R-trees and laminations for free groups III: currents and dual R-tree metrics

We study the map which associates to a current its support (whichis a lamination). We show that this map is Out(F_N)-equivariant,not injective, not surjective and not continuous. However itis semi-continuous and almost surjective in a suitable sense.Given an -tree T (with dense orbits) in the boundary of outerspace and a current µ carried by the dual lamination ofT, we define a dual pseudo-distance d_µ on T. When thetree and the current come from a measured geodesic laminationon a surface with boundary, the dual distance is the originaldistance of the tree T. In general, such a good correspondencedoes not occur. We prove that when the tree T is the attractivefixed point of a non-geometric irreducible, with irreduciblepowers, outer automorphism, the dual lamination of T is uniquelyergodic and the dual distance d_µ is either zero or infinitethroughout T.     

A sufficient condition for the subordination principle in ergodic optimization

Let T : X X be a continuous surjection of a topologicalspace, and let f : X be upper semi-continuous. Wewish to identify those T-invariant measures µ which maximize f dµ. We call such measures f-maximizing, and denotethe maximum by ß(f). The study of such measures andtheir properties has recently been dubbed ergodic optimization.A first step to understanding the structure of a function'smaximizing measures is to establish the following subordinationprinciple defined by T. Bousch: if µ and are T-invariantmeasures such that supp supp µ and µ is f-maximizing,then is also f-maximizing. Previous authors have approachedthis result by constructing a continuous function g : X such that f – ß(f) g T – g. We providea sufficient condition for the subordination principle whichhas advantages when the space X is noncompact.     

On the Algorithmic Complexity of Minkowski's Reconstruction Theorem

In 1903 Minkowski showed that, given pairwise different unitvectors µ₁, ..., µ_m in Euclidean n-space Rⁿ whichspan Rⁿ, and positive reals µ₁, ..., µ_m such that^m_i=1µ_iµ_i = 0, there exists a polytope P in Rⁿ, uniqueup to translation, with outer unit facet normals µ₁, ...,µ_m and corresponding facet volumes µ₁, ..., µ_m.This paper deals with the computational complexity of the underlyingreconstruction problem, to determine a presentation of P asthe intersection of its facet halfspaces. After a natural reformulationthat reflects the fact that the binary Turing-machine modelof computation is employed, it is shown that this reconstructionproblem can be solved in polynomial time when the dimensionis fixed but is #P-hard when the dimension is part of the input. The problem of ‘Minkowski reconstruction’ has variousapplications in image processing, and the underlying data structureis relevant for other algorithmic questions in computationalconvexity.     

Weakly Almost Periodic Functions on N with a Negative Base

Weakly almost periodic compactifications have been seriouslystudied for over 30 years. In the pioneering papers of de Leeuwand Glicksberg [4] and [5], the approach adopted was operator-theoretic.The current definition is more likely to be created from theperspective of universal algebra (see [1, Chapter 3]). For adiscrete group or semigroup S, the weakly almost periodic compactificationwS is the largest compact semigroup which (i) contains S asa dense subsemigroup, and (ii) has multiplication continuousin each variable separately (where largest means that any othercompact semigroup with the properties (i) and (ii) is a quotientof wS). A third viewpoint is to envisage wS as the Gelfand spaceof the C*-algebra of bounded weakly almost periodic functionson S (for the definition of such functions, see below). In this paper, we are concerned only with the simplest semigroup(N, +). The three approaches described above give three methodsof obtaining information about wN. An early striking resultabout wN, that it contains more than one idempotent, was obtainedby T. T. West using operator theory [13]. He considered theweak operator closure of the semigroup {T, T², T³, ...} of iteratesof a single operator T on the Hilbert space L²(µ) fora particular measure µ on [0, 1]. Brown and Moran, ina series of papers culminating in [2], used sophisticated techniquesfrom harmonic analysis to produce measures µ that permittedthe detection of further structure in wN; in particular, theyfound 2^cdistinct idempotents. However, for many years, no otherway of showing the existence of more than one idempotent inwN was found. The breakthrough came in 1991, and it was made by Ruppert [11].In his paper, he created a direct construction of a family ofweakly almost periodic functions which could detect 2^c differentidempotents in wN. His method was very ingenious (he used aunique variant of the p-adic expansion of integers) and rathercomplicated. Our main aim in this paper is to construct weaklyalmost periodic functions which are easy to describe and soappear more ‘natural’ than Ruppert's. We also showthat there are enough functions of our type to distinguish 2^cidempotentsin wN.     

Non-Regular Tangential Behaviour of a Monotone Measure

A Radon measure µ on Rⁿ is said to be k-monotone if is a non-decreasing function on (0,) for every x Rⁿ. (If µ is the k-dimensional Hausdorffmeasure restricted to a k-dimensional minimal surface then thisimportant property is expressed by the monotonicity formula.)We give an example of a 1-monotone measure µ in R² withnon-unique and non-conical tangent measures at a point. Furthermore,we show that µ can be the one-dimensional Hausdorff measurerestricted to a closed set A R². 2000 Mathematics Subject Classification49Q05, 49Q20 (primary), 28A75, 53A10 (secondary).     

Convergence estimates for the numerical approximation of homoclinic solutions

Received on 21 November 1995. Revised on 12 July 1996. This article is concerned with the numerical computation ofhomoclinic solutions converging to a hyperbolic or semi-hyperbolicequilibrium of a system u = f(u, µ). The approximationis done by replacing the original problem with a boundary valueproblem on a finite interval and introducing an additional phasecondition to make the solution unique. Numerical experimentshave indicated that the parameter µ is much better approximatedthan the homoclinic solution. This was proved in Schecter (1995IMA J. Numer Anal. 15, 23–60) for phase conditions satisfyingan additional ‘niceness’ assumption, which is unfortunatelynot satisfied for the phase condition most commonly used innumerical experiments and which actually suggested the super-convergenceresult. Here, this result is proved for arbitrary phase conditions.Moreover, it is shown that it suffices to approximate the originalboundary value problem to first order when considering semi-hyperbolicequilibria, extending a result of Schecter (1993 SIAM J. NumerAnal. 30, 1155–78). Permanent address: WIAS, MohrenstraBe 39, 10117 Berlin, Germany     

Small Deviations of Riemann-Liouville Processes In lq Spaces With Respect To Fractal Measures

We investigate Riemann–Liouville processes R_H, with H> 0, and fractional Brownian motions B_H, for 0 < H <1, and study their small deviation properties in the spacesL_q([0, 1], µ). Of special interest here are thin (fractal)measures µ, that is, those that are singular with respectto the Lebesgue measure. We describe the behavior of small deviationprobabilities by numerical quantities of µ, called mixedentropy numbers, characterizing size and regularity of the underlyingmeasure. For the particularly interesting case of self-similarmeasures, the asymptotic behavior of the mixed entropy is evaluatedexplicitly. We also provide two-sided estimates for this quantityin the case of random measures generated by subordinators. While the upper asymptotic bound for the small deviation probabilityis proved by purely probabilistic methods, the lower bound isverified by analytic tools concerning entropy and Kolmogorovnumbers of Riemann–Liouville operators. 2000 MathematicsSubject Classification 60G15 (primary), 47B06, 47G10, 28A80(secondary).     

Sharp Geometric Poincare Inequalities for Vector Fields and Non-Doubling Measures

We derive Sobolev–Poincaré inequalities that estimatethe L^q(d µ) norm of a function on a metric ball when µis an arbitrary Borel measure. The estimate is in terms of theL¹(d ) norm on the ball of a vector field gradient of the function,where d dx is a power of a fractional maximal function of µ.We show that the estimates are sharp in several senses, andwe derive isoperimetric inequalities as corollaries. 1991 MathematicsSubject Classification: 46E35, 42B25.     

A Class of Functional Equations on a Locally Compact Group

Let G be a locally compact group not necessarily unimodular.Let µ be a regular and bounded measure on G. We study,in this paper, the following integral equation, E(µ) This equation generalizes the functional equation for sphericalfunctions on a Gel'fand pair. We seek solutions in the spaceof continuous and bounded functions on G. If is a continuousunitary representation of G such that (µ) is of rank one,then tr((µ)(x)) is a solution of E(µ). (Here, trmeans trace). We give some conditions under which all solutionsare of that form. We show that E(µ) has (bounded and)integrable solutions if and only if G admits integrable, irreducibleand continuous unitary representations. We solve completelythe problem when G is compact. This paper contains also a listof results dealing with general aspects of E(µ) and propertiesof its solutions. We treat examples and give some applications.     

Uniqueness of solutions to weak parabolic equations for measures

We study uniqueness of solutions of parabolic equations formeasures µ(dt dx) = µ_t(dx)dt of the type L*µ = 0, satisfying µ_t as t 0, where each µ_tis a probability measure on ^d, L = _t + a^ij(t, x)_xixj + bⁱ(t,x)_xj is a differential operator on (0, T) x ^d and is a giveninitial measure. One main result is that uniqueness holds underuniform ellipticity and Lipschitz conditions on a^ij but forbⁱ merely local integrability and coercivity conditions aresufficient.     

Application to Dirichlet Series of Maximal Large Sieve Inequality

In this note, the maximal Large Sieve inequality has been usedto obtain following estimate: where D(s)=, s=+it, is a Dirichletseries, dµ is a nonnegative Borel measure on R and U =[–Clog_–1N] (say T1). In addition, the higher-dimensionalversion of (*) has been obtained.     

On the Centred Hausdorff Measure

Let v be a measure on a separable metric space. For t, q R,the centred Hausdorff measures µ^h with the gauge functionh(x, r) = r^t(vB(x, r))^q is studied. The dimension defined bythese measures plays an important role in the study of multifractals.It is shown that if v is a doubling measure, then µ^h isequivalent to the usual spherical measure, and thus they definethe same dimension. Moreover, it is shown that this is trueeven without the doubling condition, if q 1 and t 0 or ifq 0. An example in R² is also given to show the surprisingfact that the above assertion is not necessarily true if 0 <q < 1. Another interesting question, which has been askedseveral times about the centred Hausdorff measure, is whetherit is Borel regular. A positive answer is given, using the aboveequivalence for all gauge functions mentioned above.     

Smoothness of the Lq-Spectrum of Self-Similar Measures with Overlaps

Let µ be the self-similar measure for a linear functionsystem S_jx=x+b_j (j=1,2,...,m) on the real line with the probabilityweight . Under the condition that satisfies the finite type condition, the L^q-spectrum (q) of µ is shown to bedifferentiable on (0,); as an application, µ is exactdimensional and satisfies the multifractal formalism.     

Stability Regions of Hill's Equation

The stability of the solutions of Hill's equation+[B—µø(t)]x= 0 is studied through the use of a fundamental matrix of solutionsevaluated after one period. This fundamental matrix is expandedin a power series in µ, but its determinant is computedhere by using series expansion rather than the Jacobi-Liouvilleformula. Using this procedure, it is shown that successive approximationsof boundaries of stability regions in the (B, µ) planecan be obtained simply, in particular without expansion of Bin power series of µ and Ø(t) in a Fourier series.A simple expression is obtained for the first approximation.Two examples are treated, with stability curves drawn up tothe second approximation.     

Bounded-Norm Matrix-Inverse Mappings

This paper introduces the concept of bounded-norm matrix-inversemappings, i.e. mappings µ : R^mxnR^nxm such that, for allnonzero mxn matrices A, the matrix µ(A) is a generalizedinverse of A and ||µ(A)||> k/s(A), where K < 0 isa constant and s(A) is the nonzero singular value of A havingsmallest absolute value. It is shown how the definition of suchmappings is motivated by the need to ensure finite terminationof the inner-iterations of generalized elimination methods forthe solution of nonlinearly constrained optimization problems.The main result of the paper is that the mapping defined byµ(A) = A^b is a bounded-norm matrix-inverse mapping, providedthat the basic inverse A^b is calculated using Gaussian eliminationwith complete pivoting. The concept of bounded-norm matrix-inversemappings is then extended to that of boundednorm least-squaresmatrix-inverse mappings. It is proved that the mapping definedby µ(A) = A^ß is a bounded-norm least-squaresmatrix-inverse mapping, provided that the basic least-squaresinverse A^ß is calculated using the QR decompositionwith column pivoting.     

Recurrence and asymptotics for orthonormal rational functions on an interval

^{Let µ be a positive bounded Borel measure on a subsetI of the real line and = {₁, ..., _n} a sequence of arbitrary ‘complex’poles outside I. Suppose {₁, ..., _n} is the sequence of rationalfunctions with poles in orthonormal on I with respect to µ. First, we are concernedwith reducing the number of different coefficients in the three-termrecurrence relation satisfied by these orthonormal rationalfunctions. Next, we consider the case in which I = [–1, 1] and µ satisfies the Erdos–Turán conditionµ' > 0 a.e. on I (where µ' is the Radon–Nikodymderivative of the measure µ with respect to the Lebesguemeasure) to discuss the convergence of _n+1(x)/_n(x) as n tendsto infinity and to derive asymptotic formulas for the recurrencecoefficients in the three-term recurrence relation. Finally,we give a strong convergence result for _n(x) under the morerestrictive condition that µ satisfies the Szeg condition(1 – x2)–1/2 log µ'(x) L1([– 1, 1]).}     

The Milnor number of a function on a space curve germ

Given a finite function germ f:(X, 0) (, 0) on a reduced spacecurve singularity (X, 0), we show that µ(f) = µ(X,0) + deg(f) – 1. Here, µ(f) and µ(X, 0) denotethe Milnor numbers of the function and the curve, respectively,and deg(f) is the degree of f. We use this formula to obtainseveral consequences related to the topological triviality andWhitney equisingularity of families of curves and families offunctions on curves.     

Representations of compact linear operators in Banach spaces and nonlinear eigenvalue problems

Let X and Y be reflexive Banach spaces with strictly convexduals, and let T be a compact linear map from X to Y. It isshown that a certain nonlinear equation, involving T and itsadjoint, has a normalised solution (an ‘eigenvector’)corresponding to an ‘eigenvalue’, and that the sameis true for each member of a countable family of similar equationsinvolving the restrictions of T to certain subspaces of X. Theaction of T can be described in terms of these ‘eigenvectors’.There are applications to the p-Laplacian, the p-biharmonicoperator and integral operators of Hardy type.     

The Kirwan Map for Singular Symplectic Quotients

Let M be a Hamiltonian K-space with proper moment map µ.The symplectic quotient X = µ^–1(0)/K is a singularstratified space with a symplectic structure on the strata.In this paper we generalise the Kirwan map, which maps the Kequivariant cohomology of µ^–1(0) to the middle perversityintersection cohomology of X, to this symplectic setting. The key technical results which allow us to do this are Meinrenken'sand Sjamaar's partial desingularisation of singular symplecticquotients and a decomposition theorem, proved in Section 2 ofthis paper, exhibiting the intersection cohomology of a ‘symplecticblowup’ of the singular quotient X along a maximal depthstratum as a direct sum of terms including the intersectioncohomology of X.