Singular Continuous Measures Research Articles

Thin spectra and singular continuous spectral measures for limit‐periodic Jacobi matrices

AbstractThis paper investigates the spectral properties of Jacobi matrices with limit‐periodic coefficients. We show that generically the spectrum is a Cantor set of zero Lebesgue measure, and the spectral measures are purely singular continuous. For a dense set of limit‐periodic Jacobi matrices, we show that the spectrum is a Cantor set of zero lower box counting dimension while still retaining the singular continuity of the spectral type. We also show how results of this nature can be established by fixing the off‐diagonal coefficients and varying only the diagonal coefficients, and, in a more restricted version, by fixing the diagonal coefficients to be zero and varying only the off‐diagonal coefficients. We apply these results to produce examples of weighted Laplacians on the multidimensional integer lattice having purely singular continuous spectral type and zero‐dimensional spectrum.

Paatero’s V(k) space II

Abstract In this article we continue our investigation of the Paatero space. We prove that the intersection of every Paatero class V(k) with every Hardy space Hp is closed in that Hp and associate singular continuous measures to elements of V(k). There have been no examples in the literature of functions in V(k) with zeros in the unit disk other than the one at the origin. We close this gap in the literature. We derive a representation of the measure associated to a function in V(k) for functions whose derivatives are rational, or algebraic, or transcendental functions in the unit disk.Finally, we consider the notion of regulated domains, introduced by Pommerenke and show that there are regulated domains whose boundary is not of bounded boundary rotation.

Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schrödinger operators

We introduce a notion of \beta -almost periodicity and prove quantitative lower spectral/quantum dynamical bounds for general bounded \beta -almost periodic potentials. Applications include the first sharp arithmetic spectral criterion for the entire family of supercritical analytic quasiperiodic Schrödinger operators and arithmetic spectral/quantum dynamical criteria for families with zero Lyapunov exponents, with applications to Sturmian potentials and the critical almost Mathieu operator. In particular, we disprove a 1994 conjecture of Wilkinson–Austin.

On a family of singular continuous measures related to the doubling map

Here, we study some measures that can be represented by infinite Riesz products of 1-periodic functions and are related to the doubling map. We show that these measures are purely singular continuous with respect to Lebesgue measure and that their distribution functions satisfy super-polynomial asymptotics near the origin, thus providing a family of extremal examples of singular measures, including the Thue–Morse measure.

Return probability and self-similarity of the Riesz walk

The quantum walk is a counterpart of the random walk. The 2-state quantum walk in one dimension can be determined by a measure on the unit circle in the complex plane. As for the singular continuous measure, results on the corresponding quantum walk are limited. In this situation, we focus on a quantum walk, called the Riesz walk, given by the Riesz measure which is one of the famous singular continuous measures. The present paper is devoted to the return probability of the Riesz walk. Furthermore, we present some conjectures on the self-similarity of the walk.

On the spacing of zeros of paraorthogonal polynomials for singular measures

We prove a lower bound on the spacing of zeros of paraorthogonal polynomials on the unit circle, based on continuity of the underlying measure as measured by Hausdorff dimensions. We complement this with the analog of the result from Breuer (2011) showing that clock spacing holds even for certain singular continuous measures.

On a class of forward-backward parabolic equations: Formation of singularities

We study the formation of singularities for the problem{ut=[φ(u)]xx+ε[ψ(u)]txxinΩ×(0,T)φ(u)+ε[ψ(u)]t=0in∂Ω×(0,T)u=u0≥0inΩ×{0}, where ϵ and T are positive constants, Ω a bounded interval, u0 a nonnegative Radon measure on Ω, φ a nonmonotone and nonnegative function with φ(0)=φ(∞)=0, and ψ an increasing bounded function. We show that if u0 is a bounded or continuous function, singularities may appear spontaneously. The class of singularities which can arise in finite time is remarkably large, and includes infinitely many Dirac masses and singular continuous measures.

Scaling properties of the thue–morse measure

The classic Thue--Morse measure is a paradigmatic example of a purely singular continuous probability measure on the unit interval. Since it has a representation as an infinite Riesz product, many aspects of this measure have been studied in the past, including various scaling properties and a partly heuristic multifractal analysis. Some of the difficulties emerge from the appearance of an unbounded potential in the thermodynamic formalism. It is the purpose of this article to review and prove some of the observations that were previously established via numerical or scaling arguments.

Fourier frames for singular measures and pure type phenomena

Let μ \mu be a positive measure on R d \mathbb {R}^d . It is known that if the space L 2 ( μ ) L^2(\mu ) has a frame of exponentials, then the measure μ \mu must be of “pure type”: it is either discrete, absolutely continuous or singular continuous. It has been conjectured that a similar phenomenon should be true within the class of singular continuous measures, in the sense that μ \mu cannot admit an exponential frame if it has components of different dimensions. We prove that this is not the case by showing that the sum of an arc length measure and a surface measure can have a frame of exponentials. On the other hand we prove that a measure of this form cannot have a frame of exponentials if the surface has a point of non-zero Gaussian curvature. This is in spite of the fact that each “pure” component of the measure separately may admit such a frame.

Minkowski’s question mark measure

Minkowski’s question mark function is the distribution function of a singular continuous measure: we study this measure from the point of view of logarithmic potential theory and orthogonal polynomials. We conjecture that it is regular, in the sense of Ullman–Saff–Stahl–Totik and moreover that it belongs to a Nevai class; we provide numerical evidence of the validity of these conjectures. In addition, we study the zeros of its orthogonal polynomials and the associated Christoffel functions, for which asymptotic formulae are derived. As a by-product, we compute upper and lower bounds to the Hausdorff dimension of Minkowski’s measure. Rigorous results and numerical techniques are based upon Iterated Function Systems composed of Möbius maps.

Arithmetic Spectral Transitions for the Maryland Model

AbstractWe give a precise description of spectra of the Maryland model urn:x-wiley:0010-3640:media:cpa21688:cpa21688-math-0001 for all values of parameters. We introduce an arithmetically defined index and show that for , urn:x-wiley:0010-3640:media:cpa21688:cpa21688-math-0004 and urn:x-wiley:0010-3640:media:cpa21688:cpa21688-math-0005 Since , this gives a complete description of the spectral decomposition for all values of parameters λ, α, and θ, making it the first case of a family where arithmetic spectral transition is described without any parameter exclusion. The set of eigenvalues can be explicitly identified for all parameters, using the quantization condition. We also establish, for the first time for this or any other model, a quantization condition for singular continuous spectrum (an arithmetically defined measure zero set that supports singular continuous measures) for all parameters.© 2017 Wiley Periodicals, Inc.

Extreme value laws for fractal intensity functions in dynamical systems: Minkowski analysis**Dedicated to the memory of Joseph Ford, on the twentieth anniversary of his departure.

Typically, in the dynamical theory of extremal events, the function that gauges the intensity of a phenomenon is assumed to be convex and maximal, or singular, at a single, or at most a finite collection of points in phase–space. In this paper we generalize this situation to fractal landscapes, i.e. intensity functions characterized by an uncountable set of singularities, located on a Cantor set. This reveals the dynamical rôle of classical quantities like the Minkowski dimension and content, whose definition we extend to account for singular continuous invariant measures. We also introduce the concept of extremely rare event, quantified by non-standard Minkowski constants and we study its consequences to extreme value statistics. Limit laws are derived from formal calculations and are verified by numerical experiments.

Spacing properties of the zeros of orthogonal polynomials on Cantor sets via a sequence of polynomial mappings

Let $\mu$ be a probability measure with an infinite compact support on $\mathbb{R}$. Let us further assume that $(F_n)_{n=1}^\infty$ is a sequence of orthogonal polynomials for $\mu$ where $(f_n)_{n=1}^\infty$ is a sequence of nonlinear polynomials and $F_n:=f_n\circ\dots\circ f_1$ for all $n\in\mathbb{N}$. We prove that if there is an $s_0\in\mathbb{N}$ such that $0$ is a root of $f_n^\prime$ for each $n>s_0$ then the distance between any two zeros of an orthogonal polynomial for $\mu$ of a given degree greater than $1$ has a lower bound in terms of the distance between the set of critical points and the set of zeros of some $F_k$. Using this, we find sharp bounds from below and above for the infimum of distances between the consecutive zeros of orthogonal polynomials for singular continuous measures.

Spectra of Cantor measures

The Cantor measures \(\mu _{q, b}\) with \(2 = q, b/q \in {\mathbb Z}\) are few of known singular continuous measures that admits a spectrum. The spectra of Cantor measures \(\mu _{q, b}\) are not unique and their structure is not well-studied. It has been observed that maximal orthogonal sets of the Cantor measure \(\mu _{q, b}\) have tree structure and they can be built by selecting maximal tree appropriately. A challenging problem is when a maximal orthogonal set becomes a spectrum. In this paper, we introduce a measurement on the tree structure associated with a maximal orthogonal set \(\Lambda \) of the Cantor measure \(\mu _{q, b}\), and we use boundedness and linear increment of that measurement to justify whether \(\Lambda \) is a spectrum or not. As applications of our justification, we provide a characterization for the expanding set \(K\Lambda \) of a spectrum \(\Lambda \) to be a spectrum again. Furthermore, we construct a spectrum \(\Lambda \) such that for some integer K, the shrinking set \(\Lambda /K\) is a maximal orthogonal set but not a spectrum.

Orthogonal polynomials for Minkowski’s question mark function

Hermann Minkowski introduced a function in 1904 which maps quadratic irrational numbers to rational numbers and this function is now known as Minkowski’s question mark function since Minkowski used the notation ?(x). This function is a distribution function on [0,1] which defines a singular continuous measure with support [0,1]. Our interest is in the (monic) orthogonal polynomials (Pn)n∈N for the Minkowski measure and in particular in the behavior of the recurrence coefficients of the three term recurrence relation. We will give some numerical experiments using the discretized Stieltjes–Gautschi method with a discrete measure supported on the Minkowski sequence. We also explain how one can compute the moments of the Minkowski measure and compute the recurrence coefficients using the Chebyshev algorithm.

Cantor polynomials and some related classes of OPRL

We explore the spectral theory of the orthogonal polynomials associated to the classical Cantor measure and similar singular continuous measures. We prove regularity in the sense of Stahl–Totik with polynomial bounds on the transfer matrix. We present numerical evidence that the Jacobi parameters for this problem are asymptotically almost periodic and discuss the possible meaning of the isospectral torus and the Szegő class in this context.

Spectral and topological properties of a family of generalised Thue-Morse sequences

The classic middle-thirds Cantor set leads to a singular continuous measure via a distribution function that is known as the Devil's staircase. The support of the Cantor measure is a set of zero Lebesgue measure. Here, we discuss a class of singular continuous measures that emerge in mathematical diffraction theory and lead to somewhat similar distribution functions, yet with significant differences. Various properties of these measures are derived. In particular, these measures have supports of full Lebesgue measure and possess strictly increasing distribution functions. In this sense, they mark the opposite end of what is possible for singular continuous measures. For each member of the family, the underlying dynamical system possesses a topological factor with maximal pure point spectrum, and a close relation to a solenoid, which is the Kronecker factor of the system. The inflation action on the continuous hull is sufficiently explicit to permit the calculation of the corresponding dynamical zeta functions. This is achieved as a corollary of analysing the Anderson-Putnam complex for the determination of the cohomological invariants of the corresponding tiling spaces.

Full measure of a set of singular continuous measures

On the space of structurally similar measures, we construct a nontrivial measure m such that the subclass of all purely singular continuous measures is a set of full m-measure.

Singular Continuous Spectrum for the Laplacian on Certain Sparse Trees

We present examples of rooted tree graphs for which the Laplacian has singular continuous spectral measures. For some of these examples we further establish fractional Hausdorff dimensions. The singular continuous components, in these models, have an interesting multiplicity structure. The results are obtained via a decomposition of the Laplacian into a direct sum of Jacobi matrices.

Sur une question de H. Brezis, M. Marcus et A.C. Ponce

Investigating a question raised in a recent paper of Brezis–Marcus–Ponce [5] we show that if μ is a singular continuous Radon measure in the open subset U of Rd, there is no continuous linear map f↦(uf,gf) from L1(μ) to H1(U)×L1(U) such that f⋅μ=Δuf+gf in the sense of distributions. A generalization is also described.