Related papers: Sparse Block-Jacobi Matrices with Exact Hausdorff …
A selfadjoined block tridiagonal matrix with positive definite blocks on the off-diagonals is by definition a Jacobi matrix with matrix entries. Transfer matrix techniques are extended in order to develop a rotation number calculation for…
An explicit formula for the mean spectral measure of a random Jacobi matrix is derived. The matrix may be regarded as the limit of Gaussian beta ensemble (G$\beta$E) matrices as the matrix size $N$ tends to infinity with the constraint that…
We introduce a class of doubly infinite complex Jacobi matrices determined by a simple convergence condition imposed on the diagonal and off-diagonal sequences. For each Jacobi matrix belonging to this class, an analytic function, called a…
Denote by $\lambda_1(A), \ldots, \lambda_n(A)$ the eigenvalues of an $(n\times n)$-matrix $A$. Let $Z_n$ be an $(n\times n)$-matrix chosen uniformly at random from the matrix analogue to the classical $\ell_ p^n$-ball, defined as the set of…
We study the multifractal analysis of dimension spectrum for almost additive potential in a class of one dimensional non-uniformly hyperbolic dynamic systems and prove that the irregular set has full Hausdroff dimension.
We consider (not self-similar) Cantor sets defined by a sequence of piecewise linear functions. We prove that the dimension of the harmonic measure on such a set is strictly smaller than its Hausdorff dimension. Some Hausdorff measure…
In this work we are interested in the self--affine fractals studied by Gatzouras and Lalley and by the author which generalize the famous general Sierpinski carpets studied by Bedford and McMullen. We give a formula for the Hausdorff…
We compute the exact Hausdorff and Packing measures of linear Cantor sets which might not be self similar or homogeneous . The calculation is based on the local behavior of the natural probability measure supported on the sets.
We study spectral properties of bounded and unbounded complex Jacobi matrices. In particular, we formulate conditions assuring that the spectrum of the studied operators is continuous on some subsets of the complex plane and we provide…
Operators with zero dimensional spectral measures appear naturally in the theory of ergodic Schr\"odinger operators. We develop the concept of a complete family of Hausdorff measure functions in order to analyze and distinguish between…
A Wasserstein spaces is a metric space of sufficiently concentrated probability measures over a general metric space. The main goal of this paper is to estimate the largeness of Wasserstein spaces, in a sense to be precised. In a first…
We develop a spectral analysis of a class of block Jacobi operators based on the conjugate operator method of Mourre. We give several applications including scalar Jacobi operators with periodic coefficients, a class of difference operators…
We introduce a class of Jacobi operators with discrete spectra which is characterized by a simple convergence condition. With any operator J from this class we associate a characteristic function as an analytic function on a suitable…
We prove that on the spectrum the integrated density of states (IDS for short) of periodic Jacobi matrices is related to the discriminant. The method is to count the number of generalized zeros of Bloch wave solutions.
We determine the extent to which certain classes of fractionally `smooth' continuous mappings between metric spaces distort various dimensions, including the Hausdorff, upper Minkowski (box-counting), and upper intermediate dimensions. Our…
We consider the 1D periodic Jacobi matrices. The spectrum of this operator is purely absolutely continuous and consists of intervals separated by gaps. We solve the inverse problem (including characterization) in terms of vertical slits on…
We compute the Hausdorff dimension of the set of $\psi$-exactly approximable vectors, in the simultaneous case, in dimension strictly larger than $2$ and for approximating functions $\psi$ with order at infinity less than or equal to $-2$.…
We study various measure theories using the classical approach and then compute the Hausdorff dimension of some simple objects and self-similar fractals. We then develop a nonstandard approach to these measure theories and examine the…
We review the motivation and fundamental properties of the Hausdorff dimension of metric spaces and illustrate this with a number of examples, some of which are expected and well-known. We also give examples where the Hausdorff dimension…
In this paper, we aim to estimate block-diagonal covariance matrices for Gaussian data in high dimension and in fixed dimension. We first estimate the block-diagonal structure of the covariance matrix by theoretical and practical estimators…