Spectral Theory of Dynamical Systems

Nadkarni, M. G.

doi:10.1007/978-981-15-6225-9

Cited by 14 publications

(19 citation statements)

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“…Moreover, since R λ is ergodic, it follows that for each ξ ∈ T, we have that either Na,9.22]. We thus obtain that…”

Section: (C F )-Systems With Infinite Invariant Measure and Irrationa...mentioning

confidence: 89%

Explicit rank-one constructions for irrational rotations

Danilenko¹,

I.²

2021

Preprint

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For each well approximable irrational θ, we provide an explicit rank-one construction of the e 2πiθ -rotation R θ on the circle T. This solves "almost surely" a problem by del Junco. For every irrational θ, we construct explicitly a rank-one transformation with an eigenvalue e 2πiθ . For every irrational θ, two infinite σ-finite invariant measures µ θ and µ ′ θ on T are constructed explicitly such that (T, µ θ , R θ ) is rigid and of rank one and (T, µ ′ θ , R θ ) is of zero type and of rank one. The centralizer of the latter system consists of just the powers of R θ . Some versions of the aforementioned results are proved under an extra condition on boundedness of the sequence of cuts in the rank-one construction.

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“…Moreover, since R λ is ergodic, it follows that for each ξ ∈ T, we have that either Na,9.22]. We thus obtain that…”

Section: (C F )-Systems With Infinite Invariant Measure and Irrationa...mentioning

confidence: 89%

Explicit rank-one constructions for irrational rotations

Danilenko¹,

I.²

2021

Preprint

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“…We should note that there are several alternative proofs of more general statements. For more on the rich theory of generalized Riesz products and how they appear in the spectral theory of dynamical systems, see the book by Nadkarni [92,Ch. 15], the papers by el Abdalaoui and Nadkarni [3], el Abdalaoui [2], and references therein.…”

Section: Spectral Measures As Riesz Productsmentioning

confidence: 99%

“…Theorem 2.5 (Nadkarni [92]). If T is an aperiodic automorphism of a probability measure space (X, µ),…”

Section: Introductionmentioning

confidence: 99%

Self-similarity and spectral theory: on the spectrum of substitutions

Bufetov¹,

Solomyak²

2021

Preprint

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In this survey of the spectral properties of substitution dynamical systems we consider primitive aperiodic substitutions and associated dynamical systems: Z-actions and R-actions, the latter viewed as tiling flows. Our focus is on the continuous part of the spectrum. For Z-actions the maximal spectral type can be represented in terms of matrix Riesz products, whereas for tiling flows, the local dimension of the spectral measure is governed by the spectral cocycle. We give references to complete proofs and emphasize ideas and connections.

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“…The theorem above gives a new generalization of Choksi-Nadkarni Theorem [18], [52]. We point out that in [10], the author generalized the Choksi-Nadkarni Theorem to the case of funny rank one group actions for which the group is compact and Abelian.…”

Section: On the Maximal Spectral Type Of Any Rank One Flow By Cs Cons...mentioning

confidence: 99%

On the spectral type of rank one flows and Banach problem with calculus of generalized Riesz products on the real line

Abdalaoui

2020

Preprint

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It is shown that a certain class of Riesz product type measures on R is realized a spectral type of rank one flows. As a consequence, we will establish that some class of rank one flows has a singular spectrum. Some of the results presented here are even new for the Z-action. Our method is based, on one hand, on the extension of Bourgain-Klemes-Reinhold-Peyrière method, and on the other hand, on the extension of the Central Limit Theorem approach to the real line which gives a new extension of Salem-Zygmund Central Limit Theorem. We extended also a formula for Radon-Nikodym derivative between two generalized Riesz products obtained by el Abdalaoui-Nadkarni and a formula of Mahler measure of the spectral type of rank one flow but in the weak form. We further present an affirmative answer to the flow version of the Banach problem, and we discuss some issues related to flat trigonometric polynomials on the real line in connection with the famous Banach-Rhoklin problem in the spectral theory of dynamical systems.

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Spectral Theory of Dynamical Systems

Cited by 14 publications

References 0 publications

Explicit rank-one constructions for irrational rotations

Explicit rank-one constructions for irrational rotations

Self-similarity and spectral theory: on the spectrum of substitutions

On the spectral type of rank one flows and Banach problem with calculus of generalized Riesz products on the real line

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