Related papers: Spectral measures with arbitrary Hausdorff dimensi…
We study the limiting spectral measure of large symmetric random matrices of linear algebraic structure. For Hankel and Toeplitz matrices generated by i.i.d. random variables $\{X_k\}$ of unit variance, and for symmetric Markov matrices…
The Hausdorff dimension of spectral measure for the graph Laplacian is shown exactly in terms of an intermittency function. The intermittency function can be estimated by using one-dimensional discrete Schr\"{o}dinger operator method.
In this paper we show that, given a planar Reifenberg flat domain with small constant and a divergence form operator associated to a real (not necessarily symmetric) uniformly elliptic matrix with Lipschitz coefficients, the Hausdorff…
We investigate a family of Riesz products and show that they can be regarded as diffraction measures of generalized Thue-Morse sequences, possibly over an infinite alphabet. These measures are closely related to the dynamical system arising…
Work in the measure algebra of the Lebesgue measure on the Cantor space: for comeager many $[A]$ the set of points $x$ such that the density of $x $ at $A$ is not defined is $\Sigma^{0}_{3}$-complete; for some compact $K$ the set of points…
We look at invariance of a.e. boundary condition spectral behavior under perturbations, $W$, of half-line, continuum or discrete Schr\"odinger operators. We extend the results of del Rio, Simon, Stolz from compactly supported $W$'s to…
We study differentiability properties of Zygmund functions and series of Weierstrass type in higher dimensions. While such functions may be nowhere differentiable, we show that, under appropriate assumptions, the set of points where the…
Let $\{x\_n\}\_{n\geq 0}$ be a sequence of $[0,1]^d$, $\{\lambda\_n\} \_{n\geq 0}$ a sequence of positive real numbers converging to 0, and $\delta>1$. Let $\mu$ be a positive Borel measure on $[0,1]^d$, $\rho\in (0,1]$ and $\alpha>0$.…
In this paper, we study divergence properties of Fourier series on Cantor-type fractal measure, also called Mock Fourier series. We give a sufficient condition under which the Mock Fourier series for doubling spectral measure is divergent…
We demonstrate Hausdorff log dimensional bounds on the continuity of the spectral measure of the strongly disordered Holstein operator.
We study a class of dynamical systems given by measure preserving actions of the group $Z^d$ or $R^d$ and generating a set of spectral measures with an extremal rate of the Fourier coefficient decay: $\Hat\sigma(n) = O(|n|^{-1/2+\epsilon})$…
We establish Euclidean-type lower bounds for the codimension-1 Hausdorff measure of sets that separate points in doubling and linearly locally contractible metric manifolds. This gives a quantitative topological isoperimetric inequality in…
We prove that for all $s\in(0,d)$ and $c\in (0,1)$ there exists a self-similar set $E\subset \mathbb{R}^d$ with Hausdorff dimension $s$ such that $\mathcal{H}^s(E)=c|E|^s$. This answers a question raised by Zhiying Wen[16].
Self-adjoint operators on infinite-dimensional spaces with continuous spectra are abundant but do not possess a basis of eigenfunctions. Rather, diagonalization is achieved through spectral measures. The SpecSolve package [SIAM Rev., 63(3)…
We consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with…
Under the assumption of a natural subadditive potential, the so called cylinder function, working on the symbol space we prove the existence of the ergodic invariant probability measure satisfying the equilibrium state. As an application we…
We consider diffeomorphisms of a compact manifold with a dominated splitting which is hyperbolic except for a "small" subset of points (Hausdorff dimension smaller than one, e.g. a denumerable subset) and prove the existence of physical…
Let $X=\bigcup\varphi_{i}X$ be a strongly separated self-affine set in $\mathbb{R}^2$ (or one satisfying the strong open set condition). Under mild non-compactness and irreducibility assumptions on the matrix parts of the $\varphi_{i}$, we…
A theorem of David and Jerison asserts that harmonic measure is absolutely continuous with respect to surface measure in NTA domains with Ahlfors regular boundaries. We prove that this fails in high dimensions if we relax the Ahlfors…
We study a capacity theory based on a definition of a Riesz potential in metric spaces with a doubling measure. In this general setting, we study the basic properties of the Riesz capacity, including monotonicity, countable subadditivity…