A Banach space-valued ergodic theorem and the uniform approximation of the integrated density of states

Abstract

In this paper we consider bounded operators on infinite graphs, in particular Cayley graphs of amenable groups. The operators satisfy an equivariance condition which is formulated in terms of a colouring of the vertex set of the underlying graph. In this setting it is natural to expect that the integrated density of states (IDS), or spectral distribution function, exists. We show that it can be defined as the uniform limit of approximants associated to finite matrices. The proof is based on a Banach space valued ergodic theorem which even allows explicit convergence estimates. Our result applies to a variety of group structures and colouring types, in particular to periodic operators and percolation-type Hamiltonians.

corrected version

References in corpus (7)

• Hamiltonians on discrete structures: Jumps of the integrated density of states and uniform convergence
• Uniform existence of the integrated density of states for models on $\ZZ^d$
• The Integrated Density of States for Random Schroedinger Operators
• Equality of Lifshitz and van Hove exponents on amenable Cayley graphs
• $L^2$-spectral invariants and quasi-crystal graphs
• Aperiodic order, integrated density of states and the continuous algebras of John von Neumann
• $L^p$-approximation of the integrated density of states for Schrödinger operators with finite local complexity

Cited by in corpus (9)

• A Glivenko-Cantelli Theorem for almost additive functions
• Computable Følner monotilings and a theorem of Brudno I
• Linear repetitivity beyond abelian groups
• $L^p$-approximation of the integrated density of states for Schrödinger operators with finite local complexity
• Uniform existence of the integrated density of states on metric Cayley graphs
• Approximation of the integrated density of states on sofic groups
• Uniform existence of the integrated density of states for randomly weighted Hamiltonians on long-range percolation graphs
• Matrix representations of multidimensional integral and ergodic operators
• Glivenko-Cantelli Theory, Ornstein-Weiss quasi-tilings, and uniform Ergodic Theorems for distribution-valued fields over amenable groups