An ergodic theorem for Delone dynamical systems and existence of the integrated density of states

DOI10.1007/BF02807400zbMath1131.37015arXivmath-ph/0310017MaRDI QIDQ2470236

Peter Stollmann, Daniel H. Lenz

Publication date: 13 February 2008

Published in: Journal d'Analyse Mathématique (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math-ph/0310017

zbMATH Keywords

uniform convergenceergodic theoremrandom operators

Mathematics Subject Classification ID

Ergodicity, mixing, rates of mixing (37A25) Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics (82B44) Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) (37N20) Random linear operators (47B80) Symbolic dynamics (37B10) Linear difference operators (47B39)

Related Items

Tower systems for linearly repetitive Delone sets, Glivenko-Cantelli theory, Ornstein-Weiss quasi-tilings, and uniform ergodic theorems for distribution-valued fields over amenable groups, Bragg spectrum, K-theory, and gap labeling of aperiodic solids, Quantum and spectral properties of the Labyrinth model, Spectra of discrete Schrödinger operators with primitive invertible substitution potentials, A Glivenko-Cantelli theorem for almost additive functions on lattices, Uniform existence of the integrated density of states on metric Cayley graphs, Characterization of the Anderson metal-insulator transition for non ergodic operators and application, The rotation number for almost periodic potentials with jump discontinuities and \(\delta \)-interactions, Hierarchical structures in Sturmian dynamical systems, On sums of semibounded Cantor sets, Delone dynamical systems and spectral convergence, Uniform approximation of the integrated density of states for long-range percolation Hamiltonians, Delone sets with finite local complexity: Linear repetitivity versus positivity of weights, Rotation numbers of linear Hamiltonian systems with phase transitions over almost periodic lattices, Discontinuities of the integrated density of states for random operators on Delone sets, Uniform existence of the integrated density of states for randomly weighted Hamiltonians on long-range percolation graphs, Almost-additive ergodic theorems for amenable groups, UNIFORM ERGODIC THEOREMS ON APERIODIC LINEARLY REPETITIVE TILINGS AND APPLICATIONS, A Banach space-valued ergodic theorem and the uniform approximation of the integrated density of states, \(L^{2}\)-spectral invariants and convergent sequences of finite graphs, Spectral structures and topological methods in mathematical quasicrystals. Abstracts from the workshop held October 1--7, 2017, A Banach space-valued ergodic theorem for amenable groups and applications, Convergence theorems for graph sequences, A central limit theorem for functions of stationary max-stable random fields on \(\mathbb{R}^d\), Uniform existence of the integrated density of states for models on \({\mathbb{Z}}^d\), Groupoids, von Neumann algebras and the integrated density of states, Hamiltonians on discrete structures: jumps of the integrated density of states and uniform convergence, Continuity properties of the integrated density of states on manifolds, Localisation for Delone operators via Bernoulli randomisation, Tilings and traces, Approximation of the integrated density of states on sofic groups, Quasicrystals, aperiodic order, and groupoid von Neumann algebras., The rotation number for almost periodic Schrödinger operators with \(\delta \)-potentials

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Log Hölder continuity of the integrated density of states for stochastic Jacobi matrices
• Remark on the continuity of the density of states of ergodic finite difference operators
• Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation
• The spectrum of a quasiperiodic Schrödinger operator
• Quasicrystals and discrete geometry. Proceedings of the 1995 fall programme at The Fields Institute, Toronto, Canada
• Geometric models for quasicrystals. I. Delone sets of finite type
• A characterization of substitutive sequences using return words
• Hierarchical structures in Sturmian dynamical systems
• Algebras of random operators associated to Delone dynamical systems
• Discontinuities of the integrated density of states for random operators on Delone sets
• Towards a characterization of self-similar tilings in terms of derived Voronoï tessellations
• A remark on Schrödinger operators on aperiodic tilings.
• Quasicrystals, aperiodic order, and groupoid von Neumann algebras.
• Pure point dynamical and diffraction spectra
• Some remarks on discrete aperiodic Schrödinger operators
• Matrix Analysis
• LATTICE GAS MODELS ON SELF-SIMILAR APERIODIC TILINGS
• Dynamics of self-similar tilings
• Repetitive Delone sets and quasicrystals
• Linearly recurrent subshifts have a finite number of non-periodic subshift factors
• Directions in Mathematical Quasicrystals
• Uniform ergodic theorems on subshifts over a finite alphabet
• NONCOMMUTATIVE GEOMETRY OF TILINGS AND GAP LABELLING
• Aperiodic order and quasicrystals: spectral properties