自仿测度的非谱准则

设μ_(M,D)是由仿射迭代函数系{φ_d(x)=M~(-1)(x+d)}_(d∈D)唯一确定的自仿测度,它的谱性或非谱性与Hilbert空间L~2(μ_(M,D))中正交指数基(也称为Fourier基)的存在性有着直接的关系.近年来自仿测度μ_(M,D)的谱性或非谱性问题的研究受到人们普遍的关注.本文给出了判定自仿测度μ_(M,D)非谱性的几个充分条件,所得结果改进推广Dutkay,Jorgensen等人的非谱准则.     

A new exact method for obtaining the multifractal spectrum of multiscaled multinomial measures and IFS invariant measures

In this paper we present a new exact method for obtaining the multifractal spectrum of multiscaled multinomial measures and invariant measures associated with iterated function systems (IFS). A multinomial measure is shown to be generated as the invariant measure of an associated IFS. Then, the multifractal spectrum of the measure is determined by a couple of parametric implicit equations. This analysis generalizes some results previously obtained for the case of single-scaled multinomial measures (e.g., the binomial measure). A geometric interpretation of this new framework working in the space of codes of the IFS gives new insight into the nature of the multifractal formalism. This paper extends the results presented in Gutiérrez et. al. Fractals 4, (1996) 17–27.     

Coherent measures and the existence of smooth Lyapunov 1-forms for flows

Let a smooth vector field V on a smooth closed manifold M be given and let Z⊂M be an isolated invariant set for the flow of V. In this situation, we give a necessary and sufficient condition for the existence of a Lyapunov 1-form for (V,Z) in terms of the relative asymptotic cycles associated with certain invariant measures of the flow.     

Some Siegel modular standard L-values, and Shafarevich-Tate groups

We explain how the Bloch-Kato conjecture leads us to the following conclusion: a large prime dividing a critical value of the L-function of a classical Hecke eigenform f of level 1, should often also divide certain ratios of critical values for the standard L-function of a related genus two (and in general vector-valued) Hecke eigenform F. The relation between f and F (Harder?s conjecture in the vector-valued case) is a congruence involving Hecke eigenvalues, modulo the large prime. In the scalar-valued case we prove the divisibility, subject to weak conditions. In two instances in the vector-valued case, we confirm the divisibility using elaborate computations involving special differential operators. These computations do not depend for their validity on any unproved conjecture.     

m-Band orthogonal vector-valued multiwavelets for vector-valued signals

In this paper, we introduce and study vector-valued multiresolution analysis with multiplicity r (VMRA) and m-band orthogonal vector-valued multiwavelets which have potential to form a convenient tool for analyzing vector-valued signals. Necessary conditions for orthonormality of vector-valued multiwavelets are presented in terms of filter banks. The existence of m-band vector-valued orthonormal multiwavelets is proved by means of bi-infinite matrix. The relationship between vector-valued multiwavelets and traditional multiwavelets are considered, and it is found that multiwavelets can be derived from row vector of vector-valued multiwavelets. The construction of vector-valued multiwavelets from several scalar-valued wavelets is proposed. Furthermore, we show how to construct vector-valued multiwavelets by using paraunitary multifilter bank, in particular, we give formulations of highpass filters when its corresponding lowpass filters satisfy certain conditions and m=2. An example is provided to illustrate this algorithm. At last, we present fast vector-valued multiwavelets transform in form of bi-infinite vector.     

An operator-valued Lyapunov theorem

We generalize Lyapunov's convexity theorem for classical (scalar-valued) measures to quantum (operator-valued) measures. In particular, we show that the range of a nonatomic quantum probability measure is a weak^?-closed convex set of quantum effects (positive operators bounded above by the identity operator) under a sufficient condition on the non-injectivity of integration. To prove the operator-valued version of Lyapunov's theorem, we must first define the notions of essentially bounded, essential support, and essential range for quantum random variables (Borel measurable functions from a set to the bounded linear operators acting on a Hilbert space).     

Invariant measures for non-primitive tiling substitutions

We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well known that in the primitive case, the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the “integer Sierpiński gasket and carpet” tilings. For such tilings, the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling.     

Vector-Valued Kernels of Bergman Type

Bergman reproducing integral formulas can be obtained for holomorphic mappings \(f{:}\,{\mathbb {B}}\rightarrow {\mathbb {C}}^n,\,{\mathbb {B}}\) the open unit ball of \({\mathbb {C}}^n\), by applying the well-known formulas for scalar-valued functions on \({\mathbb {B}}\) to each coordinate function of f, provided those coordinate functions each lie in an appropriate Bergman space. Here, we consider an alternative formulation whereby f is reproduced as the integral of the product of a fixed vector-valued kernel and the scalar expression \(\langle f(z),z \rangle ,\,z\in {\mathbb {B}}\), where \(\langle \cdot ,\cdot \rangle \) is the Hermitian inner product in \({\mathbb {C}}^n\). We provide two different classes of vector-valued kernels that reproduce holomorphic mappings lying in spaces properly containing the weighted vector-valued Bergman spaces. An analysis of these larger spaces is given. The first set of kernels arises naturally from the scalar-valued Bergman kernels, while the second yields the orthogonal projection onto an isomorphic space of scalar-valued functions in the unweighted case.     

Remarks on limit sets of infinite iterated function systems

If an iterated function system (IFS) is finite, it is well known that there is a unique non-empty compact invariant set K and that K?=???(I ^??), where ?? is the coding map. For an infinite IFS, there are two different sets generalising K, namely ??(I ^??) and its closure ${\overline{\pi(I^\infty)}}$ . In this paper we investigate the relations between these sets and their Hausdorff dimensions. In particular, we show how to construct an IFS for any pair of prescribed dimensions for ??(I ^??) and ${\overline{\pi(I^\infty)}\setminus \pi(I^\infty)}$ . Moreover, we investigate a set which depends only on the first iteration of an IFS, and characterise its relation to the abovementioned sets. This also extends and clarifies recent results by Mihail and Miculescu, who investigated the coding map for an infinite IFS and a condition for this map to be onto. Finally, we study the special case of one-dimensional IFS and show that in terms of the relations of the abovementioned sets these systems exhibit some very special features which do not generalise to higher dimensional situations.     

Invariant measures for parabolic IFS with overlaps and random continued fractions

We study parabolic iterated function systems (IFS) with overlaps on the real line. An ergodic shift-invariant measure with positive entropy on the symbolic space induces an invariant measure on the limit set of the IFS. The Hausdorff dimension of this measure equals the ratio of entropy over Lyapunov exponent if the IFS has no ``overlaps.' We focus on the overlapping case and consider parameterized families of IFS, satisfying a transversality condition. Our main result is that the invariant measure is absolutely continuous for a.e. parameter such that the entropy is greater than the Lyapunov exponent. If the entropy does not exceed the Lyapunov exponent, then their ratio gives the Hausdorff dimension of the invariant measure for a.e. parameter value, and moreover, the local dimension of the exceptional set of parameters can be estimated. These results are applied to a family of random continued fractions studied by R. Lyons. He proved singularity above a certain threshold; we show that this threshold is sharp and establish absolute continuity for a.e. parameter in some interval below the threshold.

     

Fourier frequencies in affine iterated function systems

We examine two questions regarding Fourier frequencies for a class of iterated function systems (IFS). These are iteration limits arising from a fixed finite families of affine and contractive mappings in Rd, and the “IFS” refers to such a finite system of transformations, or functions. The iteration limits are pairs (X,μ) where X is a compact subset of Rd (the support of μ), and the measure μ is a probability measure determined uniquely by the initial IFS mappings, and a certain strong invariance axiom. The two questions we study are: (1) existence of an orthogonal Fourier basis in the Hilbert space L²(X,μ); and (2) explicit constructions of Fourier bases from the given data defining the IFS.     

Dimension theory of iterated function systems

Let {S_i} be an iterated function system (IFS) on ?^d with attractor K. Let (Σ, σ) denote the one‐sided full shift over the alphabet {1, …, ??}. We define the projection entropy function h_π on the space of invariant measures on Σ associated with the coding map π : Σ → K and develop some basic ergodic properties about it. This concept turns out to be crucial in the study of dimensional properties of invariant measures on K. We show that for any conformal IFS (respectively, the direct product of finitely many conformal IFSs), without any separation condition, the projection of an ergodic measure under π is always exactly dimensional and its Hausdorff dimension can be represented as the ratio of its projection entropy to its Lyapunov exponent (respectively, the linear combination of projection entropies associated with several coding maps). Furthermore, for any conformal IFS and certain affine IFSs, we prove a variational principle between the Hausdorff dimension of the attractors and that of projections of ergodic measures. © 2008 Wiley Periodicals, Inc.     

Some Remarks on Normal Measures and Measurable Cardinals

We prove two theorems which in a certain sense show that the number of normal measures a measurable cardinal κ can carry is independent of a given fixed behavior of the continuum function on any set having measure 1 with respect to every normal measure over κ . First, starting with a model V ⊨ “ZFC + GCH + o(κ) = δ*” for δ* ≤ κ⁺ any finite or infinite cardinal, we force and construct an inner model N ⊆ V [G] so that N ⊨ “ZF + (∀δ < κ) [DC_δ] + ¬AC_κ + κ carries exactly δ* normal measures + 2^δ = δ⁺⁺ on a set having measure 1 with respect to every normal measure over κ”. There is nothing special about 2^δ = δ here, and other stated values for the continuum function will be possible as well. Then, starting with a modelV ⊨ “ZFC + GCH + κis supercompact”, we force and construct models of AC in which, roughly speaking, regardless of the specified behavior of the continuum function below κ on any set having measure 1 with respect to every normal measure over κ, κ can in essence carry any number of normal measures δ* ≥ κ⁺⁺.     

The Density of States Measure of the Weakly Coupled Fibonacci Hamiltonian

We consider the density of states measure of the Fibonacci Hamiltonian and show that, for small values of the coupling constant V, this measure is exact-dimensional and the almost everywhere value d _V of the local scaling exponent is a smooth function of V, is strictly smaller than the Hausdorff dimension of the spectrum, and converges to one as V tends to zero. The proof relies on a new connection between the density of states measure of the Fibonacci Hamiltonian and the measure of maximal entropy for the Fibonacci trace map on the non-wandering set in the V-dependent invariant surface. This allows us to make a connection between the spectral problem at hand and the dimension theory of dynamical systems.     

Invariant subspaces of finite codimension in Banach spaces of analytic functions

We give a characterization of invariant subspaces of finite codimension in Banach spaces of vector-valued analytic functions in several variables, where invariant refers to invariance under multiplication by any polynomial. We obtain very weak conditions under which our characterization applies, that unifies and improves a number of previous results. In the vector-valued case, the results are new even for one complex variable. As a concrete application in several variables, we consider the Bergman space on a strictly pseudo-convex domain, and we improve previous results (assuming C^∞-boundary) to the case of C²-boundary.     

直觉模糊集之间的距离测度

介绍了直觉模糊集之间距离测度公理化的定义,引入了一些新的距离测度并给出了这些距离测度性质的证明.讨论了相似测度与距离测度之间的关系.最后,指出直觉模糊集距离测度在模式识别中的应用.     

Expanding measures

We prove that any C1+α transformation, possibly with a (non-flat) critical or singular region, admits an invariant probability measure absolutely continuous with respect to any expanding measure whose Jacobian satisfies a mild distortion condition. This is an extension to arbitrary dimension of a famous theorem of Keller (1990) [33] for maps of the interval with negative Schwarzian derivative.Given a non-uniformly expanding set, we also show how to construct a Markov structure such that any invariant measure defined on this set can be lifted. We used these structure to study decay of correlations and others statistical properties for general expanding measures.     

On p-adic vector measure spaces

For a separating algebra R of subsets of a set X, E a complete Hausdorff non-Archimedean locally convex space and m:R→E a bounded finitely additive measure, we study some of the properties of the integrals with respect to m of scalar-valued functions on X. The concepts of convergence in measure, with respect to m, and of m-measurable functions are introduced and several results concerning these notions are given.     

The approximate calculation of invariant measures of diffusions via finite difference approximations to degenerate elliptic partial differential equations

If $\text{L}$ is the (possibly degenerate) differential generator of a diffusion process whose measures converge to a unique invariant measure μ, then formally the value γ in $\text{F}$V(x)+k(x)?γ=0 is ∝ k(x) μ(dx). A finite difference approximation is used to solve the differential equation. The coefficients in the finite difference equation are one-step transition probabilities for some Markov chain whose (suitable) continuous time interpolations converge weakly to the diffusion. Under reasonable conditions, the invariant measure of the sequence of chains converges weakly to the weak sense density of μ. as the finite difference intervals go to zero. The approximating measure can be taken to be the invariant measure (or Cesaro sum of the n-step transition probabilities for the chain), of the chain, suitably weighted.     

Invariant measures for frequently hypercyclic operators

We investigate frequently hypercyclic and chaotic linear operators from a measure-theoretic point of view. Among other things, we show that any frequently hypercyclic operator T acting on a reflexive Banach space admits an invariant probability measure with full support, which may be required to vanish on the set of all periodic vectors for T  ; that there exist frequently hypercyclic operators on the sequence space _c0$c_0$ admitting no ergodic measure with full support; and that if an operator admits an ergodic measure with full support, then it has a comeager set of distributionally irregular vectors. We also give some necessary and sufficient conditions (which are satisfied by all the known chaotic operators) for an operator T to admit an invariant measure supported on the set of its hypercyclic vectors and belonging to the closed convex hull of its periodic measures. Finally, we give a Baire category proof of the fact that any operator with a perfectly spanning set of unimodular eigenvectors admits an ergodic measure with full support.