Linear Repetitivity, I. Uniform Subadditive Ergodic Theorems and Applications

Damanik, David; Lenz, Daniel

doi:10.1007/s00454-001-0033-z

Cited by 20 publications

(27 citation statements)

References 21 publications

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“…For example, the articles [15,17,19] study the LR property from the point of view of combinatorics on words, whereas [14,32,38] discuss its implications within the theory of tilings. For example, the articles [15,17,19] study the LR property from the point of view of combinatorics on words, whereas [14,32,38] discuss its implications within the theory of tilings.…”

Section: Introductionmentioning

confidence: 99%

“…LR has been shown to have consequences in mathematical disciplines as diverse as combinatorics [15,19], ergodic theory [14,32,34], and spectral theory of Schrödinger operators [35]. LR has been shown to have consequences in mathematical disciplines as diverse as combinatorics [15,19], ergodic theory [14,32,34], and spectral theory of Schrödinger operators [35].…”

Section: Introductionmentioning

confidence: 99%

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Linearly Recurrent Circle Map Subshifts and an Application to Schrödinger Operators

Adamczewski

Damanik²

2002

Ann. Henri Poincaré

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We discuss circle map sequences and subshifts generated by them. We give a characterization of those sequences among them which are linearly recurrent. As an application we deduce zero-measure spectrum for a class of discrete onedimensional Schrödinger operators with potentials generated by circle maps.Our present study is motivated by the paper [35]. Consider discrete onedimensional Schrödinger operators (Hψ)(n) = ψ(n + 1) + ψ(n − 1) + V (n)ψ(n)( 1 ) in 2 (Z), where the potential V : Z → R is given byHere, λ = 0 is the coupling constant, α ∈ (0, 1) irrational is the rotation number, and β ∈ (0, 1) and θ ∈ [0, 1) are arbitrary numbers. These potentials are called circle map potentials in the mathematical physics community (cf. [23, 24, 25]) and codings of rotations by people working in combinatorics on words or symbolic dynamics. The operator (1) with potential (2) has been studied in many papers;

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Linearly Recurrent Circle Map Subshifts and an Application to Schrödinger Operators

Adamczewski

Damanik²

2002

Ann. Henri Poincaré

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“…This paper would not have been possible without the collaboration leading to [5]. The author would like to thank the referee for several valuable remarks and suggestions.…”

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confidence: 99%

Uniform ergodic theorems on subshifts over a finite alphabet

Lenz¹

2002

Ergod. Th. Dynam. Sys.

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Link to this article: http://journals.cambridge.org/abstract_S0143385702000111 How to cite this article: DANIEL LENZ (2002). Uniform ergodic theorems on subshifts over a nite alphabet. Ergodic Theory and Dynamical Systems, 22, pp 245-255Abstract. We investigate uniform ergodic type theorems for almost additive and subadditive functions on a subshift over a finite alphabet. We show that every uniquely ergodic subshift admits a uniform ergodic theorem for Banach-space-valued almost additive functions. We then give a necessary and sufficient condition on a minimal subshift to allow for a uniform subadditive ergodic theorem. This provides, in particular, a sufficient condition for unique ergodicity.

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“…Extensions of [77] to linearly recurrent systems, including higherdimensional ones, were found by Damanik and Lenz [40].…”

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confidence: 91%

Strictly ergodic subshifts and associated operators

Damanik¹

2007

Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday

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Dedicated to Barry Simon on the occasion of his 60th birthday.Abstract. We consider ergodic families of Schrödinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure. These properties have indeed been established for large classes of operators of this type over the course of the last twenty years. We review the mechanisms leading to these results and briefly discuss analogues for CMV matrices.

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Linear Repetitivity, I. Uniform Subadditive Ergodic Theorems and Applications

Cited by 20 publications

References 21 publications

Linearly Recurrent Circle Map Subshifts and an Application to Schrödinger Operators

Linearly Recurrent Circle Map Subshifts and an Application to Schrödinger Operators

Uniform ergodic theorems on subshifts over a finite alphabet

Strictly ergodic subshifts and associated operators

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