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For a large class of tilings of R d , including the Penrose tiling in two dimension as well as the icosahedral ones in 3 dimension, the continuous hull Ω T of such a tiling T inherits a minimal R d-lamination structure with flat leaves and a transversal Γ T which is a Cantor set. In this case, we show that the continuous hull can be seen as the projective… (More)

An analogue of the Riemannian Geometry for an ultrametric Cantor set (C, d) is described using the tools of Noncommutative Geometry. Associated with (C, d) is a weighted rooted tree, its Michon tree [28]. This tree allows to define a family of spectral triples (C Lip (C), H, D) using the ℓ 2-space of its vertices, giving the Cantor set the structure of a… (More)

We give a new proof of correlation estimates for arbitrary moments of the resol-vent of random Schrödinger operators on the lattice that generalizes and extends the correlation estimate of Minami for the second moment. We apply this moment bound to obtain a new n-level Wegner-type estimate that measures eigenvalue correlations through an upper bound on the… (More)

Let (A, H, D) be a spectral triple, namely: A is a C *-algebra, H is a Hilbert space on which A acts and D is a selfadjoint operator with compact resolvent such that the set of elements of A having a bounded commutator with D is dense. A spectral metric space, the noncommutative analog of a complete metric space, is a spectral triple (A, H, D) with… (More)

- J Bellissard, B Simon
- 1982

For a dense G, of pairs (,I, a) in R', we prove that the operator (Hu)(n) = u(n + 1) + u(n-1) + I cos(2nan + 0) u(n) has a nowhere dense spectrum. Along the way we prove several interesting results about the case a =p/q of which we mention: (a) If qB is not an integral multiple of A, then all gaps are open, and (b) If q is even and 0 is chosen suitably,… (More)

- J. Bellissard, A. van Elst
- 2008

We give an overview of the Integer Quantum Hall Effect. We propose a mathematical framework using Non-Commutative Geometry as defined by A. Connes. Within this framework, it is proved that the Hall conductivity is quantized and that plateaux occur when the Fermi energy varies in a region of localized states.

We develop a mathematical framework allowing to study anomalous transport in homogeneous solids. The main tools characterizing the anomalous transport properties are spectral and diffusion exponents associated to the covariant Hamiltonians describing these media. The diffusion exponents characterize the spectral measures entering in Kubo's formula for the… (More)

Using the results obtained by the non commutative geometry techniques applied to the Harper equation, we derive the areas distribution of random walks of length N on a two-dimensional square lattice for large N , taking into account finite size contributions. Let us consider on a square lattice all closed paths of length N starting at the origin. For such a… (More)

- J. Bellissard
- 2002

This work proposes a very simple random matrix model, the Flip Matrix Model, liable to approximate the behavior of a two dimensional electron in a weak random potential. Its construction is based on a phase space analysis, a suitable discretization and a simplification of the true model. The density of states of this model is investigated using the… (More)

Let T be an aperiodic and repetitive tiling of R d with finite local complexity. We present a spectral sequence that converges to the K-theory of T with page-2 iso-morphic to the Pimsner cohomology of T. It is a generalization of the Serre spectral sequence to a class of spaces which are not fibered. The Pimsner cohomology of T generalizes the cohomology of… (More)