1,721,027 research outputs found
Minkowski's question mark measure
No full textMinkowski'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 UllmanâSaffâStahlâTotik 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
Direct and inverse computation of Jacobi matrices of infinite iterated function systems
No full textWe introduce a new set of algorithms to compute the Jacobi matrices associated with invariant measures of infinite iterated function systems, composed of one–dimensional, homogeneous affine maps. We demonstrate their utility in the study of theoretical problems, like the conjectured almost periodicity of such Jacobi matrices, the singularity of the measures, and the logarithmic capacity of their support. Since our technique is based on a reversible transformation between pairs of Jacobi matrices, it can also be applied to solve an inverse/approximation problem. The proposed algorithms are tested in significant, highly sensitive cases: they perform in a stable fashion, and can reliably compute Jacobi matrices of large order
Fourier-Bessel functions and the many asymptotics of orthogonal polynomials of singular continuous measures
No full textWe study the Fourier transform of polynomials in an orthogonal
family, taken with respect to the orthogonality measure. Mastering the asymptotic properties of these transforms, that we call Fourier--Bessel functions, in the argument, the order, and in
certain combinations of the two is required to solve a number of
problems arising in quantum mechanics. We present known results, new approaches and open conjectures, hoping to justify our belief that the importance of these investigations extends beyond the application just mentioned, and may involve interesting discoveries
The global statistics of return times: return time dimensions versus generalized measure dimensions
No full textWe investigate the relations holding among generalized dimensions of invariant measures in dynamical systems and similar quantities defined by the scaling of global averages of powers of return times. Because of a heuristic use of Kac theorem, these latter have been used in place of the former in numerical and experimental investigations; to mark this distinction, we call them return time dimensions. We derive a full set of inequalities linking measure and return time dimensions and we comment on their optimality with the aid of two maps due to von Neumann -- Kakutani and to Gaspard -- Wang. We conjecture the behavior of return time dimensions in a typical system. We only assume ergodicity of the dynamical system under investigation
Dynamical Systems and Numerical Analysis: the Study of Measures generated by Uncountable I.F.S.
No full textA Stable Stieltjes Technique to Compute Jacobi MatricesAssociated with Singular Measures
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