Self-Similar Measures with Scaling of Spectra and Beurling Dimension

Abstract

We study the geometric properties of random multiplicative cascade measures defined on self-similar sets. We show that such measures and their projections and sections are almost surely exact-dimensional, generalizing Feng and Hu’s result [11] for self-similar measures. We introduce a technique that uses projection properties of fractal percolation to establish dimension conservation results for sections of deterministic self-similar sets. For example, let K be a self-similar subset of R^2 with Hausdorff dimension dim_H⁡K>1 such that the rotational components of the underlying similarities generate the full rotation group. Then for all ϵ>0, writing θ_0 for projection onto the line L_θ in direction θ, the Hausdorff dimensions of the sections satisfy dim_H⁡d(K∩π_θ^(-1) x)>dim_H⁡〖K-1-ϵ〗 for a set of x∈L_θ of positive Lebesgue measure, for all directions θ except for those in a set of Hausdorff dimension 0. We analyze all orthonormal bases of exponentials on the Cantor set defined by Jorgensen and Pedersen. A complete characterization for all maximal sets of orthogonal exponentials is obtained by establishing a one-to-one correspondence with the spectral labelings of the infinite binary tree. With the help of this characterization we obtain a sufficient condition for a spectral labeling to generate a spectrum (an orthonormal basis). This result not only provides us an easy and efficient way to construct various of new spectra for the Cantor measure but also extends many previous results in the literature. In fact, most known examples of orthonormal bases of exponentials correspond to spectral labelings satisfying this sufficient condition. For {D_k }_(k=1)^∞ be a sequence of digit sets in N and let {b_k }_(k=1)^∞ be a sequence of integer numbers bigger than 1. We call the family {f_(k,D_k ) (x)=b_k^(-1) (x+d): d∈D_k,k≥1} a Moran iterated function system (IFS), which is a natural generalization of an IFS. For 0<ρ<1 and N>1 an integer, let μ be the self-similar measure defined by μ(•)=∑_(i=0)^(N-1)▒〖1/N μ(ρ^(-1) (•)-i) 〗. We prove that L^2 (μ) has an exponential orthonormal basis if and only if ρ=1/q for some q>0 and N divides q. For A be a d×d integral expanding matrix and let S_j (x)=A^(-1) (x+d_j ) for some d_j∈Z^d,j=1,…,m. The iterated function system (IFS) {S_j }_(j=1)^m generates self-affine measures and scale functions. In general this IFS has overlaps, and it is well known that in many special cases the analysis of such measures or functions is facilitated by expressing them in vector-valued forms with respect to another IFS that satisfies the open set condition. We examine Fourier frames and, more generally, frame measures for different probability measures. We prove that if a measure has an associated frame measure, then it must have a certain uniformity in the sense that the weight is distributed quite uniformly on its support. To be more precise, by considering certain absolute continuity properties of the measure and its translation, we recover the characterization on absolutely continuous measures g dx with Fourier frames obtained. Moreover, we prove that the frame bounds are pushed away by the essential infimum and supremum of the function g. This also shows that absolutely continuous spectral measures supported on a set Ω, if they exist, must be the standard Lebesgue measure on Ω up to a multiplicative constant. We consider equally-weighted Cantor measures μ_(q,b) arising from iterated function systems of the form b^(-1) (x+i),i=0,1,…,q-1, where q<b. We classify the (q,b) so that they have infinitely many mutually orthogonal exponentials in L^2 (μ_(q,b) ). In particular, if q divides b, the measures have a complete orthogonal exponential system and hence spectral measures. Improving the construction, we characterize all the maximal orthogonal sets Λ when q divides b via a maximal mapping on the q−adic tree in which all elements in Λ are represented uniquely in finite b−adic expansions and we can separate the maximal orthogonal sets into two types: regular and irregular sets. For a regular maximal orthogonal set, we show that its completeness in L^2 (μ_(q,b) ) is crucially determined by the certain growth rate of non-zero digits in the tail of the b−adic expansions of the elements. For 1≤m<n be integers, and let K⊂R^n be a self-similar set satisfying the strong separation condition, and with dim⁡K=s>m. We study the a.s. values of the s-m-dimensional Hausdorff and packing measures of K∩V, where V is a typical n-m-dimensional affine subspace. We present some one-parameter families of homogeneous self-similar measures on the line such that, the similarity dimension is greater than 1 for all parameters and the singularity of some of the self-similar measures from this family is not caused by exact overlaps between the cylinders. We construct a planar homogeneous self-similar measure, with strong separation, dense rotations and dimension greater than 1, such that there exist lines for which dimension con-servation does not hold and the projection of the measure is singular. A spectrum of a probability measure μ is a countable set Λ such that {exp⁡(-2πiλ•),λ∈Λ} is an orthogonal basis for L^2 (μ). We consider the problem when a countable set become the spectrum of the Cantor measure. Starting from tree labeling of a maximal orthogonal set, we introduce a new quantity to measure minimal level difference between a branch of the labeling tree and its subbranches. Then we use boundedness and linear increment of that level difference measurement to justify whether a given maximal orthogonal set is a spectrum or not. This together with the tree labeling of a maximal orthogonal set provides fine structures of spectra of Cantor measures. Given a Borel probability measure μ on R and a real number p. We call pa spectral eigenvalue of the measure μ if there exists a discrete set Λ such that the sets E(Λ)≔{e^2πiλx ∶λ∈Λ} " and" E(pΛ)≔{e^2πipλx ∶λ∈Λ} are both orthonormal basis for Hilbert space L^2 (μ). We consider the equally-weighted Cantor measures μ_(p,q) generated by the iterated function system (IFS) {f_i (x)=x/p+i/q}_(i=0)^(q-1), where 2≤q∈Z and q<p∈R. It is known that if q divides p, then μ_(p,q) is a spectral measure with a spectrum Λ_(p,q)={0,1,…,q-1}+p{0,1,…,q-1} +p^2 {0,1,…,q-1}+⋯("finite sum" ) (Dai, He and Lai (2013)[362]). For μ be a Borel probability measure with compact support. We consider exponential type orthonormal bases, Riesz bases and frames in L^2 (μ). We show that if L^2 (μ) admits an exponential frame, then μ must be of pure type. We also classify various μ that admits either kind of exponential bases, in particular, the discrete measures and their connection with integer tiles. We study the Beurling dimension of Bessel sets or sequence and frame spectra of some self-similar measures on R^d.