the spectrum of the laplacian on forms over flat manifolds in this article we prove that the spectrum of the laplacian on k forms over a non compact flat manifold is always a connected closed interval of the non negative real line the proof is based on a detailed decomposition of the structure of flat manifolds

the hodge cohomology of a conformally compact metric on etudie le laplacien de hogde agissant sur des k formes differentielles pour une classe de varietes de riemann completes a courbure sectionnelle negative proche de l infini ces varietes ont des compactifications c sur lesquelles la metrique est conforme a une metrique lisse jusqu au bord avec un facteur conforme 2 une fonction definissante pour le bord

a mourre estimate and related bounds for hyperbolic manifolds with cusps of non maximal rank we consider the laplace operator on quotients of hyperbolic n dimensional space by a geometrically finite discrete group of hyperbolic isometries with parabolic subgroups of non maximal rank using methods developed by the first two authors we prove a mourre estimate and commutator estimates on the laplacian which imply absolute continuity of the spectrum and quantitative resolvent estimates these estimates will be used elsewhere to study the scattering matrix and eisenstein series and their meromorphic continuations and should be useful in studying trace formulas for these discrete groups

the spectrum of schrodinger operators and hodge laplacians on conformally cusp manifolds we describe the spectrum of the k form laplacian on conformally cusp riemannian manifolds the essential spectrum is shown to vanish precisely when the k and k 1 de rham cohomology groups of the boundary vanish we give weyl type asymptotics for the eigenvalue counting function in the purely discrete case in the other case we analyze the essential spectrum via positive commutator methods and establish a limiting absorption principle this implies the absence of the singular spectrum for a wide class of metrics we also exhibit a class of potentials v such that the schrodinger operator has compact resolvent although v tends to in most of the infinity we correct a statement from the literature regarding the essential spectrum of the laplacian on forms on hyperbolic manifolds of finite volume and we propose a conjecture about the existence of such manifolds in dimension four whose cusps are rational homology spheres

resolvent of the laplacian on geometrically finite hyperbolic manifolds for geometrically finite hyperbolic manifolds n 1 we prove the meromorphic extension of the resolvent of laplacian poincare series einsenstein series and scattering operator to the whole complex plane we also deduce the asymptotics of lattice points of in large balls of n 1 in terms of the hausdorff dimension of the limit set of

multifractality in forgetful memories learning rules of a forgetful memory generate their synaptic efficacies through iterative procedures that operate on the input data random patterns we analyse invariant distributions of the synaptic couplings as they arise asymptotically and show that they exhibit fractal or multifractal properties we also discuss their dependence upon the learning rule and the parameters specifying it and indicate how the nature of the invariant distribution is related to the network performance

hodge cohomology of gravitational instantons we study the space of l2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type these metrics generalize the geometric structures at infinity of several different well known classes of metrics including asymptotically locally euclidean manifolds the known types of gravitational instantons and also poincare metrics on q rank 1 ends of locally symmetric spaces and on the complements of smooth divisors in kahler manifolds the answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold the l2 signature formula implied by our result is closely related to the one proved by dai 25 and more generally by vaillant 67 and identifies dai s invariant directly in terms of intersection cohomology of differing perversities this work is also closely related to a recent paper of carron 12 and the forthcoming paper of cheeger and dai 17 we apply our results to a number of examples gravitational instantons among them arising in predictions about l2 harmonic forms in duality theories in string theory

the spectrum of kleinian manifolds abstract we obtain the plancherel theorem for l 2 g where g is a classical simple lie group of real rank one and g is convex cocompact discrete subgroup and deduce its consequences for the spectrum of locally invariant differential operators on bundles over kleinian manifolds as the main tool we develop a geometric version of scattering theory which in particular contains the meromorphic continuation of the eisenstein series for this situation the central role played by invariant distribution sections supported on the limit set is emphasized

sur la l 2 cohomologie des varietes a courbure negative we give a topological interpretation of the space of l2 harmonic forms on finite volume manifolds with sufficiently pinched negative curvature we give examples showing that this interpretation fails if the curvature is not sufficiently pinched and that our result is sharp with respect to the pinching constants the method consists first in comparing l2 cohomology with weighted l2 cohomology thanks to previuos works done by t ohsawa and then in identifying these weighted spaces

iterated function systems and the global construction of fractals iterated function systems i f ss are introduced as a unified way of generating a broad class of fractals these fractals are often attractors for i f ss and occur as the supports of probability measures associated with functional equations the existence of certain p balanced measures for i f ss is established and these measures are uniquely characterized for hyperbolic i f ss the hausdorff besicovitch dimension for some attrac tors of hyperbolic i f ss is estimated with the aid of p balanced measures what appears to be the broadest framework for the exactly computable moment theory of p balanced measures that of linear i f ss and of probabilistic mixtures of iterated riemann surfaces is presented this extensively generalizes earlier work on orthogonal polynomials on julia sets an example is given of fractal reconstruction with the use of linear i f ss and moment theory

approximations of fractal sets abstract five approximations of fractal sets a are compared the first three were described by hutchinson barnsley demko and williams respectively two new other approximations are still efficient in all circumstances for any algorithm error estimates in the approximations are found numerical comparisons are also done through classical examples of attractors in the plane

the spectrum of the hodge laplacian for a degenerating family of hyperbolic three manifolds we consider a sequence m i of compact hyperbolic manifolds converging to a complete hyperbolic manifold mq with cusps the laplace operator acting on the space of l2 differential forms on m0 has continuous spectrum filling the half line 0 oo one expects therefore that the spectra of this operator on m accumulate to produce the continuous spectrum of the limiting manifold we prove that this is the case and obtain a sharp estimate of the rate of accumulation

cohomologie l 2 des varietes qale we give a topological interpretation of the space of l 2 harmonis forms of some qale manifolds introduced by d joyce we introduce a analytical criterium which make possible the used of mayer vietoris sequence

power domains and iterated function systems we introduce the notion of weakly hyperbolic iterated function system ifs on a compact metric space which generalises that of hyperbolic ifs based on a domain theoretic model which uses the plotkin power domain and the probabilistic power domain respectively we prove the existence and uniqueness of the attractor of a weakly hyperbolic ifs and the invariant measure of a weakly hyperbolic ifs with probabilities extending the classic results of hutchinson for hyperbolic ifss in this more general setting we also present finite algorithms to obtain discrete and digitised approximations to the attractor and the invariant measure extending the corresponding algorithms for hyperbolic ifss we then prove the existence and uniqueness of the invariant distribution of a weakly hyperbolic recurrent ifs and obtain an algorithm to generate the invariant distribution on the digitised screen the generalised riemann integral is used to provide a formula for the expected value of almost everywhere continuous functions with respect to this distribution for hyperbolic recurrent ifss and lipschitz maps one can estimate the integral up to any threshold of accuracy

geometrically finite poincare einstein metrics we construct new examples of einstein metrics by perturbing the conformal infinity of geometrically finite hyperbolic metrics and by applying the inverse function theorem in suitable weighted holder spaces

solution of an inverse problem for fractals and other sets abstract the problem of providing succinct approximate descriptions of given bounded subsets of rn can be solved by application of the contraction mapping principle

analytic continuation of the single eigenvalue probability density function an exact expression for the single eigenvalue probability density function for any dimension of a gaussian orthogonal ensemble is found by analytic continuation it is shown that the form can be used to derive a number of exact results in a simple way in odd dimensions it is shown that the single eigenvalue probability density function can be written as a sum of two densities

pseudodifferential operator calculus for generalized q rank 1 locally symmetric spaces i abstract this paper is the first of two papers constructing a calculus of pseudodifferential operators suitable for doing analysis on q rank 1 locally symmetric spaces and riemannian manifolds generalizing these this generalization is the interior of a manifold with boundary where the boundary has the structure of a tower of fibre bundles the class of operators we consider on such a space includes those arising naturally from metrics which degenerate to various orders at the boundary in directions given by the tower of fibrations as well as q rank 1 locally symmetric spaces examples include ricci flat metrics on the complement of a divisor in a smooth variety constructed by tian and yau in this first part of the calculus construction parametrices are found for fully elliptic differential a operators which are uniformly elliptic operators on these manifolds that satisfy an additional invertibility condition at infinity in the second part we will consider operators that do not satisfy this condition