Direct and inverse computation of Jacobi matrices of infinite iterated function systems

Abstract

We 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

Orthogonal polynomials of equilibrium measures supported on Cantor sets

“…Typically, it is found that the associated spectral measures are singular continuous, and in certain cases (like the so-called Fibonacci matrix [85,24]) supported on Cantor sets with self-similar geometry. Conversely, one can start with IFS balanced (not equilibrium) measures on such sets, and ask what are the properties of the associated Jacobi matrices-that can be computed numerically [31,56,65]. It is still an open problem to assess whether almost periodicity of some sort characterizes these matrices, as conjectured in [57].…”

“…This provides a new algorithm for the difficult numerical problem of computing the Jacobi matrix J(µ). Since this latter can also be obtained by different, non iterative techniques [31,56,65], we can measure the convergence of the matrix entries of J(µ n ) as a function of the order n, to assess the relative performance the new algorithm.…”

“…3, one also finds that A(ǫ) ∼ Cǫ η , with η ≃ .286 and C a quantity independent of ǫ and n. Taking into account that the computational complexity of the revised RKPW algorithm is approximately of BN ǫ N arithmetical operations (with B a small constant, see Appendix B), we can conclude that estimating the Jacobi matrix of µ of size N ǫ within ǫ via this procedure has a cost that scales roughly as ǫ −η/β N 1+1/β ǫ ≃ ǫ −.31 N 2.09 ǫ , that is, slightly more than quadratic in the size N ǫ and slowly increasing with respect to the precision 1/ǫ. Remark that the recursive algorithms for IFS with a finite number of maps [56,65] require an order of N 2 ǫ operations to compute the same matrix, but this computation is exact in principle, in practice affected by a slowly increasing error [65]. The new algorithm 3 is therefore not optimal for balanced measures, but it can be extended to equilibrium ones, for which no alternative techniques are available, except for an algebraic procedure that can be set up using a Padé scheme [55].…”

Refinable functions, functionals, and iterated function systems

“…It seems possible to extend the proof in the original paper to the case of non-equal coefficients {α n }. Needless to say, this technique is classical in these problems [37,43,44].…”

“…A remarkable exception to this state of things is provided by the study of Bernoulli convolutions [26,41,44], where the main theoretical question to be answered is about the absolute continuity versus singularity of the invariant measure. A renewed interest in this question has been brought recently by the study of IFS with uncountably many maps [36,37]: preliminary results on a specific example show that a transition from singular behavior to absolute continuity and to increasing regularity of the density of an IFS measure takes place as a parameter is varied. Crucial for our aim, this density is a refinable function.…”