Quantitative ergodic theorems and their number-theoretic applications

Gorodnik, Alexander; Nevo, Amos

doi:10.1090/s0273-0979-2014-01462-4

Cited by 26 publications

(27 citation statements)

References 129 publications

(157 reference statements)

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“…We say that the action of G on Y has a spectral gap if q 0 < ∞ (see [23,Section 3.2]). Theorem 6.1 (Nevo [38]).…”

Section: Mean Ergodic Theorem For Averaging Operatorsmentioning

confidence: 99%

Quantum ergodicity and Benjamini–Schramm convergence of hyperbolic surfaces

Masson¹,

Sahlsten²

2017

Duke Math. J.

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Abstract. We present a quantum ergodicity theorem for fixed spectral window and sequences of compact hyperbolic surfaces converging to the hyperbolic plane in the sense of Benjamini and Schramm. This addresses a question posed by Colin de Verdière. Our theorem is inspired by results for eigenfunctions on large regular graphs by Anantharaman and the first-named author. It applies in particular to eigenfunctions on compact arithmetic surfaces in the level aspect, which connects it to a question of Nelson on Maass forms. The proof is based on a wave propagation approach recently considered by Brooks, Lindenstrauss and the first-named author on discrete graphs. It does not use any microlocal analysis, making it quite different from the usual proof of quantum ergodicity in the large eigenvalue limit. Moreover, we replace the wave propagator with renormalised averaging operators over discs, which simplifies the analysis and allows us to make use of a general ergodic theorem of Nevo. As a consequence of this approach, we require little regularity on the observables.

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“…We say that the action of G on Y has a spectral gap if q 0 < ∞ (see [23,Section 3.2]). Theorem 6.1 (Nevo [38]).…”

Section: Mean Ergodic Theorem For Averaging Operatorsmentioning

confidence: 99%

Quantum ergodicity and Benjamini–Schramm convergence of hyperbolic surfaces

Masson¹,

Sahlsten²

2017

Duke Math. J.

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“…Remarkably, a version of the first equality in still holds in many non‐amenable lcsc groups, despite the fact that such groups do not admit any Følner sequences whatsoever. We provide some explicit examples of this phenomenon, which are inspired by the work of Gorodnik and Nevo on ergodic theorem of lattices in semisimple Lie groups , in Theorem below. For a more systematic treatment of approximation theorems for non‐amenable groups in the context of the so‐called spherical auto‐correlation we refer the reader to the sequel .…”

Section: Introductionmentioning

confidence: 99%

Aperiodic order and spherical diffraction, I: auto-correlation of regular model sets

Björklund

Hartnick

Pogorzelski

2017

Proc. London Math. Soc.

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ABSTRACT. We introduce and study model sets in commutative spaces, i.e. homogeneous spaces of the form G/K where G is a (typically non-abelian) locally compact group and K is a compact subgroup such that (G, K) is a Gelfand pair. Examples include model sets in hyperbolic spaces, Riemannian symmetric spaces, regular trees and generalized Heisenberg groups. Continuing our work from [6] we associate with every regular model set in G/K a Radon measure on K\G/K called its spherical auto-correlation. We then define the spherical diffraction of the regular model set as the spherical Fourier transform of its spherical auto-correlation in the sense of Gelfand pairs. The main result of this article ensures that the spherical diffraction of a uniform regular model set in a commutative space is pure point. In fact, we provide an explicit formula for the spherical diffraction of such a model set in terms of the automorphic spectrum of the underlying lattice and the underlying window. To describe the coefficients appearing in this formula, we introduce a new type of integral transform for functions on the internal space of the model set. This integral transform can be seen as a shadow of the spherical Fourier transform of physical space in internal space and is hence referred to as the shadow transform of the model set. To illustrate our results we work out explicitly several examples, including the case of model sets in the Heisenberg group.

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“…There is a vast literature on ergodic theorems, and the theory is far developed, so we do not attempt to give an overview. We just refer to [21,22] for ergodic theorems on abstract groups or subgroups of lattices.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

Pattern Size in Gaussian Fields from Spinodal Decomposition

Bianchi¹,

Blömker²,

Wacker³

2017

SIAM J. Appl. Math.

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We study the two-dimensional snake-like pattern that arises in phase separation of alloys described by spinodal decomposition in the Cahn-Hilliard model. These are somewhat universal patterns due to an overlay of eigenfunctions of the Laplacian with a similar wavenumber. Similar structures appear in other models like reaction-diffusion systems describing animal coats' patterns or vegetation patterns in desertification.Our main result studies random functions given by cosine Fourier series with independent Gaussian coefficients, that dominate the dynamics in the Cahn-Hilliard model. This is not a cosine process, as the sum is taken over domains in Fourier space that not only grow and scale with a parameter of order 1/ε, but also move to infinity. Moreover, the model under consideration is neither stationary nor isotropic.To study the pattern size of nodal domains we consider the density of zeros on any straight line through the spatial domain. Using a theorem by Edelman and Kostlan and weighted ergodic theorems that ensure the convergence of the moving sums, we show that the average distance of zeros is asymptotically of order ε with a precisely given constant.

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Quantitative ergodic theorems and their number-theoretic applications

Cited by 26 publications

References 129 publications

Quantum ergodicity and Benjamini–Schramm convergence of hyperbolic surfaces

Quantum ergodicity and Benjamini–Schramm convergence of hyperbolic surfaces

Aperiodic order and spherical diffraction, I: auto-correlation of regular model sets

Pattern Size in Gaussian Fields from Spinodal Decomposition

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