irrational rotations. It is apparently the rst time that commutator methods are applied in the context of rotation algebras, for the study of their generators.

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“…Anzai skew products or more generally operators V g have a well known property called the purity law. Precisely, each operator V g has either Lebesgue or continuous singular or discrete spectrum (see [6] and [10]). …”

Section: (T × T λ ⊗ λ) Moreover the Operator U T ϕ : H M → H M Is Umentioning

confidence: 99%

Some examples of cocycles with simple continuous singular spectrum

Frączek

2001

Studia Math.

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Abstract.We study spectral properties of Anzai skew products T ϕ :where α is irrational and ϕ : T → T is a measurable cocycle. Precisely, we deal with the case where ϕ is piecewise absolutely continuous such that the sum of all jumps of ϕ equals zero. It is shown that the simple continuous singular spectrum of T ϕ on the orthocomplement of the space of functions depending only on the first variable is a "typical" property in the above-mentioned class of cocycles, if α admits a sufficiently fast approximation.

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Cocycles on the Circle. II

Cited by 25 publications

References 1 publication

Non singular transformations and spectral analysis of measures

Non singular transformations and spectral analysis of measures

Commutator methods for unitary operators

Some examples of cocycles with simple continuous singular spectrum

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