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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)
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)
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)