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Title of Journal: Commun Math Phys

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Abbravation: Communications in Mathematical Physics

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Springer-Verlag

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DOI

10.1002/mop.26931

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1432-0916

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Quasicrystals and Almost Periodicity

Authors: JeanBaptiste Gouéré
Publish Date: 2005/02/18
Volume: 255, Issue: 3, Pages: 655-681
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Abstract

We give in this paper topological and dynamical characterizations of mathematical quasicrystals Let Open image in new window denote the space of uniformly discrete subsets of the Euclidean space Let Open image in new window denote the elements of Open image in new window that admit an autocorrelation measure A Patterson set is an element of Open image in new window such that the Fourier transform of its autocorrelation measure is discrete Patterson sets are mathematical idealizations of quasicrystals We prove that S ∈ Open image in new window is a Patterson set if and only if S is almost periodic in Open image in new window Open image in new window where Open image in new window denotes the Besicovitch topology Let χ be an ergodic random element of Open image in new window We prove that χ is almost surely a Patterson set if and only if the dynamical system has a discrete spectrum As an illustration we study deformed model sets

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Other Papers In This Journal:
  1. Statistics of Wave Functions for a Point Scatterer on the Torus
  2. Hausdorff Dimension of Julia Sets of Feigenbaum Polynomials with High Criticality
  3. Stability and Duality in $${\mathcal{N}\,{=}\,2}$$ Supergravity
  4. Exponential Mixing for the 3 D Stochastic Navier–Stokes Equations
  5. Binding of Polarons and Atoms at Threshold
  6. Probabilistic Weyl Laws for Quantized Tori
  7. On Perturbations of the Schwarzschild Anti-De Sitter Spaces of Positive Mass
  8. Wall-Crossing, Free Fermions and Crystal Melting
  9. Standing Ring Blow up Solutions to the N-Dimensional Quintic Nonlinear Schrödinger Equation
  10. Laminar Boundary Layers in Convective Heat Transport
  11. There are no Magnetically Charged Particle-like Solutions of the Einstein Yang-Mills Equations for Models with an Abelian Residual Group
  12. Duality of 2D Gravity as a Local Fourier Duality
  13. Nonlinear Asymptotic Stability of the Lane-Emden Solutions for the Viscous Gaseous Star Problem with Degenerate Density Dependent Viscosities
  14. Extended Scaling Relations for Planar Lattice Models
  15. Infinite-Dimensional Schur–Weyl Duality and the Coxeter–Laplace Operator
  16. Phase Boundaries in Algebraic Conformal QFT
  17. On the Singularity of the Free Energy at a First Order Phase Transition
  18. Free Analog of Pressure and Its Legendre Transform
  19. Continuity of the Lyapunov Exponents for Quasiperiodic Cocycles
  20. On the Correlation Function of the Characteristic Polynomials of the Hermitian Wigner Ensemble
  21. Stationary Random Metrics on Hierarchical Graphs Via $${(\min,+)}$$-type Recursive Distributional Equations
  22. Thermalization and Canonical Typicality in Translation-Invariant Quantum Lattice Systems
  23. Higgs Bundles and ( A , B , A )-Branes
  24. Inviscid Incompressible Limits of the Full Navier-Stokes-Fourier System
  25. Global Well-Posedness Issues for the Inviscid Boussinesq System with Yudovich’s Type Data
  26. Dispersive Estimates for Schrödinger Operators in Dimension Two
  27. A Monoidal Category for Perturbed Defects in Conformal Field Theory
  28. On Static Shells and the Buchdahl Inequality for the Spherically Symmetric Einstein-Vlasov System
  29. Holomorphic Factorization for a Quantum Tetrahedron
  30. High Energy Limits of Laplace-Type and Dirac-Type EigenFunctions and Frame Flows
  31. The Operator Product Expansion for Perturbative Quantum Field Theory in Curved Spacetime
  32. The Szegő Kernel on a Sewn Riemann Surface
  33. Wave Relations
  34. An Application of Mirror Extensions
  35. Diffusion in a Weakly Random Hamiltonian Flow
  36. Spherical T-Duality
  37. A Fuchsian Matrix Differential Equation for Selberg Correlation Integrals
  38. Lax Pair Equations and Connes-Kreimer Renormalization
  39. SNA’s in the Quasi-Periodic Quadratic Family
  40. On the Geometry of Closed G 2 -Structures
  41. Irreducible Characters of General Linear Superalgebra and Super Duality
  42. Connecting Solutions of the Lorentz Force Equation do Exist
  43. Toric G 2 and Spin (7) Holonomy Spaces from Gravitational Instantons and Other Examples
  44. The Smoluchowski-Kramers Limit of Stochastic Differential Equations with Arbitrary State-Dependent Friction
  45. Dirac Reduction for Poisson Vertex Algebras
  46. Dispersive Estimates for Schrödinger Equations with Threshold Resonance and Eigenvalue
  47. Multipermutation Solutions of the Yang–Baxter Equation
  48. The Shape of Expansion Induced by a Line with Fast Diffusion in Fisher-KPP Equations
  49. A Lower Bound on Blowup Rates for the 3D Incompressible Euler Equation and a Single Exponential Beale-Kato-Majda Type Estimate
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  52. Equivariant Comparison of Quantum Homogeneous Spaces
  53. Regular Strongly Typical Blocks of $${\mathcal{O}^{\mathfrak {q}}}$$
  54. Exponential Equations Related to the Quantum ‘ ax + b ’ Group
  55. Twisted Logarithmic Modules of Vertex Algebras
  56. Singular Continuous Spectrum for Singular Potentials
  57. How to Remove the Boundary in CFT – An Operator Algebraic Procedure
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  61. Triangular Invariants, Three-Point Functions and Particle Stability on the de Sitter Universe
  62. On positive functionals on algebras of test functions for quantum fields
  63. Large Deviations for the Yang-Mills Measure on a Compact Surface
  64. Vanishing Viscosity Solutions of the Compressible Euler Equations with Spherical Symmetry and Large Initial Data
  65. Periodicity and the Determinant Bundle
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