Spectral Properties of Schrödinger Operators Associated with Almost Minimal Substitution Systems

Eichinger, Benjamin; Gohlke, Philipp

doi:10.1007/s00023-020-00975-5

Cited by 4 publications

(3 citation statements)

References 34 publications

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“…Substitutions on infinite alphabets and tilings with infinite local complexity (ILC) have been steadily gaining attention in the study of symbolic dynamics and aperiodic order [10,16,19,28,34,35]. Indeed, infinite alphabets must naturally be considered when recoding non-uniformly recurrent sequences by return words [11], or when performing the balanced-pair and overlap algorithms for a non-Pisot substitution [21]. In the context of automatic sequences, constant-length substitutions on infinite alphabets are the natural setting for regular sequences-sequences admitting a finitely-generated k-kernel [2,Thm.…”

Section: Introductionmentioning

confidence: 99%

Substitutions on compact alphabets

Mañibo¹,

Rust²,

Walton³

2022

Preprint

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We develop a general theory of continuous substitutions on compact Hausdorff alphabets. Focussing on implications of primitivity, we provide a self-contained introduction to the topological dynamics of their subshifts. We then reframe questions from ergodic theory in terms of spectral properties of the corresponding substitution operator. The standard Perron-Frobenius theory in finite dimensions is no longer applicable. To overcome this, we exploit the theory of positive operators on Banach lattices. As an application, we identify computable criteria that guarantee quasi-compactness of the substitution operator and hence unique ergodicity of the associated subshift.

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Section: Introductionmentioning

confidence: 99%

Substitutions on compact alphabets

Mañibo¹,

Rust²,

Walton³

2022

Preprint

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“…Remark Substitutions covered in Theorem 4.14 generate sequences exhibiting a generalised Toeplitz structure; see [13, Sec. 4.3] and [32] on generalised Toeplitz sequences over compact alphabets.…”

Section: Diffraction For Substitutions On Infinite Alphabetsmentioning

confidence: 99%

Spectral properties of substitutions on compact alphabets

Mañibo

Rust

Walton

2023

Bulletin of London Math Soc

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We consider substitutions on compact alphabets and provide sufficient conditions for the diffraction to be pure point, absolutely continuous and singular continuous. This allows one to construct examples for which the Koopman operator on the associated function space has specific spectral components. For abelian bijective substitutions, we provide a dichotomy result regarding the spectral type of the diffraction. We also provide the first example of a substitution that has countably infinite Lebesgue spectral components and countably infinite singular continuous components. Lastly, we give a non‐constant length substitution on a countably infinite alphabet that gives rise to substitutive Delone sets of infinite type. This extends the spectral theory of substitutions on finite alphabets and Delone sets of finite type with inflation symmetry.

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“…The potentials considered here are defined by the iteration of one invertible map T , and hence by a Z-action. This setting arises, for example, in the study [12] of Schrödinger operators with potentials generated by almost primitive (but non-primitive) substitutions; see [22] for a discussion of the infinite invariant measures arising in that context.…”

Section: Introductionmentioning

confidence: 99%

Ergodic Schrödinger operators in the infinite measure setting

et al. 2021

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We develop the basic theory of ergodic Schrödinger operators, which is well known for ergodic probability measures, in the case of a base dynamics on an infinite measure space. This includes the almost sure constancy of the spectrum and the spectral type, the definition and discussion of the density of states measure and the Lyapunov exponent, as well as a version of the Pastur-Ishii theorem. We also give some counterexamples that demonstrate that some results do not extend from the finite measure case to the infinite measure case. These examples are based on some constructions in infinite ergodic theory that may be of independent interest.

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Spectral Properties of Schrödinger Operators Associated with Almost Minimal Substitution Systems

Cited by 4 publications

References 34 publications

Substitutions on compact alphabets

Substitutions on compact alphabets

Spectral properties of substitutions on compact alphabets

Ergodic Schrödinger operators in the infinite measure setting

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